DERIVED LOG ALBANESE SHEAVESFEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO
Abstract. We define higher pro-Albanese functors for every effective log motive over a
field k of characteristic zero, and we compute them for every smooth log smooth scheme
X = (X, ∂X). The result involves an inverse system of the coherent cohomology of theunderlying scheme as well as a pro-group scheme Alblog (X) that extends Serre’s semiabelian Albanese variety of X − |∂X|. This generalizes the higher Albanese sheaves of
Ayoub, Barbieri-Viale and Kahn.

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Reciprocity sheaves and logarithmic motives with rational coefficients . . . . . . . . . 6
3. Categories of rational maps and universal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4. The Albanese functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5. The derived Albanese functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6. Some computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7. Laumon 1-motives and compact objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Appendix A. Pro-left derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1. Introduction
Let k be a perfect field and let X be a smooth, projective and geometrically connectedk-scheme. A very classical tool in the study of the geometry of X is given by the Albanesevariety AlbX of X, the universal Abelian variety receiving a map from X (up to the
choice of a base point). When X is a smooth curve, the Albanese variety coincides with
the Jacobian variety Jac(X) of X, and essentially every invariant of X can be recoveredfrom it. In higher dimension, the Albanese variety is still an important tool for gathering
information about the Chow group CH0 (X) of zero cycles of X. Extending the Albanese
map by linearity, there is in fact a well-defined morphism (now independent on the choiceof a base point)(1.0.1)aX : CH0 (X)0 → AlbX (k).

Much is known, at least conjecturally, about this map. If X is proper over an algebraicallyclosed field, a famous theorem of Rojtman [Roj80] asserts that aX is an isomorphism on
torsion subgroups (at least modulo p-torsion in characteristic p > 0, which was later fixedby Milne [Mil82]). If k is finite, the kernel of aX can be explicitly determined by geometricclass field theory [KS83]. If k is the algebraic closure of a finite field, then a theorem of
Kato and Saito (see again [KS83]) asserts that aX is in fact an isomorphism, a statementthat is conjectured to be true even when k = Q as consequence of the Bloch-Beilinson
conjectures. This is far from being true over the complex numbers, as shown by Mumford.
F.B. is supported by the PRIN “Geometric, Algebraic and Analytic Methods in Arithmetic”.
A.M. is supported by the Swiss National Science Foundation (SNSF), project 200020 178729.
S.S. is supported by the JSPS KAKENHI Grant (20H01791).
1

2

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITOWhen X is no longer proper, both sides of (1.0.1) need to be modified. It is already clear
from the case of curves [Ser75] that one can consider a more general class of commutative
algebraic groups as the targets of maps from X, including Abelian varieties, tori, and theirextensions, i.e. semi-Abelian varieties. Serre [Ser60] (see also [FW84]) showed that the
problem of finding a universal map to a semi-Abelian variety has always a solution. The
corresponding universal object is now known as Serre’s Albanese variety: it agrees withthe usual Albanese variety if X is proper.
Using Serre’s semi-Abelian Albanese variety it is possible to extend the Albanese morphism to every smooth quasi-projective variety1, replacing the Chow group CH0 (X) witha different quotient of the free Abelian group of zero cycles of X, namely the Suslin homology group H0S (X). As observed by Spieß and Szamuely [SS03], every semi-Abelianvariety, seen as étale sheaf on the big site Sm(k), has a natural structure of étale sheafwith transfers, i.e. it enjoys an extra functoriality with respect to the category of finite
correspondences Cor(k) introduced by Suslin and Voevodsky. Since every map from anaffine space to a torus or to an Abelian variety is constant, such sheaves are moreoverA1 -homotopy invariant.
These two facts are essentially enough to show that the assignment X 7→ AlbX (here
AlbX is the non-connected algebraic group whose neutral component is exactly Serre’s
semi-Abelian Albanese2) can be promoted by left Kan extension to a motivic “realization”
functor
(1.0.2)eff
L Alb : DMét(k, Q) −→ D(HI≤1,ét (k, Q))

eff(k, Q), i.e. the fulldefined on the ∞-category of Voevodsky’s effective motives DMéttrsubcategory of the derived ∞-category of étale sheaves with transfers D(Shvét(k, Q))
1whose objects are A -local complexes, taking values in the derived ∞-category of the
Abelian category HI≤1,ét (k, Q) of 1-motivic sheaves with rational coefficients: this is thefull subcategory of étale sheaves with transfers generated under colimits by lattices (i.e.
étale sheaves L such that L(k) ∼= Zr ) and semi-abelian varieties (see [ABV09, Prop. 1.3.8]).
It is naturally a full subcategory of the abelian category of homotopy invariant sheaves
with transfers.
This result, due to Ayoub and Barbieri-Viale [ABV09, Thm. 2.4.1] (extending BarbieriViale and Kahn [BVK16] to non necessarily geometric motives) has several consequences.
First, it provides a construction of an Albanese map for arbitrary motives (in particular,
for every separated k-scheme of finite type, not necessarily smooth or proper), giving
for example vast generalizations of the theorem of Rojtman [BVK16, 13]. Second, theAlbanese functor is now a derived functor: it has higher homotopy groups Li Alb(M ) =eff
πi (L Alb(M )) for every M ∈ DMét(k, Q), encoding information such as the Néron-Severigroup of a variety (see [BVK16, Thm. 9.2.3]). Moreover, the functor L Alb in (1.0.2) caneff
be identified with the left adjoint of the derived functor iDMof the natural embedding
≤1
eff

tr
HI≤1,ét (k, Q) ⊂ Shvét(k, Q). Using this fact, one can show that iDM
is fully faithful,≤1eff
and its essential image coincides with the stable ∞-category DM≤1 (k, Q) generated by
the motives of curves. If we restrict ourselves to compact objects, D(HI≤1,ét (k, Q))ωcoincides with the (bounded) derived category of the Abelian category of Deligne 1-motives
introduced in [Del74]. In fact, the properties of (1.0.2) are essential in the “motivic” proofof Deligne’s conjectures on 1-motives, see [BVK16, Part 4] and [Vol13].

Our goal in this paper is to extend the picture sketched above in order to include amore general kind of algebraic groups in the definition of the Albanese variety. Thanks to
Chevalley’s structure theorem, any connected commutative algebraic group over a perfectfield can be written as an extension of an Abelian variety by an affine smooth group scheme,
1at least after inverting the exponential characteristic of the ground field, in an appropriate sense.
2This is necessary in order to make the assignment independent from the choice of a base point

DERIVED LOG ALBANESE SHEAVES

3

which splits as a product of a torus and a unipotent commutative group. As observed
by Serre, however, the problem of finding a universal map from a smooth variety X toan arbitrary commutative algebraic group does not have a solution in general (namely,when X is not proper), whence the classical restriction to semi-Abelian varieties. A
solution does, however, exist, if a bound on the dimension of the tangent spaces of themaps is imposed. Let us assume that k has characteristic zero (and keep this assumption
throughout the rest of the Introduction, see Proposition 3.17). Faltings and Wüstholz[FW84] realized that when X admits a smooth compactification X with normal crossing
boundary D, it is possible to use any finite dimensional subspace of the vector space
H 0 (X, Ω1X ) to give such a bound. A natural choice is to use for n ≥ 1 the subspaces
1 (nD)) of regular 1-forms on X having poles of order at most n along D. The
H 0 (X, ΩX
resulting universal object Alb(X,nD) depends on the pair X(n) := (X, nD) in a functorial
way. This gives a generalized Albanese morphismaX ⊗ Q : Qtr (X) → AlbX(n) ⊗Z Q,
which is a surjective morphism of étale sheaves with transfers with rational coefficients(here Qtr (X) denotes the étale sheaf of Q-vector spaces represented by X). The generalized
Albanese AlbX(n) is an extension of Serre’s semi-Abelian Albanese of X (independent on
the choice of the compactification X) by a unipotent group. If X is a curve, AlbX(n) =Jac(X, nD) is exactly the generalised Jacobian variety of Rosenlicht and Serre [Ser75],
and in higher dimension it is the generalised Albanese with modulus considered in [BS19],[BK18] (see also [Rus08], [Rus13]).
By varying n, we get a pro-object in the category of commutative algebraic groups up
to isogeny “ limn ”AlbX(n) , which satisfies an obvious universal property, see Prop. 3.18.
←−
In fact, we can give a finer result. Let RSCét,≤1 (k, Q) be the full (abelian) subcategorytrof the category of étale sheaves with transfers Shvét(k, Q) generated under colimits bycommutative connected k-group schemes of finite type and lattices. Note that we clearly
have HIét,≤1 (k, Q) ⊂ RSCét,≤1 (k, Q). Write Comp(X) for the category of normal compactifications X of X such that the complement X − X is the support of an effective
Cartier divisor.
Theorem 1.1. (see Thm. 4.27) Assume that the characteristic of k is zero. The embedtr
ding RSCét,≤1 (k, Q) ⊆ Shvét(k, Q) has a pro-left adjoint:(1.1.1)tr(k) → pro- RSCét,≤1 ,Alb : Shvét

induced by colimit fromQtr (X) 7→ “ lim ”AlbX(n)←−nwhere X = (X, D) for a choice of X ,→ X as above such that the support D of X − X isa normal crossing divisor.
It is natural to ask for a derived version of the above Theorem, in the spirit of the resultof Ayoub, Barbieri-Viale and Kahn. However, since unipotent group schemes are A1 eff
contractible, (1.1.1) cannot be extended to DMét(k, Q) in a non-trivial way, i.e. without
simply collapsing to the subcategory HIét,≤1 (k, Q), recovering (1.0.2).
Our solution to this difficulty is to extend the construction to a framework in which
A1 -contractibility is no longer a problem. This is achieved by passing from the world ofalgebraic geometry to the world of logarithmic algebraic geometry, in the sense of Fontaine,
Illusie, Kato and others. Over a field k (seen as log scheme with trivial log structure),
we can, roughly speaking, replace schemes with log pairs X := (X, ∂X), where X is theunderlying k-scheme and ∂X is a log structure supported on a Cartier divisor (the so-called
compactifying log structure associated to the open embedding of schemes X −|∂X| ,→ X).
For X a smooth log-smooth scheme (i.e. a log scheme such that the underlying scheme
is smooth and the log structure is supported on a normal crossing divisor, see 2.1.1), we

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FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

can modify a bit the previous construction and consider a system (X, |∂X|red + nD)n≥1 ,where X is a smooth compactification of X (which always exists under our characteristic
zero assumption) such that the reduced divisor |∂X|red associated to the log structure on
X extends to a divisor on X and such that D = X − X is an effective Cartier divisor withthe property that D + |∂X|red has normal crossings. We have then a pro-algebraic group(1.1.2)

Alblog (X) := “ lim ”Alb(X,|∂X|red +nD) ∈ pro- RSCét,≤1←−nwhich is an invariant of the log scheme X. Note that we recover the previous pro-Albanesein the case where X = (X, triv), i.e. the scheme X seen as log scheme with trivial log
structure.
In order to exploit this formalism, we need another observation. Any commutative groupscheme G (not just semi-Abelian varieties) gives rise to an étale sheaf with transfers (stilltr(k, Z)
denoted G) on Sm(k). As such, it belongs to the subcategory RSCét (k, Z) of Shvétof reciprocity sheaves. Its objects satisfy the property that each section a ∈ F (X) for any
X ∈ Sm(k) “has bounded ramification”, i.e. that the corresponding map a : Ztr (X) → F
factors through a quotient h0 (X) associated to a pair X = (X, D) where X is a propercompactification of X and D is an effective Cartier divisor such that X = X −|D| (we referto such a pair as a Cartier compactification of X). Thanks to [Sai21], every reciprocity
sheaf F is logarithmic, i.e. it can be extended in a unique way to a functor Log(F ) definedon the category SmlSm(k) of smooth log smooth log schemes over k (see also [BM21] foran alternative construction). In fact, we have that (with rational coefficients3)
Log(F ) ∈ logCIdét ⊂ Shvltr
dét (k, Q)
where Shvltr
dét (k, Q) is the category of dividing étale sheaves with log transfers introduced
in [BPØ20, Section 3] (see also [BPØ21]), i.e. sheaves for a certain Grothendieck topology
on the category lSm(k) of log smooth log schemes over k, equipped with an extra transfer
structure with respect to an extension of Voevodsky’s category of finite correspondences.
The topology is generated by étale covers of the underlying schemes together with admissible blow-ups with center contained in the locus where the log structure is non-trivial.
The category logCIdét is the Grothendieck abelian category [BM21, Thm. 5.7] of strictly := (P1 , ∞)-invariant sheaves (here (P1 , ∞) denotes the log scheme P1 with compactifying log structure given by the open embedding A1 ,→ P1 ). Again by [BM21, Thm.
5.7], it is the heart of a t-structure, called the homotopy t-structure, on the ∞-categoryof effective log motives logDMeff (k, Q) [BPØ20], i.e. the full subcategory of the derived∞-category D(Shvltr
dét (k, Q)) consisting of -local complexes. By Saito’s theorem, we haveactually a fully faithful embedding (see (2.10.2))
CIω≤1: RSCét,≤1 (k, Q) ,→ logCIdét ⊂ Shvltr
dét (k, Q),(1.1.3)and passing to the derived categories, a functor
eff

L

logDM

ω≤1
: D(RSCét,≤1 (k, Q)) → D(Shvltr−→
logDMeff (k, Q),dét (k, Q)) −where L is the localization functor.
If we put these facts together, we see that for each (X, ∂X) ∈ SmlSm(k), each smoothcompactification (X, |∂X| + D) ∈ Comp(X) and n ≥ 1, we can construct a strictly CI (Albinvariant sheaf ω≤1(X,|∂X|red +nD) ) defined on the category of log smooth log schemes
over k. With this input, the assignment X 7→ Alblog (X) can be promoted to a functor
defined on the motivic category in the following way.
3Note that the result holds integrally if we replace the étale topology with the Nisnevich topology

DERIVED LOG ALBANESE SHEAVES

5Theorem 1.2 (Theorems 5.1 and 5.15, Proposition 5.16). Assume that the characteristic
logDMeffof k is zero. The functor ω≤1
has a pro-left adjoint, the log motivic Albanese functor:
L Alblog : logDMeff (k, Q) → Pro-D(RSCét,≤1 (k, Q)).
which fits in a commutative diagram:
L Alb≤1

D(HI≤1 (k, Q))j

Pro-D(RSC≤1 (k, Q))
L Alblog

L Alb

DMeff
≤1 (k, Q)

DMeff (k, Q)ω∗

logDMeff (k, Q),

where L Alb≤1 is the (restriction of the) functor L Alb of Ayoub, Barbieri-Viale and Kahn
(1.0.2), ω ∗ is the natural comparison functorω ∗ : DMeff (k, Q) → logDMeff (k, Q),
eff

logDM
is fullywhich is fully faithful by [BPØ20, Thm.8.2.16]. Moreover, the functor ω≤1efffaithful and its essential image is the full stable ∞-subcategory of logDM (k, Q) generatedby ω ∗ DMeff≤1 (k, Q) and Ga [n].

The proof of the above theorem is fairly technical, and requires new ingredients compared to the argument given in [ABV09] (among which some very explicit computations).
We would like to stress that the formalism of stable ∞-categories is essential to generalize
the usual construction of derived functors (via resolutions) to pro-adjoint functors between
derived categories, as we explain in Appendix A.
For X ∈ SmlSm(k), we determine the homotopy groups πi L Alblog (X) completely:
Theorem 1.3 (Theorem 6.1). Let X ∈ SmlSm(k) geometrically connected and (X, D) a
Cartier compactification of X. Then we have that
for 2 ≤ i ≤ dim(X)“ lim ”(H i (X, OX (nD))∨ ) ⊗k Ga


←−



“ lim ” (H 1 (X,OX (nD))/H 1 (X,OX ))∨ ⊗k Ga for i = 1←−
πi L Alblog (X) ∼=⊕NS∗ (X−|∂X|)

log


Alb (X) (see (1.1.2))
for i = 00otherwise.
where NS∗ (X − |∂X|) is the dual torus to the Néron-Severi group of X − |∂X|, and for V
a k-vector space, V ∨ denotes its linear dual.
As an application, we can identify the compact objects of logDMeff≤1 (k, Q): our result generalizes [BVK16] on Deligne 1-motives to the case of étale Laumon 1-motives (or“Deligne 1-motives with additive part”, see 7):
Theorem 1.4 (Theorem 7.16). Assume that k has characteristic zero. The functor
logDMeffω≤1preserves compact objects and it induces an equivalence∼

Db (Ma1,ét ⊗ Q) −→ logDMeff≤1,gm (k, Q)
where the right hand side is the ∞-subcategory of compact objects of logDMeff≤1 (k, Q)
1.1. Outline. We now give a brief outline of the contents of the various sections of thispaper.
In Section 2, we give a quick reminder of the theory of reciprocity sheaves and modulus
sheaves with transfers as developed in [KMSY21a], [KMSY21b], [KSY21] and [Sai20]. We
also give a quick recollection of the material in [BPØ20] and [BM21] on logarithmic motives
and we prove some basic result with rational coefficients. In Section 3, we construct the
Albanese map with modulus as a universal object in the category of reciprocity sheaves

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FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

and compare it with the usual Albanese map and the Albanese group scheme of [BK18].
In Section 4 we introduce the categories of n-reciprocity sheaves by a suitable modificationof the techniques of [ABV09]. We prove that the category of 0-reciprocity sheaves agrees
with the category of 0-motivic sheaves of [ABV09]. We show the existence of a pro-left
adjoint Alblog of the fully faithful embedding of 1-motivic sheaves into the category of
dividing étale sheaves with log transfers, or “logarithmic sheaves” for short (again withrational coefficients).
Section 5 is the most technical one: we prove that the category of logarithmic sheaves
admits enough BC-admissible objects (in the sense of the Appendix A) and deduce theexistence of a pro-left derived functor L Alblog . The techniques in particular are fairly
different from the corresponding one in [ABV09], although the general structure of theproof is similar. Next, we prove that the functor L Alblog factors through logDMeff andthat on DMeff≤1 it agrees with the motivic Albanese map of [ABV09] (note that this resultis optimal, see Remark 6.10). After that, we compute L Alblog (Ga ) thanks to an explicit
resolution (the Breen-Deligne resolution of the algebraic group Ga ), deducing the full
faithfulness of the inclusion D(RSC≤1,ét )(k, Q) → logDMeff (k, Q).
In Section 6, we perform several computations, and we identify precisely L Alb(X) for
X ∈ SmlSm(k), proving Theorem 1.3. We also pose some questions about the behaviourof the higher pro-Albanese sheaves in some special geometric situations. In Section 7 we
consider the category of étale Laumon 1-motives, and prove that they are motivic in thesense that their bounded derived category agrees with the category of compact objects inthe category of logarithmic 1-motives, as explained in Theorem 1.4.
Finally, in the Appendix A we introduce the notion of BC-admissible objects in astable ∞-category and generalise the notion of a derived functor to pro-adjuntions between
derived stable ∞-categories which are not in general induced by Quillen adjunctions.
Acknowledgements. The authors are deeply grateful to Joseph Ayoub for the many
insights and suggestions. F.B. wishes to thank Luca Barbieri-Viale for several usefulconversations. This project started while the first and the last named author were visiting
the Institut Mittag-Leffler in Djursholm, Sweden, during the special trimester program“Algebro-Geometric and Homotopical Methods”, in the Spring 2017. Another part of thisproject was written while the first author was a Postdoc at the University of Regensburg,
Germany and the last author was a visiting fellow there, both supported by the SFB 1085
“Higher Invariants”.
Warning 1.5. In the whole paper, we will commit the following abuse of notation: for Ga smooth commutative group scheme, we still write G for the associated étale sheaf withtransfers. For a ring Λ, we often write G ∈ Shvét (X, Λ) for the sheaf G ⊗Z Λ. Notice thatif Λ is torsion free, the functor ⊗Z Λ is exact, hence if
1→H→G→Q→1
is an exact sequence of commutative algebraic groups, then0 → H ⊗Z Λ → G ⊗Z Λ → Q ⊗Z Λ → 0
is an exact sequence of étale sheaves with transfers.
2. Reciprocity sheaves and logarithmic motives with rational coefficients
We work over a fixed ground field k, which is assumed to be perfect. Let Λ be a(commutative) ring of coefficients. In this section, we recall the main results on reciprocity
sheaves and logarithmic motives and we state some general results on the categories withrational coefficients.
Let Sm(k) be the category of separated smooth schemes of finite type over k, and let
Cor(k) be the additive category of finite correspondences. It has the same objects as
Sm(k), and for X, Y ∈ Sm(k), the hom group Cor(X, Y ) is the free abelian group on theDERIVED LOG ALBANESE SHEAVES

7

set of integral closed subschemes of X × Y which are finite and surjective over a connectedcomponent of X (see [MVW06, Def. 1.1]). We denote by PShtr (k, Λ) the category ofadditive presheaves of Λ-modules on Cor(k), whose objects are called presheaves withtransfers. For X ∈ Sm(k), we let Λtr (X) = Cor(−, X) ⊗Z Λ be the representable object.
tr
For τ the Nisnevich or the étale topology, we let Shvtr
τ (k, Λ) ⊆ PSh (k, Λ) be the categoryof τ -sheaves with transfers and we letaVτ : PShtr (k, Λ) → Shvtr
τ (k, Λ)be Voevodsky’s τ -sheafification functor: it is induced by the classical sheafification functor defined on the category of presheaves of Λ-modules without transfers. Let HI ⊆PShtr (k, Λ) be the category of A1 -invariant presheaves, i.e. objects F such that the pro'jection X × A1 → X induces an isomorphism F (X × A1 ) −→ F (X) for every X ∈ Sm(k).
trtr
Set HIτ = HI ∩ Shvτ (k, Λ) ⊆ Shvτ (k, Λ).
We recall the following result:
Proposition 2.1 ([MVW06], Cor. 14.22, Prop. 14.23). Let Λ be a Q-algebra. Then forevery F ∈ PShtr (k, Λ) we have aNis F = aét F . Moreover, for all smooth X and n > 0 we
have
n
n
HNis
(X, F ) = Hét
(X, F ).
2.1. The ∞-category of logarithmic motives. We recall the construction of the ∞category of logarithmic motives of [BPØ20] and some properties. The standard reference
for log schemes is [Ogu18]. We denote by lSm(k) the category of fine and saturated (fsfor short) log smooth log schemes over Spec(k), considered as a log scheme with trivial
log structure.
2.1.1. Log geometry. For X ∈ lSm(k), we write X ∈ Sch(k) for the underlying k-scheme.
We also write ∂X for the (closed) subset of X where the log structure of X is not trivial.
Let SmlSm(k) be the full subcategory of lSm(k) having for objects X ∈ lSm(k) suchthat X is smooth over k. By e.g. [BPØ20, A.5.10], if X ∈ SmlSm(k), then ∂X is a strict
normal crossing divisor on X and the log scheme X is isomorphic to (X, ∂X), i.e. to thecompactifying log structure associated to the open embedding (X \ ∂X) → X.
A morphism f : X → Y of fs log schemes is called strict if the log structure on X is thepullback log structure from Y . Geometrically, if both X and Y are objects in SmlSm(k),this amounts to require that there is an equality ∂X = f ∗ (∂Y ) as reduced normal crossing
divisors on X. For τ a Grothendieck topology on Sch(k), the strict topology sτ on
SmlSm(k) is the Grothendieck topology generated by covers {ei : Xi → X} such thatei : Xi → X is a τ -cover and each ei is strict. Recall from [BPØ20, 3.1.4] that a cartesiansquare of fs log schemes
Y0

g0

f0

X0

Y
f

g

X

is a dividing distinguished square (or elementary dividing square) if Y 0 = X 0 = ∅ and f is alog modifications, in the sense of F. Kato [Kat21] (see [BPØ20, A.11.9] for more details on
log modifications). The collection of dividing distinguished squares forms a cd structure
on SmlSm(k), called the dividing cd structure. For τ a Grothendieck topoogy on Sch(k),the dividing topology dτ on SmlSm(k) is the topology on SmlSm(k) generated by thestrict topology sτ and the dividing cd structure.
From now until the end of the section, we will consider τ ∈ {Nis, ét}.
2.1.2. Correspondences and transfers. Following [BPØ20], we denote by lCor(k) the category of finite log correspondences over k. It is a variant of the Suslin–Voevodsky category

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FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

of finite correspondences Cor(k). It has the same objects as SmlSm(k)4, and morphisms
are given by the free abelian subgrouplCor(X, Y ) ⊆ Cor(X − ∂X, Y − ∂Y )generated by elementary correspondences V o ⊂ (X −∂X)×(Y −∂Y ) such that the closure
V ⊂ X × Y is finite and surjective over (a component of) X and such that there exists
a morphism of log schemes V N → Y , where V N is the fs log scheme whose underlying
scheme is the normalization of V and whose log structure is given by the inverse image log
structure along the composition V N → X × Y → X. See [BPØ20, 2.1] for more details,
and for the proof that this definition gives indeed a category.
Additive presheaves (of Λ-modules) on the category lCor(k) will be called presheaves(of Λ-modules) with log transfers. Write PShltr (k, Λ) for the resulting category. As usual,for X ∈ lCor(k) we denote by Λltr (X) the representable presheaf lCor(−, X) ⊗Z Λ.
^
As in [BM21], we let SmlSm(k)be the category of fs log smooth k-schemes X which
are essentially smooth over k, i.e. X is a limit limi∈I Xi over a filtered set I, where←−
Xi ∈ SmlSm(k) and all transition maps are strict étale (i.e. they are strict maps of log
schemes such that the underlying maps fij : X i → X j are étale). For X ∈ SmlSm(k) andx ∈ X, we put(2.1.1)^
Xxh = (X hx , ∂Xxh ) ∈ SmlSm(k)where (X)hx denotes the henselization of X at x and (∂X)hx denotes the pullback of ∂X^
along the henselization map. For F ∈ PShltr (k, Λ) and X ∈ SmlSm(k)such that X =limi∈I Xi for Xi ∈ SmlSm(k) we put as usual F (X) := limi∈I F (Xi ).
−→
←−
ltr
We denote by Shvltr
(k,
Λ)
⊂
PSh
(k,
Λ)
the
subcategoryof dτ -sheaves. By [BPØ20,dτ
ltr
Prop. 4.5.4] and [BPØ20, Thm. 4.5.7], the inclusion Shvdτ (k, Λ) ⊂ PShltr (k, Λ) admits
an exact left adjoint adτ (see [BPØ20, Prop. 4.2.10]), and the category Shvltrdτ (k, Λ) is
a Grothendieck abelian category ([BPØ20, Prop. 4.2.12]). For τ ∈ {Nis, ét}, [BPØ20,
Theorem 5.1.8] implies that for F ∈ PShltr (k, Λ) and X ∈ lSm(k),(2.1.2)i(X, adτ F ) =Hdτilim Hsτ(Y, asτ F ).
−→Sm

Y ∈Xdiv

Sm is the filtered category of log modifications Y → X such that Y ∈ SmlSm(k).
where Xdiv

The following statement can be shown by imitating the proof of [MVW06, Prop. 14.23]
using (2.1.2).
Proposition 2.2. Let Λ be a Q-algebra and let F be an object of PShltr (k, Λ). Then there
is a natural isomorphismnn(X, adét F ),HdNis
(X, adNis F ) = Hdétfor all X ∈ SmlSm(k) and n ≥ 0.
Finally, recall from [BPØ20, (4.3.4)] that there is an adjunction
ω]log(2.2.1)

Shvltrdτ (k, Λ)

∗
ωlog

Shvtr
τ (k, Λ)∗ F (X) = F (X −where for Y ∈ Cor(k), ω]log F (Y ) = F (Y, triv) and for X ∈ lCor(k), ωlog|∂X|). We will need later the following immediate result.
ltr
Proposition 2.3. For all A ∈ Shvtr
τ and B ∈ Shvdτ , we have that
Hom ltr(B, ω ∗ A) ∼(ω log B, A).
= ω ∗ Hom tr
Shvdτ (k,Λ)

log

log

Shvτ (k,Λ)]

4Notice that this notation conflicts with the notation of [BPØ20] where the objects were the same aslSm(k), although the categories of sheaves are the same in light of [BPØ20, Lemma 4.7.2]

DERIVED LOG ALBANESE SHEAVES

92.1.3. Log motives. In light of Proposition 2.2, from now until the end of the section, we
will consider one of the following situations:• τ is the Nisnevich topology• τ is the étale topology and Λ is a Q-algebra.
Let D(Shvltrdτ (k, Λ)) be the derived stable ∞-category of the Grothendieck abelian catltr
egory Shvdτ (k, Λ) as in [Lur17, Section 1.3.5]: it is equivalent to the underlying ∞category of the model category Cpx(PShltr (k, Λ)) with the dτ -local model structure usedin [BPØ20] and [BM21].
∗ ) of (2.2.1) induces the following adjunction of ∞-categories
The adjunction (ω]log , ωlogof sheaves (see [BPØ20, 4.3.4]):(2.3.1)Lω]log : D(Shvltrdτ (k, Λ))∗D(Shvtrτ (k, Λ)) : Rωlog .

Finally (see [BPØ20, Section 5.2]), let  := (P1 , ∞). Notice that ω() = A1 .
Definition 2.4. The stable ∞-category logDMeff (k, Λ) is the localization of the stable∞-category D(Shvltr
τ (k, Λ)) with respect to the class of maps
(adτ Λ( × X))[n] → (adτ Λ(X))[n]for all X ∈ lSm(k) and n ∈ Z. We leteffL(dτ,) : D(Shvltrdτ (k, Λ)) → logDM (k, Λ)

be the localization functor. For X ∈ SmlSm(k), we will let M (X) = Ldτ, (Λltr (X)).
The interested reader can verify that this is equivalent to the underlying ∞-category of
the model category Cpx(PShltr (k, Λ)) with the (, dτ )-local model structure of [BPØ20,
Def. 5.2.1] and [BM21, Def. 2.9]. The derived (triangulated) category of effective log
motives logDMeff (k, Λ) is by definition the homotopy category of logDMeff (k, Λ).
We recall the following result, which follows naturally from [BM21, Thm. 5.7]:
Theorem 2.5. The standard t-structure of D(Shvltr (k, Λ) induces an accessible t-structure
on logDMeff (k, Λ) compatible with filtered colimits in the sense of [Lur17, Def. 1.3.5.20],
called the homotopy t-structure.
We denote by logCIdτ its heart5, which is then identified with the category of strictly-invariant dτ -sheaves and it is a Grothendieck abelian category. The inclusion
iltr : logCIdτ ,→ Shvltrdτ (k, Λ)0admits both a left adjoint hltr
0 : F 7→ π0 (L(dτ,) (F [0])) and a right adjoint hltr (see [BM21,
Proposition 5.8]), in particular it is exact.

2.2. Comparison with Voevodsky motives. In this subsection, we assume that k
admits resolution of singularities (see e.g. [BPØ20, Def. 7.6.3] for a precise definition).
This assumption is always satisfied if ch(k) = 0.
Definition 2.6. Let X ∈ Sm(k). A smooth Cartier compactification or simply a Cartiercompactification of X is a pair (X, D) where X ∈ Sm(k) is proper and D ⊆ X is aneffective Cartier divisor with simple normal crossing such that X − |D| ∼= X.
Note that if k admits resolution of singularities, every X ∈ Sm(k) admits a (smooth)
Cartier compactification. This definition is slightly different from the one used in [KMSY21a]and [KMSY21b], where the total space X is not required to be smooth over k, but simply
normal. Under our assumption on k, this difference is irrelevant.
5In [BM21], it is denoted by CIltr

10

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

By [BPØ20, Prop. 8.2.12], the adjunction of (2.3.1) descends to an adjunction:L ω]log : logDMeff (k, Λ)(2.6.1)∗ .
DMeff (k, Λ) : R ωlog

where the right hand side is the ∞-category of Voevodsky motives. By [BPØ20, Thm.
∗ is fully faithful and for X ∈ Sm(k) and8.2.16 and Thm. 8.2.17], the functor R ωlog
(X, D) a Cartier compactification we have a natural equivalence∗R ωlog
M (X) ' M (X, ∂X)∗ is the full subcatwith ∂X supported on |D|. In particular, the essential image of R ωlog
egory spanned by M (X) with X ∈ SmlSm(k) and X proper. Finally, by [BM21, Prop.
5.12] it is t-exact with respect to the homotopy t-structure of 2.5 on logDMeff (k, Λ) andthe Morel-Voevodsky t-structure on DMeff (k, Λ). In particular, we have a fully faithful∗ ) HI → logCI
functor (still denoted by ωlogτdτ between the hearts that commutes with
the inclusions. The following result will be crucial in the proof of Proposition 5.10:∗ : HI → logCI
Lemma 2.7. The functor ωlogτdτ admits a right adjoint (in particular it
commutes with all colimits) and is monoidal.
∗ is fully faithful and by [BM21, Proposition 7.3] the composition logCI
Proof. Since ωlog
dτ ⊆
ω]log

tr
Shvltrdτ −−→ Shvτ is also fully faithful, we have that for F ∈ HIτ and G ∈ logCIdτ ,∗∗F, ω]log G) = HomHI (F, h0A1 (ω]log G)),F, G) = HomShv (ω]log ωlog
HomlogCIdτ (ωlog

where h0A1 : Shvtr
τ → HIτ is right adjoint to the inclusion (see [RS21, 4.34]). This proves
log
0∗ , which implies that ω ∗ commutes with all colimits.
that hA1 ω] is right adjoint to ωlog
log∗
Since the functor ωlog commutes with all colimits, it is enough to show that for X, Y ∈Sm(k) we have
1

1

1

A
A
∗
∗
∗
(hA
ωlog
0 (X × Y )) = ωlog h0 (X) ⊗logCI ωlog h0 (Y ),where h0A1 : Shvtr
τ → HIτ is left adjoint to the inclusion (the 0-th Suslin homology sheaf).
Thanks to [BPØ20, Proposition 8.2.4], for any choice of a smooth Cartier compactification
X → X and Y ⊆ Y , by putting as log structure ∂X and ∂Y associated with the simple
normal crossing divisor X − X and Y − Y , we have equivalences in logDMeff :1∗R ωlog
M A (X) = M (X, ∂X)

1∗
and R ωlog
M A (Y ) = M (Y , ∂Y ).

By taking π0 , [BM21, Proposition 5.12] implies that1ltr
ω ∗ hA
0 (X) = h0 (X, ∂X)

1ltr
and ω ∗ hA
0 (Y ) = h0 (Y , ∂Y ).

Hence, we have that
1

1

∗ A
ltr
ltr
ltr
ω ∗ hA
0 (X) ⊗logCI ω h0 (Y ) = h0 (X, ∂X) ⊗logCI h0 (Y , ∂Y ) = h0 ((X, ∂X) × (Y , ∂Y )).

Finally, since the underlying scheme of (X, ∂X) × (Y , ∂Y ) is X × Y , which is proper,
and the subscheme where the log structure is trivial is X × Y , we have that the log
scheme (X, ∂X) × (Y , ∂Y ) is a Cartier compactification of X × Y , hence again by [BPØ20,
Proposition 8.2.4] we have
1

∼ ∗ A
hltr
0 ((X, ∂X) × (Y , ∂Y )) = ω h0 (X × Y ),which concludes the proof.

DERIVED LOG ALBANESE SHEAVES

112.3. The abelian category of reciprocity sheaves. We recall the construction of theabelian category of reciprocity sheaves via modulus sheaves of [KMSY21a] and [KMSY21b]
as done in [KSY21] and some properties.
A pair X = (X, D) where X is proper scheme of finite type over k and D is an effective
Cartier divisor on X is called a proper modulus pair if X −|D| ∈ Sm(k). Let X = (X, DX ),Y = (Y, DY ) be proper modulus pairs and Γ ∈ Cor(X − |DX |, Y − |DY |) be a primeNcorrespondence. Let Γ ⊆ X × Y be the closure of Γ, and let Γ → X × Y be thenormalization. We say that Γ is admissible if (DX )ΓN ≥ (DY )ΓN as Weil divisors, where
N(E)ΓN denotes the pullback of the divisor E to the normalization Γ . By [KMSY21a,
Proposition 1.2.7], proper modulus pairs and admissible correspondences define an additive
category, denoted MCor(k). For X = (X, D) and n ≥ 0, we let X(n) := (X, nD).
We denote by MPST(k, Λ) or simply MPST the category of additive presheaves of Λmodules on MCor(k), whose objects are called proper modulus presheaves with trasnfers.
For X ∈ MCor(k), we let Λtr (X) = MCor(−, X) ⊗Z Λ ∈ MPST be the representableobject.
Definition 2.8. For X ∈ Sm(k), we let Comp(X) be the cofiltered category given bymodulus pairs (X, D) given by Cartier compactifications of X (see [KMSY21a, Lemma
1.8.2] and Definition 2.6).
There is a functor:
ω : MCor(k) → Cor(k)(X, D) 7→ X − |D|,

which induces adjoint functors (cf. [KMSY21a, Pr. 2.2.1]):
ω! : MPST(k, Λ)

PShtr (k, Λ) : ω ∗where ω ∗ is fully faithful. For X = (X, D) ∈ MCor(k), we haveω ∗ F (X) = F (ω(X)) = F (X − |D|).
The functor ω! is given by left Kan extension, so that for X ∈ Sm(k) and any choice of
X ∈ Comp(X), we have
(2.8.1)

ω! F (X) =lim−→

Y∈Comp(X)'

F (Y) −→ lim F (X(n) ).
−→n+ = 0),
where the displayed isomorphism follows from [Sai20, Lemma 1.27 (1)] (with X∞
which implies that we have an isomorphism in pro- MPST:“ lim ”Ztr (X(n) ) ∼lim ”Y∈Comp(X) Ztr (Y).
= “←
←−
−
n

As in the logarithmic case, let  := (P1 , ∞) ∈ MCor(k) and for any X = (X, D) ∈MCor(k) let (see [KMSY21a])X ⊗  := (X × P1 , X × ∞ + D × P1 ).
We say F ∈ MPST is -invariant if for any X ∈ MCor(k), the projection p : X ⊗  → X
induces an isomorphismp∗ : F (X) → F (X ⊗ ).
We let CI be the full subcategory of MPST consisting of all -invariant objects. By[KSY21, Lemma 2.1.2], it is a Serre subcategory of MPST and that the inclusion functor0i : CI → MPST has a left adjoint h
0 and a right adjoint h given for F ∈ MPST and
X ∈ MCor(k) by∗∗h
0 (F )(X) = Coker(i0 − i1 : F (X ⊗ ) → F (X)),

h0 (F )(X) = Hom(h
0 (X), F ),

12

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

where for a ∈ k the section ia : X → X ⊗  is induced by the map k[t] → k[t]/(t − a) ∼= k.
We write RSC(k, Λ) ⊆ PShtr (k, Λ) for the essential image of CI under ω! . It is an abeliansubcategory of PShtr (k, Λ).
Remark 2.9. By (2.8.1), for F ∈ PShtr (k, Λ) the following conditions are equivalent:(i) F ∈ RSC(k, Λ),(ii) for every X ∈ Sm(k) and every section a : Ztr (X) → F , there exists Y ∈Comp(X) such that a factors through Ztr (X) → ω! h
0 (Y),(iii) for every X ∈ Sm(k) and every section a : Ztr (X) → F , for any choice of X ∈(n) ).
Comp(X) there exists n such that a factors through Ztr (X) → ω! h
0 (XFor τ the Nisnevich or the étale topology, we let RSCτ (k, Λ) := RSC(k, Λ)∩Shvtr
τ (k, Λ).
The objects of RSC(k, Λ) (resp. RSCτ (k, Λ)) are called reciprocity presheaves (resp. τ reciprocity sheaves) of Λ-modules. By [Sai20, Thm. 0.1], the Nisnevich sheafification
restricts to a functor
aVNis : RSC(k, Λ) → RSCNis (k, Λ),which makes RSCNis a Grothendieck abelian category (see [KSY21, Corollary 2.4.2]). Notice in particular that RSCNis is closed under sub-objects and quotients in Shvtr
Nis (k, Λ),trand that the inclusion functor i : RSCNis → ShvNis (k, Λ) is exact. As in [KSY21, Theorem 2.4.3 (1)], we denote by ρ the right adjoint to the inclusion i.
By [Sai20, Theorem 0.2], each F ∈ RSCNis (k, Λ) satisfies global injectivity, i.e. for every
X ∈ Sm connected with generic point ηX , the restriction map gives an injective map:
F (X) ,→ F (ηX ).
By Proposition 2.1, if Λ is a Q-algebra the étale sheafification coincides with the Nisnevichsheafification, hence it restricts to a functor(2.9.1)

RSC(k, Λ)aV
Nis

RSCNis (k, Λ)'

RSCét (k, Λ),Vaétin particular, if Λ is a Q-algebra, RSCét (k, Λ) is a Grothendieck abelian category andevery F ∈ RSCét (k, Λ) satisfies global injectivity.
We have two important examples of reciprocity sheaves:(1) Let G ∗ be the category of smooth commutative k-group schemes, locally of finite
type over k (i.e. G ∈ G ∗ such that the connected component of the identity G0 is
a commutative algebraic gruop and π0 (G) is finitely generated). It is classical thatthe corresponding étale sheaf has a unique structure of sheaf with transfers (see[SS03, Lemma 3.2]) and by [KSYR16, Theorem 4.4] it has reciprocity in the senseof [KSYR16, Definition 2.1.3]. This defines a functor G ∗ → RSCét , generalizing∗ → HI considered in [BVK16].
the functor Gsabét(2) For C = (C, C∞ ) ∈ MCor(k) with dim(C) = 1, let Pic(C, C∞ ) denote the relative
Picard group scheme. We would like to underline that C∞ is not supposed to be
reduced. By [RY16, Thm. 1.1] combined with [Sai20, Corollary 0.3], we have that
(2.9.2)ω! h
0 (Ztr (C)) = Pic(C, C∞ ).

We end this subsection recalling the following result:
Proposition 2.10. If Λ is a Q-algebra, the “forgetting transfers” functor RSCét (k, Λ) →Shvét (k, Λ) is fully faithful and exact.
Proof. The argument for homotopy invariant sheaves with transfers given in [BVK16, 3.9]
works here as well, replacing the reference to Voevodsky’s purity theorem for homotopy
invariant sheaves with the global injectivity provided by [Sai20, Theorem 0.2].
DERIVED LOG ALBANESE SHEAVES

132.4. Reciprocity sheaves and logarithmic motives. In this subsection, we continue
to assume that k satisfies resolution of singularities. Let τ and Λ be as in 2.1.3.
log
We denote by logRecτ ⊆ Shvtr
τ the essential image of the restriction of ω] to logCIdτ .
CI := h0 ω ∗ are an equivaAs observed in [BM21, Definition 7.4], the functors ω]log and ωlog
ltr log
lence of categories between logCIdτ and logRecτ . By [Sai21], there exists a fully faithfuland exact functor
Log : RSCNis → logCIdNissuch that ω] Log ' id. This implies (see [BM21, Theorem 7.6]) that the category RSCNis
is a full subcateogry of logRecNis and the functor Log coincides with the composition:(2.10.1)

CI
ωlog

RSCNis ⊆ logRecNis −−→ logCIdNis ,→ Shvltr
dNis .

If Λ is a Q-algebra, by Proposition 2.2, we have similarly a fully faithful and exact functor:(2.10.2)

CIωlog: RSCét(2.9.1)(2.10.1)∼= logCIdét ,→ Shvltr= RSCNis → logCIdNis ∼dét .

Remark 2.11. Since logRecτ ⊆ Shvtr
τ is fully faithful, the inclusion RSCτ ⊆ logRecτ
has a right adjoint given by the functor ρ of [KSY21, Theorem 2.4.3 (1)]. The composiρtion logCIdτ ∼→ RSCτ gives then a right adjoint to the inclusion RSCτ ⊆= logRecτ −logCIdτ . Since the last inclusions of (2.10.1) and (2.10.2) also have right adjoints h0ltr(which coincides under our assumption on Λ), the functors (2.10.1) and and (2.10.2) haveright adjoints as well, in particular they commute with all colimits.
CI
Remark 2.12. For every F ∈ logCIdτ , the equivalence ω]log : logCIdτ ' logRecτ : ωlog
induces an isomorphism of sheaves:
CI logF ∼ω] F.
= ωlog(2.12.1)∗ ω log F ∈ logCI , hence
If ω]log F ∈ HIτ , we have that ωlog
τ
]
(∗)

∗
∗
CI logω]log F,ω]log F ' ωlog
ω] F = h0ltr ωlog
ωlog
∗ ω log F ∈ logCI , which
where the equivalence (∗) strictly depends on the fact that ωlogdτ]

is not true unless ω]log F ∈ HIτ . We conclude that for every F ∈ logCIdτ such thatω]log F ∈ HIτ , we have an isomorphism of sheaves:∗F ∼ω]log F.
= ωlog(2.12.2)

We finish this section with the following analogue of Proposition 2.3:
Proposition 2.13. For all A, B ∈ logCIdτ , then:ω]log HomShvltr (A, B) ∼(ω]log A, ω]log B).
= HomShvtr
τdτ∗ is fully faithful, we have an exact
Proof. By [BM21, Theorem 5.10] and the fact that ωlog
sequence∗
0 → B → ωlog
ω]log B → Q → 0

with ω]log Q = 0, which induces an exact sequence:∗ω]log B) → ω]log HomShvltr (A, Q).
0 → ω]log HomShvltr (A, B) → ω]log HomShvltr (A, ωlogdτdτ

By Proposition 2.3, we have that∗ω] HomShvltr (A, ωlog
ω]log B) ∼(ω]log A, ω]log B),= HomShvtr
τ
dτ

dτ

14

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

hence it is enough to show that ω] HomShvltr (A, Q) = 0. For X ∈ Sm(k) we have thatdτω] HomShvltr (A, Q)(X) = HomShvltr (A, HomShvltr (Λltr (X, triv), Q))
dτ

dτ

dτ

= HomlogRecτ (A, h0ltr HomShvltr (Λltr (X, triv), Q)),dτ

where h0ltr : Shvltr
dτ → logCIdτ is the right adjoint of the inclusion. Since for all Y ∈ Sm(k)we have thatω]log HomShvltr (Λltr (X, triv), Q)(Y ) = Q(X × Y, triv) = ω]log Q(X × Y ) = 0,dτ

by Lemma 2.14 below we conclude that h0ltr HomShvltr (Λltr (X, triv), Q) = 0, which condτ
cludes the proof.
log
0
Lemma 2.14. For all G ∈ Shvltr
dτ such that ω] G = 0, we have hltr G = 0.

Proof. By purity [BM21, Theoremn 5.10], it is enough to show that h0ltr Q(ηX , triv) = 0
for any generic point ηX of X ∈ Sm(k). We have thath0ltr Q(ηX , triv) = Hom(hltr
0 (Λltr (ηX , triv)), Q)
and as observed in the proof of [BM21, Lemma 7.2], it follows from [BPØ20, Proposition
8.2.2 and 8.2.4] that there is a surjective map
1ltrΛtr (ηX , triv) ' ω ∗ Λtr (ηX )  ω ∗ hA
0 (ηX ) ' h0 (Λltr (ηX , triv)).

In particular,(Λtr (ηX ), ω]log Q) = 0
h0ltr Q(ηX , triv) ,→ HomShvltr (Λltr (ηX , triv), Q) = HomShvtr
τdτ

as required.


3. Categories of rational maps and universal problems

3.1. Commutative groups schemes and torsors under them. We recall some wellknown facts on commutative group schemes over a perfect field k and we fix some notation.
Let G ∗ be again the category of smooth commutative k-group schemes, locally of finite
type over k (for short, a commutative k-group scheme). Write G for the subcategory ofsmooth commutative algebraic k-groups (i.e. objects of G ∗ which are of finite type). Given
G ∈ G ∗ , let G0 be the connected component of the identity in G. Recall (see [BVK16,
Definition 1.1.2] or [DG80, Proposition 5.1.4]) the following definition.
Definition 3.1. A group scheme L ∈ G ∗ is called discrete if L0 = Spec(k) and the abeliangroup L(k) is finitely generated (equivalently, if L is étale over k). A discrete k-groupscheme L ∈ G ∗ is called a lattice if L(k) is torsion free.
As in [BVK16], we denote by t M0 the subcategory of G ∗ consisting of discrete k-groupschemes. By [BVK16, Lemma 1.1.3] it is a Serre subcategory of G ∗ , hence it is an Abeliancategory. We denote by M0 the full subcategory of lattices. By [DG80, Proposition 5.1.8],
for G ∈ G ∗ , there is an exact sequence
0 → G0 → G → π0 (G) → 0
where π0 (G) is an étale k-group, which is universal for homomorphisms from G to discretegroups. The fibers of G → π0 (G) are the irreducible components of G.
Definition 3.2. Let G ∈ G be a smooth commutative algebraic k-group. By k-torsor
under G or for G we mean a k-scheme P , locally of finite type over k, equipped with anaction P × G → P such that the induced morphism (s, g) 7→ (s, sg) : P × G → P × P
is an isomorphism. Write PG for the category of k-torsors under G: morphisms between
torsors are G-equivariant k-morphisms.

DERIVED LOG ALBANESE SHEAVES

15

3.1.1. `Given a k-torsor P under G ∈ G, we can construct a commutative k-group scheme
PG = n∈Z P ∨n ∈ G ∗ using the sum of torsors ∨G (see [Mil80, III.4.8.b]) following [Ram01,1.2]. It fits in a short exact sequenceaP
0 → G → PG −−→
Z → 0,

presenting PG as extension of Z by the group G. Moreover, we can identify the torsor P
with the fiber of the section 1 ∈ Z along the map aP , so that we have a natural inclusion
P ,→ PG .
3.1.2. Write P for the category whose objects are pairs (P, G), where P ∈ PG for a
G ∈ G a smooth commutative connected algebraic group. A morphism in P is the datumof a pair of morphisms (f 1 , f 0 ) : (P, G) → (P 0 , G0 ), where f 0 : G → G0 is a k-morphismof algebraic groups and f 1 : P → P 0 is f 0 -equivariant. If X is a k-scheme, we write X\Pfor the comma category over X: its objects are triples (u, P, G), where (P, G) ∈ P andu : X → P is a k-morphisms. Morphisms in X\P are defined in the obvious way.
Definition 3.3. A fibration to torsors is the datum, for each X ∈ Sm, of a full category
MX of X\P, contravariantly functorial in X. Similarly, a fibration to torsors for proper
modulus pairs is the datum, for each X = (X, X∞ ) ∈ MCor, of a full subcategory MX of
X\P, where X = X \ X∞ ∈ Sm, contravariantly functorial in X for maps in MSmfin (see[KMSY21a, Definition 1.3.3.(2)]).
The initial object (if it exists) of MX is called the M-Albanese torsor of X. By
definition, it is the datum of a smooth commutative connected algebraic group Alb0M (X),a k-torsor Alb1M (X) under Alb0M (X) and a k-morphism X → Alb1M (X) which is universal
for maps in MX . The algebraic group Alb0M (X) is called the M-Albanese variety of X.
Similarly, if M− is a fibration to torsors for proper modulus pairs, the initial object (if itexists) of MX is called the M-Albanese torsor of X. The corresponding algebraic group,
Alb0M (X) will be called the M-Albanese variety of X.
Example 3.4. For any X, let SAbX be the full subcategory of X\P consisting of maps
to torsors P under semi-Abelian varieties. In this case, the existence of an initial object
for SAbX was proven by Serre [Ser60] in the case the base field k is algebraically closed.
In [Wit08, Appendix A], a Galois descent argument is used to show that the Albanesevariety, the Albanese torsor and the universal map(1)

X → AlbSAb (X)always exist, without any assumption on k.
If X is smooth and proper over k, the semi-abelian variety Alb0SAb (X) is in fact an
Abelian variety, and coincides with the classical Albanese variety of X, dual (as abelianvariety) to the Picard variety Pic0,red.
X
3.2. A universal construction. We will discuss a number of situations in which the
M-Albanese torsor of a proper modulus pair X exists. In fact, we will consider different
fibrations to torsors M− for proper modulus pairs, giving sufficient conditions for the
initial object to exist. In the end, all the fibrations that we consider will turn out to beequivalent, giving then a unique notion of Albanese torsor for a proper modulus pair X.
Definition 3.5. Let X = (X, X∞ ) be a proper geometrically integral modulus pair, and11write U (X) = U (X, X∞ ) for the k-vector space H 0 (X, ΩX,cl(X∞ )), where ΩX,cldenotesthe subsheaf of closed forms. Let MΩ
X be the full subcategory of X\P consisting of triples(u, P, G) with the following property: Let k ⊂ k be an algebraic closure of k and let
uk : Xk → Pk ∼= Gk be the base change of u to k. Then (u, P, G) ∈ MΩ
X if and only if∗(uk ) Ω(Gk ) ⊆ U (X k , X∞,k ), where Ω(Gk ) denotes the space of invariant differential forms
on Gk . The assignment X 7→ MΩ
X defines a fibration to torsors for proper modulus pairs
in the sense of Definition 3.3

16

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

Remark 3.6. It follows immediately that (u, P, G) ∈ X\P belongs to MΩ
X if and only ifu∗L Ω(GL ) ⊆ U (X L , X∞,L )for any algebraically closed field L ⊂ k.
Theorem 3.7. For any X ∈ MCor, the MΩ
X -Albanese torsor of X exists.
Proof. Suppose that k = k is algebraically closed. Since any k-torsor over an algebraic
group G ∈ G is trivial in this case, the category MΩ
X is equivalent to the category ofmorphisms u : X → G from X to algebraic groups satisfying the condition u∗ (Ω(G)) ⊆0
U (X). Morphisms in MΩ
X are k-morphisms f : G → G of torsors commuting with the00structural morphisms u : X → G and u : X → G (where we view G and G0 acting onthemselves). Although the morphism f is not a homomorphism of algebraic groups ingeneral, it can be written as f = f0 + τ , where f0 is a group homomorphism and τ is atranslation.
Following [Ser60], in order to check whether an initial object for MΩ
X exists, it is enoughΩ,gto restrict to the subcategory MX of morphisms which are generating (see [Ser60, Definition 1]). For them, one has the following simple
Lemma 3.8 (Lemma 6, p.198, [FW84]). Let u : X → G ∈ MΩ
X and suppose that u is
generating in the sense of [Ser60, Definition 1]. Then the pullback mapu∗ : H 0 (G, Ω1G )inv = Ω(G) → H 0 (X, Ω1X )is injective.
By e.g. [Ser75, Prop. III.16], the dimension of any G ∈ G agrees with the dimension of
the k-vector space Ω(G) of invariant differential forms. The previous Lemma implies thatfor any u : X → G ∈ MΩ,g
X , one has dim G ≤ dim U (X). Since by [Ser60, Corollaire, p.05],
a necessary and sufficient condition for the initial object to exist is a uniform bound on
the dimensions of the groups appearing in MΩ,g
X , we conclude.
Suppose now that k is any perfect field and let k be an algebraic closure of k. Write
X k for the base change X ⊗k k and Xk for the pair (X k , (X∞ )k ). According to the above
argument, the category MΩ
X admits a universal object,
k

Ω
albΩ
X : Xk → AlbX .
kk

The descent to the base field k can be done following the proof of Serre [Ser75, V.22] in
Ω,(1)
Ω,(0)
the case of generalized Jacobians of curves to get a triple (albΩ, AlbX ) defined
X , AlbX
over k.

3.3. Cutting curves. Assume now that X = (X, X∞ ) ∈ MCor is such that X is smoothover k. A finite morphism ν : C → X, with C a normal and geometrically integral curve,
is admissible for X if ν(C) 6⊆ X∞ . In this case, write C∞ for the effective Cartier divisorν ∗ X∞ on C. Write C for the open subset C \ C∞ . We will use the following Lemma,taken from [BS19].
Lemma 3.9 (Lemma 10.14, [BS19]). Let γ be the restriction map
Y1
H 0 (C, Ω1C )/H 0 (C, ΩC
γ : H 0 (X, Ω1X ) →(ν ∗ X∞ )),ν : C→X

where the product runs over the set of admissible curves ν : C → X. Then the kernel of γ
agrees with H 0 (X, Ω1X (X∞ )).
If u : X → P is a k-morphism from X to a torsor P for an algebraic group G ∈ G, we
get by composition a morphismuuC : C := C ×X X → X −→ P.
Write ν ∗ (u, P, G) for the corresponding object in C\P.

DERIVED LOG ALBANESE SHEAVES

17

Lemma 3.10. A triple (u, P, G) ∈ X\P belongs to MΩ
X if and only if for any admissible
curve ν : C → X, we have ν ∗ (u, P, G) ∈ MΩ.
(C,C )∞

Proof. The necessity of the condition is clear. According to Definition 3.5 and Remark 3.6,
the statement of the Lemma can be checked over an algebraic closure k of k, so that we
can assume k = k. As above, the category MΩ
X is equivalent to the category of morphisms
ψ : X → G from X to algebraic groups satisfying the condition ψ ∗ (Ω(G)) ⊆ U (X). Let
now ψ : X → G be a k-morphism, and let ω ∈ Ω(G). We have to show that ψ ∗ (ω) ∈1 (X )) (note that ψ ∗ (ω) is automatically closed), i.e. that η := ψ ∗ (ω) has poles
H 0 (X, ΩX∞
along |X∞ | of order bounded by the multiplicity of X∞ , assuming that this condition issatisfied after restriction admissible to curves. But this is precisely the content of Lemma
3.9.

Proposition 3.11 (see [FW84]). Let X be a smooth proper modulus pair over a field kof characteristic 0. Let k be an algebraic closure of k. Then the MΩ
X -Albanese torsor
Ω,(1)
Ω,(0)
Ω
(albX , AlbX , AlbX ) is characterized by the property that the pullback mapΩ,(0)uk∗ : Ω(AlbX)k → H 0 (Xk , Ω1X )k1 (X )) ⊗ k.
is injective, with image equal to the subspace H 0 (X, ΩX∞
k

3.4. The universal regular quotient of the Chow group of zero cycles. We startby recalling the definition of the Kerz-Saito Chow group of 0-cycles with modulus (see[KS16]). For an integral scheme C over k and for E a closed subscheme of C, we set\
××G(C, E) =Ker OC,x→ OE,xx∈E=××lim Γ(U, ker(OC
)),
→ OE
−→

E⊂U ⊂C

where U runs over the set of open subsets of C containing E (the intersection taking placein the function field k(C)× ). We say that a rational function f ∈ G(C, E) satisfies themodulus condition with respect to E.
Let X = (X, X∞ ) ∈ MSm be a proper modulus pair and write X for the complement
X \ |X∞ |. Let Z0 (X) be the free abelian group on the set of closed points of X. Let C
be an integral normal curve over k and let ϕC : C → X be a finite morphism such that
ϕC (C) 6⊂ X∞ (so C is admissible in the sense of 3.3). The push forward of cycles along
the restriction of ϕC to C = C ×X X gives a well defined group homomorphism
τC : G(C, ϕ∗C (X∞ )) → Z0 (X),sending a function f to the push forward of the divisor divC (f ).
Definition 3.12 (Kerz-Saito). We define the Chow group CH0 (X) = CH0 (X|X∞ ) of
0-cycles of X with modulus X∞ as the cokernel of the homomorphismM∗τ:
G(C, ϕC(X∞ )) → Z0 (X),ϕC : C→X

where the sum runs over the set of finite morphisms ϕC : C → X from admissible curves.
Definition 3.13. Let X ∈ Sm(k) be geometrically integral and X ∈ Comp(X) (seeDefinition 2.8). Let CH0 (X) be the Chow group of 0-cycles with modulus of X. Let MCH
X
be the full subcategory of X\P consisting of triples (u, P, G) with the following property.
For any algebraically closed field L ⊃ k, write uL : Z0 (XL )0 → PL (L) ∼= GL (L) for the

18

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

induced morphism on zero-cycles of degree zero. Then (u, P, G) ∈ MCH
X if and only if uLfactors as
Z0 (XL )0uL&/ GL (L),CH0 (XL )0
where CH0 (XL )0 is the image of Z0 (XL )0 .

Proposition 3.14. Assume that X is a smooth, proper and geometrically integral moduCH,(1)CH,(0)CH, AlbX) of X exists. If
lus pair over k. Then the MCH
X -Albanese torsor (albX , AlbX
(1)(0)
dim X = 1, it agrees with the Rosenlich-Serre generalized Jacobian (ϕX∞ , JacX,X , JacX,X )∞∞of [Ser75, V.4.20].
Proof. If dim X = 1, this is precisely the content of [Ser75, V.Theorem 1], and the very
definition of modulus for a rational map and local symbols. For the general case, as in the
proof of Theorem 3.7, it is enough to show the existence in the case k = k is algebraicallyclosed (the descent argument is identical). Similarly, we can restrict to the category MCH,gX
of morphisms which are generating. According to [Ser60, Corollaire, p.05] it is then enoughto show that there exists a uniform bound on the dimensions of the groups appearing inΩMCH,g. We do this by showing that MCH
X is a (full) subcategory of MX . The requiredXbound will be then provided by Lemma 3.8.
Suppose then that that u : X → G is a k-morphism from X = X\|X∞ | to a commutative
connected algebraic group G such that the induce map on zero cycles factors throughCH0 (X). Let ϕ : C → X be a finite morphism from normal integral curve C such that
ϕ(C) 6⊂ X∞ . Put C = C ×X X and C∞ = ϕ∗ (X∞ ). Let uC : C → G be the compositionu ◦ ϕ. Then we haveuC (divC (f )) = u(ϕ∗ (divC (f ))) = 0 in G(k) for any f ∈ G(C, C∞ ).
In particular, the divisor C∞ is a modulus in the sense of Rosenlicht-Serre for the rationalmap (still denoted uC ) uC : C 99K G. Therefore we have then a factorization ([Ser75, V,
Theorem 2])
C
,


a

JacC,C∞

uC

#
ũC

/G

where a : C → JacC,C∞ is the universal map from C to its generalized Jacobian (withrespect to a chosen k-rational point). But now we haveu∗C (Ω(G)) = a∗ (ũ∗C (Ω(G))) ⊆ a∗ (Ω(JacC,C∞ )) = H 0 (C, Ω1C ⊗C OC (C∞ )),where the last equality follows from [Ser75, V, Proposition 5]. This implies that (uC , G) ∈MΩ
for any admissible curve and so, by Lemma 3.10, we deduce that (u, G) ∈(C,C )∞MΩ
X.



Remark 3.15. Suppose that k = k is algebraically closed and let X be a smooth proper
integral modulus pair. Then the morphism
ρX : CH0 (X)0 → AlbX (k)CH,(0)

to the Albanese variety AlbX = AlbX
of X induced by albX is surjective and regular.
We can reformulate the universal property in MCH
X by saying that AlbX is the universal
regular quotient of the Chow group of zero cycles with modulus. As such it agrees, aposteriori, with the Albanese variety Alb(X|X∞ ) of [BK18, Theorem 1.1]. If k = C andDERIVED LOG ALBANESE SHEAVES

19d|X∞ | is a strict normal crossing divisor, it agrees with the generalized Jacobian JX|X∞

(for d = dim X) studied in [BS19, 10.2]. Despite the fact that it is defined starting froma different modulus condition on algebraic cycles, it agrees also with the Albanese variety
with modulus Alb(X, X∞ ) defined by Russell [Rus13]. This is a consequence of Lemma
3.10 and the fact that both notions agree with the Rosenlicht-Serre generalized Jacobianin the one-dimensional case6.
3.4.1. Let X be as above. From the proof of Proposition 3.14, we deduce immediately
the existence of a natural surjective map of torsors
Ω,(1)

ρΩ,CH
: AlbX
X

CH,(1)→ AlbXΩ,(0)equivariant with respect to a surjective homomorphism of algebraic k groups AlbX→
CH,(0)Ω,CHΩCH◦ albX . We will see below that those maps are isomorAlbX
such that albX = ρX
phisms when the characteristic of the base field is zero.
To further relate the Chow groups of zero cycles with modulus of a pair X with the MΩ
X
Albanese construction of 3.2, we also recall the following
Proposition 3.16 ([Ser75, III, Proposition 10] or [KSYR16, Proposition 4.3.1]). Let G
be a commutative algebraic group over a field K of characteristic zero. Let C be a proper
normal curve over K, C an open dense subscheme of C and ψ : C → G a K-morphism.
Let D be an effective divisor on C supported on C \ C such thatψ ∗ (Ω(G)) ⊂ H 0 (C, Ω1C ⊗C OC (D)).
Then we have ψ(divC (f )) = 0 in G(K) for any f ∈ G(C, D).
Proposition 3.17. Let k be a field of characteristic zero. Then the MCH
X -Albanese torsor
Ω,(1)
Ω,(0)
Ω
Ω
of X agrees with the MX -Albanese torsor (albX , AlbX , AlbX ) of Theorem 3.7.
Ω
Proof. It is enough to show that the two categories MCH
X and MX have the same objects(as they are both full subcategories of X\P). According to Definitions 3.13 and 3.5,
it is enough to show the statement under the assumption that k = k is algebraicallyis a fullclosed. We already know thanks to the proof of Proposition 3.14 that MCH
X
subcategory of MΩ
(this
does
not
requirektobeofcharacteristic
0).
To
prove
the
other
X
Ω
inclusion, let u : X → G ∈ MX . Again by Lemma 3.10, for any X-admissible morphism
ϕ : C → X from a normal integral curve, the composition uC : C ×X X → X → G satisfiesu∗C (Ω(G)) ⊆ H 0 (C, Ω1C ⊗C OC (C∞ )), where C∞ denotes as before the pullback ϕ∗ (X∞ ).
By Proposition 3.16 above, we have uC (divC (f )) = u(ϕ∗ divC (f )) = 0 in G(k) for any
f ∈ G(C, C∞ ), so that the map induced by u on the group of zero cycles of degree zero
Z0 (X)0 factors through CH0 (X)0 . The same argument applies to any base-change to L ⊃ kalgebraically closed, so that (u, G) ∈ MCH
X as required.

3.5. The Albanese scheme with modulus. We can now extend the construction of
Ramachandran [Ram01] to the modulus setting. Let X ∈ Sm(k) be geometrically integral
and X ∈ Comp(X). Thanks to Theorem 3.7, we have a map, defined over k,(3.17.1)

Ω,(1)

albX

Ω,(1)

: X → AlbX

universal for morphisms from X to torsors under commutative algebraic groups in MΩ
X (inthe following, we shall say “to torsors under commutative algebraic groups with modulus`Ω,(1) ⊗n∗X”). Let AlbΩ
) constructed in 3.1.1. The
X ∈ G be the k-group schemen∈Z (AlbX
6An independent (and explicit) proof of the fact that over C the generalized Jacobian J d

X|X∞agrees with

Alb(X, X∞ ) defined by Russell has been given by T. Yamazaki [Yam17], using Hodge-theoretic methods.

20

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITOΩ,(1)universal map (3.17.1) composed with the natural inclusion of AlbX
in AlbΩ
X gives thena canonical morphism
aX : X → AlbΩ
X
which is now universal for morphisms to k-group schemes in the appropriate sense. By
construction, the k-group scheme AlbΩ
X is an extensionΩ,(0)0 → AlbX→ AlbΩ
X → Z → 0.

∼
If X has a k-rational point, the extension is split, i.e. we have an isomorphism AlbΩ
X =
Ω,(0)
AlbX ×Z. This happens in particular when k is algebraically closed, and correspondsΩ,(1) ∼Ω,(0)to the fact that we can trivialize the torsor AlbX
= AlbX . Recall now the following
Proposition, which follows from [KSYR16, Theorem 4.1.1] (while the transfer structure
follows from [SS03, Proof of Lemma 3.2])
Proposition 3.18. The k-group scheme AlbΩ
X , regarded as étale sheaf on Sm(k), has acanonical structure of sheaf with transfers, and as such it has reciprocity in the sense of[KSYR16, Definition 2.1.3].
Thanks to the Proposition, there is a unique map of preshaves with transfersaX : Ztr (X) → AlbΩ
X
extending the map aX : X → AlbΩX defined above.
3.5.1. When X is moreover smooth over k, we can apply Proposition 3.14 to get a map,
CH,(1)
CH,(1)
defined over k, albX
: X → AlbX
, universal for morphisms to torsors in MCH
X .
We can repeat the constructions of the previous point to get yet another k-group scheme∗
AlbCH
X ∈ G together with a canonical morphism
CHaCH
X : X → AlbX .

The group AlbCH
X has by the same argument of Proposition 3.18 a canonical structureof étale sheaf with transfers, with reciprocity in the sense of [KSYR16, Definition 2.1.3].
This gives us a unique map of presheaves with transfers
CHaCH
X : Ztr (X) → AlbX

extending aCH
X . When the base field has characteristic zero, the two constructions agreeΩby Proposition 3.17, so that we canonically identify AlbCH
X with AlbX . In general, weCHhave a surjective map of étale sheaves AlbΩ
X → AlbX .
3.6. The maximal semi-abelian quotient. Let X ∈ Sm(k) be geometrically connected. Serre’s Albanese map of Example 3.4 can be extended to a unique map ofpresheaves with transfers:
Ztr (X) → AlbX

(3.18.1)

where AlbX is the semi-abelian Albanese scheme of X, defined by [Ram01] or [SS03,
Lemma 3.2] using the same recipe of Section 3.5. Since AlbX is semi-abelian, it is a
homotopy invariant étale sheaf with transfers. Thus, taking sections over any field L ⊃ k,we have a factorization of the map (3.18.1) through1hA
0 (XL ) → AlbX (L),1

where hA
0 (XL ) denotes the zeroth Suslin homology group of XL = X ⊗k L. If X admits a(0)(0)
k-point, the scheme AlbX decomposes as Z × AlbX , where AlbX denotes Serre’s semiabelian Albanese variety of X. In particular, we get for any L ⊃ k algebraically closed an
induced (surjective) map on the degree zero part(3.18.2)

1

(0)

0
hA
0 (XL ) → AlbXL (L) = AlbXL (L).

DERIVED LOG ALBANESE SHEAVES

21

Let X ∈ Comp(X). By e.g. [BCKS17, Proposition 2.6], there is a natural surjection
1
CH0 (X) → hA
0 (X), which can be composed with (3.18.2) to give a surjective homomorphism
CH0 (XL )0 → AlbXL (L).
(1)

(0)

By Definition 3.13, we have then that the object (albX , AlbX , AlbX ) belongs to MCH
X .
CH,(0)
By Proposition 3.14, the universal property of AlbX
gives a unique surjection
CH,(0)

AlbX
CH,(1)(0)→ AlbX(1)(and similarly for AlbX
and AlbX ), which factors through the semi-abelian quotient
CH,(0)
CH,(0)AlbSAb,X of AlbX
. It is straightforward to show that if X has a k-rational point, theCH,(0)(0)algebraic groups AlbSAb,X and AlbX are isomorphic. We have therefore the following
Proposition 3.19. Let X be as above, and suppose that X has a k-rational point. Then
CH,(0)
the semi-abelian part of AlbX
agrees with Serre’s semi-abelian Albanese variety of X.
3.7. Universal problem for presheaves with transfers. We continue with the notations of 3.5. As observed in Remark 2.9 (ii), for any F ∈ RSC and for every section
g : Ztr (X) → F , there exists X ∈ Comp(X) such that g factors through ω! h
0 (X). In thiscase, we say that g has modulus X. We apply this to the case AlbCH.
X
Proposition 3.20. Let X ∈ Sm(k) be geometrically connected and X ∈ Comp(X). Then
the canonical map aX : Ztr (X) → AlbCH
X factors through ω! h0 (X) and it is universal withrespect to this property: for any smooth commutative k-group scheme G, seen as étale
reciprocity sheaf, and for any section g : Ztr (X) → G with modulus X, there is a uniquemorphism g̃ : AlbCH
X → G in PST such thatg

Ztr (X)

G

aX

g̃

AlbCH
X .
Proof. We first prove that aX : Ztr (X) → AlbX factors through ω! h
0 (X). We have toshow that for any smooth k-scheme S, the map Ztr (X)(S) → AlbX (S) factors throughω! h 
0 (X)(S). Since AlbX , as any commutative k-group scheme, satisfies global injectivity,
it is enough to check the factorization after passing to the function field k(S) of S, and infact even to its algebraic closure. Let then K ⊃ k be an algebraically closed field, and lookat the map Ztr (X)(K) → AlbXK (K). Let π0 (X) be the spectrum of the integral closureof k in Γ(X, OX ). The assignment X 7→ π0 (X) is universal for morphisms from X intoétale k-schemes. Since X is geometrically integral by assumption, we have Ztr (π0 (X)) = Z(as étale sheaves), and the map Ztr (XK ) → AlbXK induces then a map(0)

Ztr (XK )0 → AlbXK ,where Ztr (XK )0 denotes the kernel of Ztr (XK ) → Ztr (π0 (XK )). We can then identify
Ztr (XK )0 (K) with the group of 0-cycles of degree zero Z0 (XK )0 of XK . Since X is aproper modulus pair, we have CH0 (XK )0 = h0 (XK )0 (K) thanks to [KSY21, Remark
2.2.3], and thus the claim follows from Proposition 3.14. The same argument proves theuniversal property as well.

From now on, we write simply AlbX for the k-group scheme AlbCH
X . We end this section
with a the following result, which will be crucial for the construction of the category of
1-reciprocity sheaves.

22

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

Lemma 3.21. Under the assumptions of Proposition 3.20, the map
aX ⊗Z Q : Qtr (X) → AlbX ⊗Z Q
is a surjective morphism of étale sheaves with transfers with rational coefficients.
Proof. Let Im(aX ) ⊂ AlbX be the image of aX in PST. Since AlbX is a smooth k-groupscheme, we have AlbX ∈ RSCNis by [KSYR16, Theorem 4.1.1]. Let C = AlbX /Im(aX ) ∈V (C ⊗ Q) = 0. By (2.9.1), we have that aV (C ⊗ Q) ∈RSC. We have to show that aét
Z
Z
ét
RSCét (k, Q), in particular it satisfies global injectivity by [Sai20, Theorem 0.2], i.e. for
any Y ∈ Sm geometrically connected with function field k(Y ) with algebraic closure k(Y )there is an injective map
VV
aét(C ⊗Z Q)(Y ) ,→ aét
(C ⊗Z Q)(k(Y )) ,→ (C ⊗Z Q)(k(Y )) = C(k(Y )) ⊗Z Q.

To complete the proof, it is then enough to show that C(k(Y )) = 0. This follows fromthe fact that for any K ⊃ k algebraically closed, the map
Ztr (X ⊗k K)(K) → AlbX (K)(0)

is surjective, since CH0 (XK )0 → AlbX (K) is surjective.



4. The Albanese functors
4.1. n-reciprocity sheaves. For any n ≥ 0, let Cor(k)≤n be the category of finite correspondences on smooth k-schemes of dimension ≤ n, and let MCor(k)≤n be the category of modulus correspondences on smooth proper modulus pairs X = (X, X∞ ) withdim(X) ≤ n. We let MPST(k≤n ) be the category of additive presheaves on MCor(k)≤n .
The natural inclusions of subcategory of objects of dimension ≤ n give rise to a standard
string of adjoint functors between the category of presheaves(σn,! , σn∗ ),(4.0.1)

σn,! : MPST(k≤n )  MPST (k) : σn∗ .

Here, we follow the convention of [KMSY21a] for the left Kan extension of the restriction
functor σn∗ . Note that this is different from the one adopted in [ABV09].
Remark 4.1. Let(4.1.1)V, σnV,∗ ),(σn,!

V: PST(k≤n )  PST(k) : σnV,∗ )(σn,!

be the analogous adjoint functors from [ABV09]. Since the functor ω clearly restricts toa functor MCor(k)≤n → Cor(k)≤n , for all F ∈ PST(k) we have thatσn∗ ω ∗ F ∼= ω ∗ σnV,∗ F in MPST(k≤n ).
By adjunction, we conclude that for all F ∈ MPST(k≤n ),V
σn,!
ω! F ∼= ω! σn,! F in PST(k).

For F ∈ MPST(k) and X ∈ Cor(k)≤n , for any modulus pair X ∈ Comp(X), we have
X ∈ MCor(k)≤n , hence σn∗ F (X) = F (X)ω! σn∗ F (X) =lim−→σn∗ F (X) = ω! F (X) = σnV,∗ (ω! F )(X).

X∈Comp(X)Finally, for every modulus pair X and every α : Y → ω(X) ∈ Cor(k), by [KMSY21b,
Theorem 1.6.2] there exists a proper modulus pair Y0 ∈ Comp(Y ) and α0 : Y0 → X ∈MCor(k) such that α = ω(α0 ). In particular, since Y ∈ MCor(k)≤n , the system {F (Y )}DERIVED LOG ALBANESE SHEAVES

23

for Y ∈ Cor(k)≤n running over the maps Y → ω(X) is cofinal in the system {F (ω(Y))}
for Y ∈ MCor(k)≤n running over the maps Y → X. Hence we have thatVV
ω ∗ σn,!
F (X) = σn,!
F (ω(X)) =lim−→

F (Y )lim−→
0

F (ω(Y0 ))(Y →ω(X))
Y ∈Cor(k)≤n=(Y →X)Y ∈Cor(k)≤n

=

lim
−→

F (ω(Y)) = σn,! ω ∗ F (X).

(Y→X)
Y∈MCor(k)≤nRemark 4.2. The functors σn,! commute with colimits of presheaves since they are left adjoint. The functor σn∗ also commutes with colimits of presheaves, since they are computed
section-wise and σn∗ F (X) = F (X) for all X ∈ MCor(k)≤n . In particular for all diagrams
lim σ σ ∗ F{Fi } in MPST, σn,! σn∗ lim Fi ∼−→ = −→ n,! n i
The following lemma is mutuated from [ABV09, Lemma 1.1.12]:∼

Lemma 4.3. The unit map Id −→ σn∗ σn,! is invertible.
Definition 4.4. We say that F ∈ MPST is n-generated (resp. strongly n-generated ) if
the counit map σn,! σn∗ F → F is surjective (resp. an isomorphism).
Remark 4.5. If F is n-generated (resp. strongly n-generated ), then ω! F is n-generated(resp. strongly n-generated) in the sense of [ABV09]. Indeed, the functor ω! is exact andV σ ∗,V ω F by Remark 4.1.
ω! σn,! σn∗ F = σn,!
n!
For example, if X = (X, X∞ ) with dim(X) ≤ n, then Ztr (X) is strongly n-generated.
The proof of the following Lemma is a diagram chase.
Lemma 4.6. Quotients and extensions of (strongly) n-generated sheaves are again (strongly)
n-generated.
Definition 4.7. Let F ∈ CI. Following [ABV09, Definition 1.1.20], we say that F is an
n-modulus presheaf if the natural map
V
∗
rec
hrec
0,ét (σn,! σn F ) → h0,ét F = aét ωCI F

is an isomorphism of étale sheaves with transfers. Here, for any G ∈ MPST, we detr
VVnote by hrec
0,ét (G) the étale sheaf with transfers aét ωCI h0 G, where aét : PST → Shvét is
Voevodksy’s étale sheafification functor and ωCI is the composition ω! ◦ i , where i isthe inclusion CI → MPST, which has a right adjoint. Notice that the functor hrec
0,ét is acomposition of left adjoints, hence it commutes with all colimits.
We write CI≤n for the full subcategory of n-modulus presheaves.
The following is identical to [ABV09, Remark 1.1.21]:
Lemma 4.8. Let F ∈ MPST be strongly n-generated. Then h
0 (F ) is an n-modulus
sheaf.
Remark 4.9. Notice that in this case, differently from [ABV09, Remark 1.1.21], if F is an
n-modulus presheaf then it is not automatic that F is the h
0 of a strongly n-generated
sheaf: we only know that the map∗h
0 σn,! σn F → FVω .
is an isomorphism after applying aét
CI

Definition 4.10. We define the category of n-reciprocity sheaves RSCét,≤n as the essenVω .
tial image of CI≤n via the functor aét
CI

24

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

The following result is immediate:tr. The following are equivalent
Lemma 4.11. Let F ∈ Shvét(i) F is an n-reciprocity sheaf;∗(ii) F ∼= hrec0,ét σn,! σn G for some G ∈ MPST.

Moreover, if the above conditions hold, we can take G ∈ CI≤n in (ii).
Remark 4.12. If F ∈ RSCét,≤n , then F is an n-generated étale sheaf in the sense oftr∗[ABV09]. Indeed, for F = hrec
0,ét σn,! σn G, then there is a surjective map in Shvét
V V V,∗
V
V
∗
∗
∼ rec
aét
σn,! σn (F ) = aét
ω! σn,! σn∗ G  aétωCI h
0,ét σn,! σn G = h0,ét σn,! σn G = F.

Recall that the category F ∈ HIét,≤n of [ABV09, Definition 1.2.20], is the full subcateV hA1 σ V σ V,∗ F → aV hA1 F = F is an isomorphism.
gory of HIét of objects F such that aét
0
n,! n
ét 0
Proposition 4.13. If F ∈ HIét,≤n , then F ∈ RSCét,≤n .
Proof. Take F ∈ HIét,≤n . By Remark 4.1 we have∗ ∗rec ∗ V V,∗
hrec0,ét σn,! σn ω F = h0,ét ω σn,! σn F.

(4.13.1)

1

∗
∗ A
∗
Notice that h
0 ω = ω h0 , so by full faithfulness of ω we conclude that
1

1

V A V V,∗V
∗ A V V,∗∗ V V,∗V ∗ V V,∗hrec
0,ét ω σn,! σn F = aét ω! h0 ω σn,! σn F = aét ω! ω h0 σn,! σn F = aét h0 σn,! σn F = F.

In view of (4.13.1) and Lemma 4.11, this implies F ∈ RSCét,≤n .



The following lemma is analogue to [ABV09, Lemma 1.1.23]
Lemma 4.14. For any G ∈ CI, the natural map∗
V,∗ V
σnV,∗ hrec
0,ét (σn,! σn G) → σn aét ωCI G(4.14.1)

tr
induced by the counit map σn,! σn∗ G → G is an isomorphism in Shvét(k≤n )

Corollary 4.15 (cfr. [ABV09, Corollary 1.1.26]). Let F ∈ RSC such that F = ωCI G
with G ∈ CI and consider the natural map
∗
V
V

∗
V

V
hrec
0,ét (σn,! σn G) = aét ωCI h0 σn,! σn G → aét ωCI h0 G = aét ωCI G = aét F

induced by the counit map σn,! σn∗ G → G. Let N be the kernel of the above map. If N is
an n-generated étale sheaf in the sense of [ABV09], then it is zero.
Proof. Since the functor σnV,∗ is exact, we have by the definition of N an exact sequenceof étale sheaves:∗V,∗ V
0 → σnV,∗ (N ) → σnV,∗ hrec
0,ét (σn,! σn G) → σn aét F,
hence by Lemma 4.14 we conclude σnV,∗ (N ) = 0. Since N is n-generated, we have a
V σ V,∗ (N ) → N , showing that N = 0. 
surjective map of étale sheaves with transfers σn,!
n4.2. 0-reciprocity sheaves. We specialize the general results of the previous section to
the case n = 0. By definition, the objects of the category MCor≤0 of smooth modulus
pairs of dimension ≤ 0 are the finite étale extensions ` ⊃ k (with empty modulus divisor).
The essential image of the restriction of the functor ω : MCor → Cor to MCor≤0 induces
an equivalence of categoriesω|≤0 : MCor≤0 ' Cor≤0whose inverse is given by
λ|≤0 : Spec(`) 7→ (Spec(`), ∅).
and induces an equivalence of categories(4.15.1)

ω|≤0,! : MPST(k≤0 ) ' PST(k≤0 ) : λ|≤0,! .

DERIVED LOG ALBANESE SHEAVES

25

Moreover, for any X ∈ MCor with X = ω(X), we haveMCor(X, (Spec(`), ∅)) ∼= Cor(ω(X), Spec(`)) ∼= Cor(π0 (X), Spec(`)) ∼= Zπ0 (|X⊗k `|) ,where π0 (X) is the spectrum of the integral closure of k in Γ(X, OX ) and π0 (|Y |) for ascheme Y denotes the set of connected components of the underlying topological space|Y |. The first isomorphism follows from the fact that Spec(`) ∈ Corprop (see [KMSY21a,
Lemma 1.5.1]), while the second and the third are classical (see [MVW06, Lecture 1] and[ABV09, 1.2.1]). From this we get an adjunction(4.15.2)λ|≤0 ◦ π0 ◦ ω : MCor  MCor≤0 : σ0 .

We let Π0 denote λ|≤0 ◦ π0 ◦ ω. Passing to the categories of presheaves, we have that thefunctor σ0,! of (4.0.1) has a left adjoint:(4.15.3)

Π0,! : MPST(k)  MPST(k≤0 ) : Π∗0 = σ0,! .

In particular, σ0,! is given explicitly by(4.15.4)

σ0,! (F )(X) = F (Π0 (X)) = F (π0 (ω(X)), ∅) for F ∈ MPST(k≤0 ).

The following Corollary shows that the category of 0-reciprocity sheaves is simply equivalent to the category of 0-motivic sheaves in the sense of Ayoub–Barbieri-Viale.
Corollary 4.16. Let G ∈ MPST(k), then σ0,! σ0∗ G ∈ CI. In particular, every 0reciprocity sheaf is strongly 0-generated in the sense of [ABV09], and we have equivalences(4.16.1)tr(k≤0 ).
HIét,≤0 ' RSCét,≤0 ' Shvét

Proof. Let X ∈ MCor and X = ω(X). By(4.15.4), we have that
σ0,! σ0∗ G(X ⊗ P1 ) = σ0∗ G(π0 (ω(X ⊗ P1 )), ∅).
On the other hand, we have that ω(X ⊗ P1 ) = X × A1 and as observed in the proof of[ABV09, Lemma 1.2.2], we have that π0 (X × A1 ) = π0 (X), hence
σ0,! σ0∗ G(X ⊗ P1 ) = σ0∗ G(π0 (ω(X ⊗ P1 )), ∅) = σ0∗ G(π0 (ω(X)), ∅) = σ0,! σ0∗ G(X).
∗ 0Hence σ0,! σ0∗ G ∈ CI proving the first assertion. Let F ∈ RSCét,≤0 and let F = hrec
0,ét σ0,! σ0 G
0∗0with G ∈ CI(k) be as in Lemma 4.11(ii). Then σ0,! σ0 G ∈ CI, so
∗ 0
V
∗ 0
V V V,∗0V
ωCI h
F = aét0 σ0,! σ0 G ' aét ω! σ0,! σ0 G = aét σ0,! σ0 ω! G .
tr
This implies that F ∈ Shvét(k≤0 ). On the other hand, by Proposition 4.13 we haveHIét,≤0

RSCét,≤0tr
Shvét(k≤0 ).

By [ABV09, Lemma 1.2.2], the composition above is an equivalence, hence we deduce theequivalences of (4.16.1).

4.3. 1-reciprocity sheaves and Albanese functors. In this subsection, we will assume
that k has characteristic zero. In particular, k satisfies resolutions of singularities and forΩany smooth proper modulus pair, we can identify AlbCH
X with AlbX , and we simply write
AlbX for the Albanese scheme of X.
Remark 4.17. If C = (C, C∞ ) denotes a 1-dimensional smooth and proper modulus pair,by Lemma 4.8, hrec
0,ét (C) is a 1-reciprocity sheaf. Moreover, as observed in (2.9.2), we have:
V

V
hrec
0,ét (C) = aét ωCI h0 (Ztr (C)) = aét h0 (C) = Pic(C, C∞ ),where Pic(C, C∞ ) denotes the relative Picard group scheme, whose connected component of the identity Pic0 (C, C∞ ) agrees with the Rosenlicht-Serre generalized Jacobian
Jac(C, C∞ ). By Proposition 3.20 and the proof of Proposition 3.14, we have Pic(C, C∞ ) =AlbC . Hence we finally have that hrec
0,ét (C) = AlbC is represented by a commutative groupscheme.

26

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

Remark 4.18. More generally, let X be a smooth and proper modulus pair. Then the sheaf
AlbX is 1-generated. In fact, AlbX can be written as extension of a semi-abelian k-groupscheme (i.e. a k-group scheme G such that G0 is a semi-abelian variety) and a unipotent
algebraic group U . In characteristic 0, the group U is a product of Ga , and Ga is a direct1summand of hrec
0,ét (P , 2∞), so that it is 1-generated. The semi-abelian k-group scheme
G is a quotient of the generalized Jacobian of a suitable curve contained in G, which is
1-generated by the previous remark (see [ABV09, 1.3]). Hence G itself is 1-generated by[ABV09, Lemma 1.1.15]. Applying again [ABV09, Lemma 1.1.15] to AlbX we get thestatement.
We deduce from the previous remarks the following analogue to [ABV09, Lemma 1.3.4].
Lemma 4.19. Let F ∈ RSCét,≤1 . Then any subsheaf of F is a 1-generated étale sheaf inthe sense of [ABV09].
Proof. We essentially follow the steps in the proof of [ABV09, Lemma 1.3.4], starting fromthe case of F = hrec
0,ét (C), for C a smooth and proper modulus pair of dimension 1. Thisis a 1-reciprocity sheaf by Remark 4.17. Let E ⊂ F be a subsheaf. Since colimits of
1-generated étale sheaves are 1-generated étale sheaves, (see 4.2), we can assume that E
is the image of a map a : Ztr (X) → hrec
0,ét (C), for X ∈ Sm. By Remark 4.17, we have thena map
a : Ztr (X) → AlbC ∼= hrec0,ét (C).
Since AlbC ∈ RSC, by Remark 2.9 there exists a smooth proper modulus pair X with
X = X◦ such that a factors through h0 (X), and by Proposition 3.20, it uniquely factors
through aX : h0 (X) → AlbX :h0 (X)
aX

a



a0

$

/ AlbC .
AlbX
By Lemma 3.21, the motivic Albanese map aX is a surjective morphism of étale sheaves,hence the image of a agrees with the image of a0 . Thus E = Im(a0 ) is a 1-generated étale
sheaf by [ABV09, Lemma 1.1.15] and Remark 4.18.
∗
Now the general case. Let E ⊂ F ∼= hrec0,ét (σ1,! σ1 G) with G ∈ MPST (cf. Lemma∗
4.11(ii)). The sheaf σ1 G ∈ MPST(k≤1 ) can be written as colimit of representable sheaves
in MPST(k≤1 ), i.e.

σ1∗ G = lim Ztr (C)≤1 ,−→∗

C = (C, C∞ ),dim(C) = 1.

C→σ1 G

Since σ1,! and hrec
0,ét commute with colimits (being left adjoints), we have then∼ hrec (σ1,! (σ ∗ G)) = lim hrec (σ1,! Ztr (C)≤1 ) = lim hrec (C).
(4.19.1)
F =
1
0,ét
−→ 0,ét−→ 0,ét∗rec
For any C ∈ (C → σ1 G)≤1 , we have in particular a map h0,ét (C) → F , and thus a map
from the fiber productlim (hrec
0,ét (C) ×F E) → E−→
C→σ1,∗ F

which is surjective (the proof of surjectivity is formal and identical to the corresponding
statement in the proof of [ABV09, Lemma 1.3.4]). Now it is enough to notice that eachrechrec
0,ét (C) ×F E ⊂ h0,ét (C) is a 1-generated étale sheaf by the previous step and the fact that
1-generated étale sheaves are stable by colimits. To conclude we apply again [ABV09,
Lemma 1.1.15].
tr
Proposition 4.20. Let F ∈ Shvét(k, Q) be an étale sheaf of Q-vector spaces which is
1-generated in the sense of [ABV09]. If it is a reciprocity sheaf, then it is a 1-reciprocity
sheaf. In particular, any subsheaf of a 1-reciprocity sheaf of Q-vector spaces is again

DERIVED LOG ALBANESE SHEAVES

27

a 1-reciprocity sheaf and the category RSCét,≤1 (k, Q) is closed under taking subobjects,
colimits and extensions in RSCét (k, Q).
Proof. By Lemma 4.19, any subsheaf of a 1-reciprocity sheaf is again 1-generated, andby [KSY21, Corollary 2.4.2] any subsheaf of a reciprocity sheaf is a reciprocity sheaf.
Then the second part of the Proposition follows from the first, since RSCét (k, Q) is antr(k, Q) (here we are using the fact that weabelian category stable by colimits in Shvétconsider Q-coefficients in order to exploit Proposition 2.1, since [KSY21, Corollary 2.4.2]
is a statement about the Nisnevich sheafification) and 1-generated étale sheaves are stableby colimits and extensions by [ABV09, Lemma 1.1.15].
tr(k, Q) be a 1-generated étale sheaf
We now prove the first assertion. Let F ∈ Shvétof Q-vector spaces and suppose that F ∈ RSC, i.e. that there exists G ∈ CI such thatF = ω! G. By Remark 4.1, we have that
V V,∗
σ1,!
σ1 F = ω! σ1,! σ1∗ G
V σ V,∗ F → F is the image via ω of the counit σ σ ∗ G → G. Since
and the counit σ1,!
!
1,! 1
1
∗
G ∈ CI, the map σ1,! σ1∗ G → G factors through h
0 σ1,! σ1 G, which induces a factorizationV ω σ σ∗Gaét
! 1,! 1V σ V σ V,∗ F
aét
1,! 1

(∗)

F

∗
hrec
0,ét (σ1,! σ1 G).

Since F is a 1-generated étale sheaf, the map (∗) is surjective, hence the induced map
∗
∗
rec
hrec
0,ét (σ1,! σ1 G) → F is surjective. Let N = ker(h0,ét (σ1,! σ1 G) → F ). By Lemma 4.11,∗
hrec
0,ét (σ1,! σ1 G) ∈ RSCét,≤1 so that N is a 1-generated étale sheaf by Lemma 4.19. Hence∗∼
N = 0 by Corollary 4.15 so that hrec
0,ét (σ1,! σ1 G) = F , which concludes the proof by Lemma
4.11.

Recall that RSCét (k, Q) is a Grothendieck abelian category by [KSY21, Corollary 2.4.2].
The following corollary is immediate from the previous proposition.
Corollary 4.21. The inclusion RSCét,≤1 (k, Q) ⊂ RSC(k, Q) (and consequentlytr
RSCét,≤1 (k, Q) ⊆ Shvét(k, Q)) is exact, and the category RSCét,≤1 (k, Q) is a Grothendieckabelian category (in particular, it has enough injectives).
The proof of the following Lemma is identical to the proof of [ABV09, Lemma 1.3.6],
using Lemma 4.19.
Lemma 4.22. Let G ∈ G ∗ be a smooth commutative k-group scheme and let F be an étalesubsheaf of G with transfers such that its sheaf of connected components π0 (F ) is zero.
Then F is represented by a closed subgroup of G.
Definition 4.23. A 1-reciprocity sheaf F ∈ RSCét,≤1 (k, Q) is called finitely generated if
there exists a commutative k-group scheme G ∈ G ∗ with π0 (G) finitely generated together
with a surjection q : G → F . If the kernel of q is itself finitely generated, we say that F isfinitely presented.
?
We write RSCét,≤1⊂ RSCét,≤1 for the full subcategory of finitely presented 1-reciprocity
sheaves. An almost word-by-word translation of [ABV09, Proposition 1.3.8], using Lemma
4.19 and Lemma 4.22 gives the following canonical presentation of every finitely presented
1-reciprocity sheaf. This result will be repeatedly used in the rest of the paper.

Proposition 4.24. Any 1-reciprocity sheaf is filtered colimit of finitely presented 1-reciprocity
sheaves. If F is a finitely presented 1-reciprocity sheaf, then there is a unique and functorial
exact sequence
0→L→G→F →028

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

tr(k≤0 )
where G ∈ G ∗ is a smooth commutative k-group scheme and L ∈ RSCét,≤0 (k) ∼= Shvét
is a torsion free finitely generated 0-motivic sheaf.

Remark 4.25. For a smooth commutative k-group scheme G ∈ G ∗ , write Gsab for its semiabelian quotient: it is a smooth commutative k-group scheme such that the connectedcomponent of the identity (Gsab )0 is a semi-abelian variety. We have by Chevalley’s
theorem an extension
0 → U → G → Gsab → 0(4.25.1)

where U is a unipotent group. Since k is of characteristic zero, U ∼= Gra where r is theunipotent rank of G. More generally, for any F finitely presented 1-motivic sheaf, by
Proposition 4.24 we have a functorial commutative diagram
0

L

G

F

0

Gsab

FA

ϕ

(4.25.2)

1

1

0

1

sab ) is unipowhere by Remark 4.25, F A is isomorphic to hA0,ét (F ), and since ker(G → G
1

1tent, by [MVW06, Example 2.23] we also conclude that C∗A (F ) ' hA0,ét (F )[0], where
1

1C∗A (F ) is the Suslin complex. Moreover, by the right exactness of hA0,ét , the kernel of the1

map Gsab → F A is a quotient of the lattice L, hence it is itself a lattice.
Remark 4.26. Every G ∈ G ∗ is a compact object in RSCét , since the string of forgetfultr→ PST → PSh(Sm(k), Set) preserves filtered colimits andfunctors RSCét ⊆ Shvétgroup schemes are compact in PSh(Sm(k), Set). In particular, since compact objects are?
stable by finite colimits, every object of RSCét,≤1
is compact in RSCét .
We can now prove the following generalization to [ABV09, Proposition 1.3.11].
tr
Theorem 4.27. Let ch(k) = 0. The embedding RSCét,≤1 (k, Q) ⊆ Shvét(k, Q) has a
pro-left adjoint:tr(k) → pro- RSCét,≤1 .
Alb : Shvét
induced by colimit fromQtr (X) 7→ “ lim ”AlbX(n)←−nfor any choice of X ∈ Comp(X) smooth.
Proof. For X ∈ Sm(k), recall from Definition 2.8 the cofiltered category of Cartiercompactifications Comp(X). It is enough to show that for any X, for any choice of
X ∈ Comp(X) with total space smooth (since ch(k) = 0, such choice exists), and for any
E ∈ RSCét,≤1 (k, Q), we havelim HomShvtr (k) (AlbX(n) , E) = HomShvtr (k) (Qtr (X), E).
ét
ét
−→
n

By Galois descent it is also enough to prove the claim for k algebraically closed. By Lemma
3.21, for any X(n) the Albanese map aX : Qtr (X) → AlbX(n) is a surjective morphism oftrShvét(k, Q) (here we are considering AlbX(n) after tensoring with Q), hence since filteredcolimits are exact, we only need to show thatlim HomShvtr (k) (AlbX(n) , E) → HomShvtr (k) (Qtr (X), E)
ét
ét
−→
n

is surjective, i.e. that for any choice of X ∈ Comp(X), every map Qtr (X) → E factors
trthrough AlbX(n) for some n. Since Qtr (X) is compact in Shvét(k), by Proposition 4.24
we can suppose that E is finitely presented. Let E = coker(L ,→ G) be as in Proposition
4.24. Then we get a long exact sequence

DERIVED LOG ALBANESE SHEAVES

29

1. . . → Hom(Qtr (X), G) → Hom(Qtr (X), E) → Hét
(X, L) → . . . .
1 (X, L) = H 1 (X, Qr ) = 0. Thus the map
Since L ∼= Qr , being k separably closed, Hét
Nis

Hom(Qtr (X), G) → Hom(Qtr (X), E)
is surjective, i.e. every map s : Qtr (X) → E factors through G. Since G ∈ RSCét (k, Q),
by Remark 2.9 (iii) for every s as above and for any choice of X ∈ Comp(X), there exists(n) ), and by Proposition 3.20, s has to factor throughn such that s factors through hrec
0,ét (X
AlbX(n) , concluding the proof.

4.4. The log Albanese functor. We now generalize this to the logarithmic setting. Let
CIωlog: RSCét (k, Q) → Shvdét (k, Q)
CI is fully faithful, exact
be the composition in (2.10.2). As observed in Remark 2.11, ωlog
and commutes with all colimits.

Theorem 4.28. Let ch(k) = 0. The fully faithful exact functor
CIωlog

RSCét,≤1 (k, Q) ⊆ RSCét (k, Q) −−→ Shvltr
dét (k, Q)
has a pro-left adjoint, called the log Albanese functor
Alblog : Shvltr
dét (k) → pro- RSCét,≤1 .
induced by colimit fromQtr (X) 7→ “ lim ”AlbX(n)←−nfor any choice of X ∈ Comp(X) smooth.
CI commutes with filtered colimits, we
Proof. We proceed as in the non-log case. Since ωlog
can reduce to prove the adjunction for maps against E finitely presented, quotient of asmooth commutative group k-scheme G by a lattice L. As before, it is enough to provethe claim with k algebraically closed, by Galois descent. Since we are considering sheavesof Q-vector spaces, we can assume that L ∼= Qr . For X ∈ SmlSm(k) we have
1
CI
1
Hdét
(X, ωlogL) ∼(X − |∂X|, Qr ) = 0.
= HétSo following the steps of the previous proof, it is enough to show that for any G commutative group scheme and X ∈ SmlSm(k) any map
CIs : Qltr (X) → ωlog
G
CI AlbCI
factors through ωlog
X(n) for some n, and conclude by full faithfulness of ωlog . On theother hand, since G ∈ RSCNis , by [BM21, Lemma 6.6 and Theorem 7.6] we have thatCI(n)
HomShvltr (Qltr (X), ωlogG) = lim HomRSCét (hrec), G).
0,ét (X−→détn

CI is fully faithful, s factors through ω CI hrec (X(n) ) for some n, hence again bySince ωlog
log 0,ét
CI , it factors through ω CI Alb
Proposition 3.20 and full faithfulness of ωlog
X(n) , proving thelog
claim.
We finish this section by proving some results on extensions in RSCét,≤1 (k, Q):
Proposition 4.29. The category RSCét,≤1 (k, Q) is closed under extensions in Shvét (k, Q).

30

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITOProof. Let F1 , F2 ∈ RSCét,≤1 . Since RSCét,≤1 (k, Q) is a full subcategory of Shvét (k, Q)
by Proposition 2.10, we have that an extension in RSCét,≤1 (k, Q) splits in Shvét (k, Q) if
and only if it splits in RSCét,≤1 (k, Q), so
Ext1RSCét,≤1 (k,Q) (F1 , F2 ) ,→ Ext1Shvét (k,Q) (F1 , F2 ).
?
. In particular, there
By Proposition 4.24, we can write F1 = lim Fi1 with Fi1 ∈ RSCét,≤1−→
is a surjective map ⊕Fi1  F1 . Let K be its kernel: we have the following commutative
diagram with exact rows:(4.29.1)

HomRSCét,≤1 (K, F2 )Ext1RSCét,≤1 (F1 , F2 )'Q

Ext1RSCét,≤1 (Fi1 , F2 )

(∗)

(∗∗)

Ext1Shvét,≤1 (F1 , F2 )HomShvét,≤1 (K, F2 )Ext1RSCét,≤1 (K, F2 )Q

Ext1Shvét,≤1 (Fi1 , Fi2 )Ext1Shvét,≤1 (K, F2 )so by the five-lemma (∗) is surjective if (∗∗) is surjective, hence we can suppose F1 ∈tr?
and filtered colimits are. Since F1 is compact in both RSCét,≤1 and Shvét
RSCét,≤1?
exact, again by 4.24 it is enough to suppose that F2 ∈ RSCét,≤1 . Let Fi = coker(Li ,→ Gi )as in Proposition 4.24, we have a commutative square with exact rows:
Ext1RSCét,≤1 (G1 , G2 )Ext1RSCét,≤1 (G1 , F2 )(1)

Ext1Shvtr (G1 , G2 )ét

Ext2RSCét,≤1 (G1 , L2 )

(3)
(2)

Ext1Shvtr (G1 , F2 )Ext2Shvtr (k,Q) (G1 , L2 ).

ét

ét

By [Mil17, Exercise 5-10] we have that G ∗ is closed by extensions in fppf sheaves.

This
implies that (1) is an isomorphism. Moreover, Ext2Shvtr (k,Q) (G1 , L2 ) = 0 by [Mil70, Corolétlary 1.], which implies that (2) is surjective, so (3) is surjective, hence an isomorphism.
We have now the following commutative diagram with exact rows:
HomRSCét,≤1 (L1 , F2 )Ext1RSCét,≤1 (F1 , F2 )Ext1RSCét,≤1 (G1 , F2 )Ext1RSCét,≤1 (L1 , F2 )''HomShvét,≤1 (L1 , F2 )Ext1Shvét,≤1 (F1 , F2 )Ext1Shvét,≤1 (G1 , F2 )and we conclude using the 5-lemma.

Ext1Shvét,≤1 (L1 , F2 ).



Lemma 4.30. For all F1 , F2 ∈ RSCét,≤1 we have
ExtiShvét (k,Q) (F1 , F2 ) = 0

for i ≥ 2.

?
Proof. By the same argument of 4.29, we can suppose F1 , F2 ∈ RSCét,≤1(namely, the?
exact sequence of (4.29.1) allows to consider F1 ∈ RSCét,≤1
and since now F1 is acompact object in Shvét,≤1 (k, Q) and filtered colimits are exact, we can assume that?
F2 ∈ RSCét,≤1). Let Fi = coker(Li ,→ Gi ) as in Proposition 4.24. Since L1 is a lattice,i
Ext (L1 , F2 ) = 0 for i ≥ 1, so we can suppose F1 = G1 . Since by [Mil70, Cor. 1] we have

ExtiShvét (k,Q) (G1 , L2 ) = ExtiShvét (k,Q) (G1 , G2 ) = 0

for i ≥ 2,

we can conclude.

Proposition 4.31. For F1 , F2 ∈ RSCét,≤1 (k, Q), we have
ExtiRSCét,≤1 (k,Q) (F1 , F2 ) ' ExtiShvét (k,Q) (F1 , F2 ) for all i ≥ 0.
In particular the category RSCét,≤1 (k, Q) has cohomological dimension 1, i.e. for i ≥ 2
we have
ExtiRSCét,≤1 (k,Q) (F1 , F2 ) = 0.

DERIVED LOG ALBANESE SHEAVES

31

Proof. The second part follows from the first in light of Lemma 4.30. Let us proceed
by induction: the case i = 0 is Proposition 2.10 and i = 1 is Proposition 4.29. Leti ≥ 2: in light of Lemma 4.30, we need to show that ExtiRSCét,≤1 (k,Q) (F1 , F2 ) = 0. Let
0 → F2 → I → B → 0 with I injective in RSCét,≤1 (k, Q). We conclude since:
(3)

(2)

(1)
0 = ExtiShvét (k,Q) (F1 , B) ∼= Exti+1= ExtiRSCét,≤1 (k,Q) (F1 , B) ∼RSCét,≤1 (k,Q) (F1 , F2 )where (1) is Lemma 4.30, (2) is the induction hypothesis and (3) comes from the fact that
I is injective.

5. The derived Albanese functor
Assume that k has characteristic zero. The goal of this section is to show the existenceof a left derived Albanese functor in the sense of Definition A.10. To ease the notation,ltr
we will write Shvtr (resp. Shvltr , resp. logCI) for Shvtr
Nis (k, Q) (resp. ShvdNis (k, Q),
ltr
trresp. logCIdNis ) and we will identify it with Shvét (k, Q) (resp. Shvdét (k, Q), resp.
logCIdét ) by Proposition 2.1 (resp. Proposition 2.2). We also write RSCét,≤1 ⊂ RSCét
CI denote the functor fromfor RSCét,≤1 (k, Q) ⊂ RSCét (k, Q). In this section we let ωlog
Theorem 4.28:(5.0.1)

RSCét,≤1

Shvltr .

By Remark 2.11 and Proposition 4.20 this functor is fully faithful, exact and commuteswith all colimits.
We announce the main theorem (cfr. [ABV09, Thm. 2.4.1]), whose proof will occupy
the rest of the Section.
Theorem 5.1. The functor Alblog of Theorem 4.28 has a pro-left derived functor L Alblog
which factors through the stable ∞-category of effective log motives, giving rise to the log
motivic Albanese functor (still denoted L Alblog ):L Alblog : logDMeff (k, Q) → Pro-D(RSCét,≤1 ).
which is a pro-left adjoint of the functor
CI )D(ωlog

eff

L

logDM

logDMeff (k, Q).
ω≤1: D(RSCét,≤1 ) −−−−−→ D(Shvltr ) −−→

5.1. Some preliminary results. We collect now some technical lemmas that will be
used in the proof of the main theorem.
Recall the pair of adjoint functors from (2.2.1)∗ : Shvtr (k, Q)
ωlog
ét

log
Shvltrdét (k, Q) : ω]

∗ F (X) = F (X − |∂X|). Recall that ω log ω ∗ = id and that by [BPØ20, Proposiwhere ωlog
log]
∗ (F ) ∼ ω CI F .
tion 8.2.8], for F ∈ HIét we have that ωlog
= log

Lemma 5.2. For any finitely presented 1-reciprocity sheaf F ∈ RSCét,≤1 , we have an
isomorphism:
1
HomShvtr (Gm , F ) ∼= HomShvtr (Gm , F A ).
Moreover, we have that
1
CI
∗
ωlog
HomShvtr (Gm , F ) ∼HomShvtr (Gm , FiA ).
= ωloglog ∗Proof. First notice that Gm ∼Gm , and that HomShvtr (Gm , F A ) ∈ HI, hence= ω] ωlog
1

1
1
CI
∗
ωlog
HomShvtr (Gm , F A ) ∼HomShvtr (Gm , F A )= ωlog

32

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

so the second isomorphism follows from the first. To prove the first isomorphism, consider
the presentation 0 → L → G → F → 0 of Proposition 4.24. Since HomShvtr (Gm , Gra ) =Ext1Shvtr (Gm , Ga ) = 0, we conclude from the sequence (4.25.1) that
HomShvtr (Gm , G) = HomShvtr (Gm , Gsab ).
By [BVK16, Lemma 3.1.4] and Proposition 2.10, we have
HomShvtr (Gm , G) ∼= HomShvtr (Gm , F ),which allows to conclude.



Let X = (X, D) be a smooth proper and geometrically integral modulus pair, and let
AlbsabX be the maximal semi-abelian quotient of AlbX . Suppose that X = X \ D has
a k-rational point. Then, by Proposition 3.19, Albsab
X agrees with Serre’s semi-abelian
Albanese variety of X, AlbX . Let U (X) be as in Definition 3.5: as observed in [BS19,
Lemma 10.7] we have an extension(5.2.1)

0 → U (X) ⊗k Ga → AlbX → Albsab
X = AlbX → 0

Since AlbX ∈ HI≤1,ét , which is a Serre subcategory of RSC≤1,ét , the natural surjectionof (5.2.1) (with rational coefficients) gives then rise to a map(5.2.2)

Ext1HIét,≤1 (AlbX , Gm ) → Ext1RSCét,≤1 (AlbX , Gm ).

Lemma 5.3. The map (5.2.2) is an isomorphism.
Proof. We need to show that
HomRSCét,≤1 (U, Gm ) = 0,and

Ext1RSCét,≤1 (U, Gm ) = 0.

which by Proposition 4.31 follow form analogous vanishing in Shvét , which are well-known(see e.g. [Ser75, VII, Proposition 7]).

∗ G , G) ∈ logCI since
Notice that for every G ∈ logCI, we have that HomShvltr (ωlogm∗∗
Gm , G) = π0 (RHomShvltr (ωlog
Gm , G))
HomShvltr (ωlog∗ G , F ) is clearly -local.
and RHomShvltr (ωlogmLemma 5.4. For any finitely presented 1-reciprocity sheaf F ∈ RSCét,≤1 , we have an
isomorphism:
1 
'
∗
∗
ωlog
Gm ⊗logCI HomShvtr (ω ∗ Gm , F ) −→ ωlogGm ⊗HI HomShvtr (Gm , F A ) .
where ⊗logCI is the tensor product of the category logCI.
Proof. By Proposition 2.13, we have that∗CI
Gm , ωlogF) ∼ω]log HomShvltr (k,Q) (ωlog= HomShvtr (k,Q) (Gm , F ).
ét

ét

1

By Lemma 5.2 the right hand side is isomorphic to HomShvtr (k,Q) (Gm , F A ), and moreoverét
it is an object of HIét (k, Q), hence by (2.12.2) we have an isomorphism
1

∗
CI
∗
∗
CI
∗
HomShvltr (ωlog
Gm , ωlogF) ∼ω]log HomShvltr (ωlog
Gm , ωlogF) ∼HomShvtr (Gm , F A ),= ωlog= ωlogso we conclude since by Lemma 2.7 we have that
1
1 
∗
∗
∗
ωlog
Gm ⊗logCI ωlog
HomShvtr (Gm , F A ) ∼Gm ⊗HI HomShvtr (Gm , F A ) .
= ωlog

DERIVED LOG ALBANESE SHEAVES

33

Remark 5.5. Thanks to Proposition 4.24, we can write every F ∈ RSCét,≤1 (k, Q) as
filtered colimit of finitely presented 1-reciprocity sheaves, F = limi Fi . Since the functor−→
CI from (5.0.1) commutes with all colimits, we have ω CI (F ) = lim ω CI F . By theωlog
log−→i log i
isomorphism∗∼ coker(Q → hlog (P1 , [0] + [∞])),
ωlog
Gm =
0
given for example by [BPØ20, 8.2.4] together with [MVW06, Theorem 7.16], we see that
∗ G is a compact object in logCI, hence in Shvltr . By Lemma 5.4 we have an isomorωlog
m
phism(5.5.1)
1 
∗
∗
CI
∗
ωlog
Gm ⊗logCI HomShvltr (ωlog
Gm , ωlogF) ∼ωlogGm ⊗HI HomShvtr (Gm , FiA ) .
= lim
−→
i

Moreover, since ω]log commutes with (filtered) colimits, we have that
(5.5.2)

1 
∗
CI
∗
Gm , ωlog
F ) = lim Gm ⊗HI HomShvtr (Gm , FiA ) .
ω]log ωlog
Gm ⊗logCI HomShvltr (ωlog−→
In particular, since HIét is closed under colimits we have that for exery F ∈ RSCét,≤1 ,∗∗CIω]log ωlog
Gm ⊗logCI HomShvltr (ωlog
Gm , ωlogF ) ∈ HIét .
Lemma 5.6. Let F ∈ RSCét,≤1 (k, Q), then:(5.6.1)j∗CI∗
Gm ⊗logCI HomShvltr (ωlog(X, ωlogF )) = 0
HdNis
Gm , ωlog

for j > 1.

Proof. Recall that for a torus T , we have that
(5.6.2)jj(X ◦ , T ) = 0(X ◦ , T ) = HZar
HNis

for j > 1.

?
. Since dNis-cohomology comBy Proposition 4.24, let F = limi Fi with Fi ∈ RSCét,≤1−→
mutes with filtered colimits, the left hand side of (5.6.1) is isomorphic to1 j∗Gm ⊗HI HomShvtr (Gm , FiA ) ).
(X, ωlog
lim HdNis−→
1

The sheaf HomShvtr (Gm , FiA ) is a 0-motivic homotopy invariant sheaf by [ABV09, Corol1∗ is exact
lary 1.3.9], hence Ti := Gm ⊗HI HomShvtr (Gm , FiA ) is a torus. Then since ωlog
and preserves injective sheaves, we have by (5.6.2) thatjj∗
(X ◦ , Ti )) = 0
HdNis(X, ωlog
Ti ) = HNis

for j > 1.


For a smooth scheme X, let NSr (X) be the group of codimension r-cycles modulo
algebraic equivalence.
Lemma 5.7. Let F ∈ RSCét,≤1 and let X = (X, D) be a proper modulus pair such thatX ◦ = X − |D| is affine and NS1 (Xk◦ ) = 0. Then we have an isomorphism∗∗
CI
Ext1RSCét (AlbX , ω]log (ωlog
Gm ⊗logCI Hom(ωlog
Gm , ωlogF ))) '
1
∗
∗
CI
HdNis
(X, ωlog
Gm ⊗logCI Hom(ωlog
Gm , ωlog(F ))).
?
∗ G ⊗
∗
CI
Proof. Let F = lim Fi with Fi ∈ RSCét,≤1
and T = ωlog
m logCI Hom(ωlog Gm , ωlog F )
−→
∗ G ⊗
∗
CI
and Ti = ωlog
m
logCI Hom(ωlog Gm , ωlog Fi ). By Remark 4.26, AlbX is a compact
object in RSCét , and filtered colimits are exact and RSCét,≤1 ⊆ RSCét is closed under
extensions, hence by (5.5.2) we have that

Ext1RSCét (AlbX , ω]log T ) ∼Ext1RSCét,≤1 (AlbX , Ti ).
= lim
−→
i

34

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

On the other hand, by Lemma 5.4 we have
1
1
∗
HdNis
(X, T ) ∼H 1 (X, ωlog
(Gm ⊗HI HomShvtr (Gm , FiA ))).
= lim−→ dNisi∗
Moreover, since ωlog is exact and preserves injective sheaves we have that

1
1
1
∗
1
HdNis
(X, ωlog(Gm ⊗HI HomShvtr (Gm , FiA ))) ∼(X ◦ , Gm ⊗HI HomShvtr (Gm , FiA )).
= HNis1

Finally, HomShvtr (Gm , FiA ) is a lattice by [ABV09, Corollary 1.3.9], hence it is enoughto show that for every lattice L we have an isomorphism
Ext1(AlbX , Gm ⊗HI L) ∼= H 1 (X ◦ , Gm ⊗HI L).
Nis

RSCét,≤1

By [ABV09, Lemma 2.4.5], since X ◦ is affine and NS1 (Xk◦ ) = 0, we have that
1
HNis
(X ◦ , Gm ⊗HI L) ∼= Ext1HIét,≤1 (AlbX ◦ , Gm ⊗HI L),

so it is enough to show that the canonical map AlbX → AlbX ◦ induces an isomorphism
Ext1(AlbX ◦ , Gm ⊗HI L) ∼(AlbX , Gm ⊗HI L).
= Ext1
HIét,≤1

RSCét,≤1

If k is algebraically closed, we have L ∼= Qr , hence the above isomorphism comes from
Lemma 5.3. A Galois descent argument (see [ABV09, Lemma 2.4.5, Step 1]) allows us todeduce the isomorphism above for any k.

5.2. Deriving the Albanese functor. We are ready to prove Theorem 5.1. The cateCI fromgories RSCét,≤1 and Shvltr are Grothendieck abelian categories and the functor ωlog
(5.0.1) is exact and commutes with filtered colimits.
The derived ∞-category D(Shvltr ) is equivalent by classical reason to the ∞-category
underlying the model category Cpx(PShltr (k, Λ)) with the dNis-local model structure
considered in [BM21]. In particular, by [BM21, Lemma 2.15], the functor iltr : D(Shvltr ) →Chdg (Shvltr ) preserves filtered colimtis. In particular, the commutative square of ∞categories:
CI )Chdg (ωlog

Chdg (Shvltr )Chdg (RSCét,≤1 )

Lltr

L≤1

D(RSCét,≤1 )CI )D(ωlogD(Shvltr )CI ) has a pro-left adjoint L Alblog . We consatisfies the hypotheses of A.4, so that D(ωlogsider the BC-admissibility with respect to this diagram. Recall from Lemma A.14 that acompact object P ∈ Shvltr is BC-admissible as an object of Chdg (Shvltr ) if and only if
for every injective object I ∈ RSCét,≤1 we have
CI
ExtnShvltr (P, ωlog(I)) = 0 for i 6= 0.

We make the following definition (see [ABV09, Def. 2.4.2]):
Definition 5.8. X ∈ SmlSm(k) is Alblog -trivial if X ◦ is affine, NS1 (Xk◦ ) = 0 andj
HZar(X, OX ) = 0

for j > 0.

Remark 5.9. If X = Spec(R) is affine and ∂X is supported on a principal divisor with
global equation f , then X ◦ = Spec(R[ f1 ]) is affine, in particular if NS1 (Xk◦ ) = 0 we havethat X is Alblog -trivial and X ◦ satisfies the hypotheses of [ABV09, Proposition 2.4.4].
The main technical input of the proof of Theorem 5.1 is the following result:
Proposition 5.10. Let X ∈ SmlSm(k) be Alblog -trivial, then the complex Qltr (X)[0] is
BC-admissible.

DERIVED LOG ALBANESE SHEAVES

35

Proof. We follow (with some modifications) the path of the proof of [ABV09, Proposition
2.4.4]. Since Qltr (X)[0] is a compact object, it is enough to prove that
CI(I)) = 0,
ExtiShvltr (Qltr (X), ωlog

for i > 0 and for every I ∈ RSCét,≤1 injective.

By [BPØ20, Proposition 4.3.2] the Ext groups in Shvltr can be computed as cohomology
groups:CI
i
CI
ExtiShvltr (Qltr (X), ωlog(I)) = HdNis(X, ωlog(I)),
i
CI (I)) = 0 for i > 0. In order to control this cohoso we need to check that HdNis(X, ωlog
mology, we look then at the adjunction map(5.10.1)∗
CI
CI∗
Gm , ωlog(I)) → ωlog
(I)ωlog
Gm ⊗logCI HomShvltr (ωlog

We will make the following claim:
Claim 5.11. Let N (resp. Q) be the kernel and the cokernel (computed in Shvltr ) of themorphism (5.10.1). Then for j > 1 we have the following vanishing:(5.11.1)

jj
HdNis(X, N ) = HdNis(X, Q) = 0.

j
CI (I)) = 0 for j > 1
Combining Claim 5.11, and Lemma 5.6, we get that HdNis(X, ωlog
and that we have a surjection
1
∗
∗
CI
1
CI
(5.11.2) HdNis(X, ωlog
Gm ⊗logCI HomShvltr (ωlog
Gm , ωlog(I))) → HdNis(X, ωlog(I)) → 0.

We are then left to show that the displayed morphism in (5.11.2) is the zero map.
CI Alb gives
For every modulus pair X ∈ Comp(X), the canonical map Qtr (X) → ωlog
X
CI
for any F ∈ RSCét,≤1 a natural map (again we are using the fact that ωlog is exact):
CI
1
CI
CI
F ),
(X, ωlog
F ) → HdNis
AlbX , ωlog
Ext1RSCét,≤1 (AlbX , F ) → Ext1Shvltr (ωlog

hence, from (5.11.2) we get a commutative diagram (cfr. with the proof of [ABV09, 2.4.4])(5.11.3)∗∗CI
Gm ⊗logCI Hom(ωlog
Ext1RSCét,≤1 (AlbX , ω]log (ωlog
Gm , ωlog
I)))

Ext1RSCét,≤1 (AlbX , I)

1
∗
∗
CI
HdNis
(X, ωlog
Gm ⊗logCI Hom(ωlog
Gm , ωlog
(I)))

1
CI
HdNis
(X, ωlog(I)).

Since I is injective in RSCét,≤1 , the term Ext1RSCét,≤1 (AlbX , I) is zero. On the other
hand, the left vertical map is an isomorphism by Lemma 5.7, which implies that (5.11.2)
is indeed the zero map. This finishes the reduction of the proof of 5.10 to Claim 5.11.
Proof of Claim 5.11. Since cohomology commutes with filtered colimits, let I = lim Fi−→
with Fi finitely generated 1-reciprocity sheaves. For all i, let Ki and Ni be the kernel andthe cokernel of the adjunction map(5.11.4)

∗
∗
CI
CI
ωlog
Gm ⊗logCI HomShvltr (ωlog
Gm , ωlog
Fi ) → ωlog
Fi .

As observed in Remark 5.5, since filtered colimits are exact we have that K = lim Ki and−→
Q = lim Qi , hence it is enough to show that for all i:−→j(5.11.5)
HdNis(X, Ni ) = 0, j > 0(5.11.6)j
HdNis(X, Qi ) = 0,

j > 0.

∗ G ⊗By Lemma 5.4 we have that the left hand side of (5.11.4) is isomorphic to ωlog
m HI

1
A
HomShvtr (Gm , Fi ) . Let Ki and Ri be the kernel and the cokernel of the adjunctionmap
11ψi : Gm ⊗HI HomShvtr (Gm , FiA ) → FiA .

36

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

We have the following diagram:(5.11.7)
0

ϕi

∗
CI
∗
Gm , ωlog
Fi )ωlog
Gm ⊗logCI HomShvltr (ωlog

Ni

CI
ωlog
Fi

Qi

0

∗
ωlog
Ri

0.

'
1 
∗
ωlog
Gm ⊗HI HomShvtr (Gm , FiA )

∗
ωlog
Ki

0

ω ∗ ψi

∗
ωlog
FiA

1

By the cancellation theorem [Voe10], HomShvtr (Gm , Gm ⊗HI M ) ∼= M for M ∈ HI, and
HomShvtr (Gm , ) is exact as an endo-functor on HI. Hence we get HomShvtr (Gm , Ki ) =HomShvtr (Gm , Ri ) = 0, in particular the sheaves Ki and Ri are birational sheaves in∗ is exact andthe sense of [KS17] (see [KS17, Proposition 2.5.2]). In particular, since ωlog
preserves injectives, by [MVW06, Proposition 14.23] and [KS17, Proposition 2.3.3] that
(5.11.8)jj∗(X, ωlog
Ki ) = HNis
(X ◦ , Ki ) = 0,
HdNis

j>0

(5.11.9)

j
j
∗
HdNis
(X, ωlog
Ri ) = HNis
(X ◦ , Ri ) = 0,

j > 0.

∗ K , ω log N is a subsheaf of K , so it is a birational sheaf, inSince Ni is a subsheaf of ωlog
i
i
i
]
log
particular it is an object of HI by [KS17, Proposition 2.3.3 (a)] so Ni ∼= ω ∗ ω Ni bylog ](2.12.2). Therefore the same argument gives the vanishing (5.11.5).
1Let Hi be the kernel of the map Fi → FiA . By a snake lemma argument on (4.25.2),
CI , wethere exists a lattice L0i and ri ≥ 0 such that Grai /L0i ∼= Hi . By the exactness of ωlogj(X, ω ∗ L0 ) = H j (X ◦ , L0 ) = 0
have that ω CI Gri /ω ∗ L0 ∼= ω CI Hi . Since L0 is a lattice, H
log

a

dNis

Nis

for j > 0 and by [RS21, Corollary 6.8] with q = 0 combined with [BM21, (7.6.1)] we havethat (see (2.1.2)):(5.11.10)

jCIGa ) =(X, ωlog
HdNisj(Y , OY ).
lim HNis−→SmY ∈Xdiv

By the comparison of Zariski cohomology with Nisnevich cohomology for coherent sheaves
Sm :we have that for all Y ∈ Xdiv
jj
(Y , OY ) ∼HNis= HZar (Y , OY ).

By definition the map Y → X is the composition of blowups in smooth centers, hence thewell know blow-up formula (see e.g. [Gro85, Corollary IV.1.1.11]) implies:jj(Y , OY ) ∼HZar= HZar (X, OX ) = 0, j > 0,

where the last vanishing comes from the fact that X was taken Alblog -trivial. In particular,
we conclude that
(5.11.11)iCI(X, ωlog
Hi ) = 0 for i 6= 0.
HdNis

From the diagram (5.11.7) and a snake lemma argument, we get the following short
exact sequence:(5.11.12)

CI
0 → ω ∗ Ki /Ni → ωlog
Hi → ker(Qi → ω ∗ Ri ) → 0.

j
Now by (5.11.8) and (5.11.5) we have that HdNis(X, ω ∗ Ki /Ni ) = 0 for j > 0, so by(5.11.11), (5.11.9) and (5.11.12) we deduce (5.11.6).


Given Proposition 5.10, we can show the following
log
Lemma 5.12. The category Shvltr
dNis (k, Q) is generated by the set of Alb -trivial objects
ltrof ShvdNis (k, Q).

Proof. The category Shvltr
dNis (k, Q) is compactly generated, and a set of compact generators
is given by Qltr (X)[i], for X ∈ SmlSm and i ∈ Z. By Proposition 5.10, it is then enough toshow that any X ∈ SmlSm can be covered (even Zariski-locally) by Xi ∈ SmlSm which

DERIVED LOG ALBANESE SHEAVES

37

are Alblog -trivial. Let (Ui , fi ) be a Zariski cover of X such that |∂X||Ui is principal. By[ABV09, Corollary 2.4.6]), we can cover each Ui by affine Uij such that NS1 ((Uij )k ) = 0,
and since |∂X|Ui is principal, |∂X|Uij is again principal. Considering the log schemes
Uij := (Uij , ∂X|Uij ), we have that NS1 ((Uij )k ) → NS1 ((Uij◦ )k ) is surjective by [Ful98,
Example 10.3.4], hence NS1 (Uij◦ ) = 0. We conclude that {Uij } is a Zariski cover of X by
Alblog -trivial log schemes.
Proof of Theorem 5.1. From Lemma 5.12 and Proposition 5.10, we have that the ∞category D(Shvltr
dNis (k, Q))) is generated by a set of compact BC-admissible compact
objects concentrated in degree zero. The existence of the derived log Albanese functor L Alblog as pro-left derived functor of Alblog follows then from Theorem A.13, and by
CI ).
constrution it is equivalent to the pro-left adjoint of the functor D(ωlog
We are left to show that the functor L Alblog factors through the localizationeff
L : D(Shvltr
dNis (k, Q)) → logDM (k, Q).

Recall that logDMeff (k, Q) is obtained as localization of D(Shvltr
dNis (k, Q)) with respect
to the class of maps:
Qltr (X × )[n] → Qltr (X)[n](CI)

for X in SmlSm(k). From the proof of Lemma 5.12, we can suppose that X is Alblog trivial (this is the exact analogue of [ABV09, 2.4.1]). We are then left to show that
L Alblog (Qltr (X × ) → Qltr (X)) is contractible for X an Alblog -trivial object. Since
Qltr (X) is BC-admissible by Proposition 5.10, we have by Remark A.9 that
L Alblog (Qtr (X)[n]) = “ lim ”AlbX(i) [n],←−for any choice of X ∈ Comp(X).

iNote also that if X is Alblog -trivial, so is X × . Indeed, (X × )◦ = X ◦ × A1 is affine ifi (X × P1 , O∼ i
X ◦ is affine, NS1 ((X ◦ × A1 )k ) ∼= NS1 (Xk◦ ) and HZar
X×P1 ) = HZar (X, OX ) for
all i. Therefore
L Alblog (Qltr (X × )[n]) = “ lim ”AlbX(i) × ,←−for any choice of X ∈ Comp(X).

i

On the other hand, by construction we have
AlbX(i) × ∼= AlbX(i) ,proving the factorization. The pro-adjunction now is formal since for X ∈ D(RSCét,≤1 ),CI )(X) ∈ D(logCI), hence it is (dNis, )-local by [BM21, Corollary 5.5],
we have that D(ωlog
so:logDMeff
CICI)(X) ' i L D(ωlog
D(ωlog)(X) ' i ω≤1(X),
hence for any Y ∈ logDMeff (k, Q)
eff

logDM
CI
)(X))
MaplogDMeff (k,Q) (Y, ω≤1(X)) ' MapD(Shvltr (k,Q)) (i Y, D(ωlogdét' MapPro-D(RSCét,≤1 (k,Q)) (L Alblog (Y ), X)
as required.



5.3. Logarithmic 1-motivic complexes. For any perfect field k and any commutative
ring Λ, recall the stable ∞-category of 1-motivic complexes DMeff≤1 (k, Λ), i.e. the fulleffstable ∞-subcategory of DM (k, Λ) generated by M (X), with dim(X) ≤ 1. If Λ = Q,
by [ABV09, Theorem 2.4.1], the composition
L Alb

eff
L Alb≤1 : DMeff≤1 (k, Q) ,→ DM (k, Q) −−−→ D(HI≤1 (k, Q))

is an equivalence. Letω ∗ : DMeff (k, Q) → logDMeff (k, Q)

38

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

be the functor from (2.6.1). We give the following definition:
Definition 5.13. For any perfect field k and any commutative ring Λ, we let logDMeff≤1 (k, Λ)
eff
eff
∗
CI
be the full stable ∞-subcategory of logDM (k, Λ) generated by ω DM≤1 and ωlog (U )[n]for a unipotent group scheme U . We will call it the stable ∞-category of log 1-motives.
Remark 5.14. By [BPØ20, Theorem 7.6.7], if k satisfies (RS) the ∞-category ω ∗ DMeff
≤1 is
eff
equivalent to the ∞-subcategory of logDM (k, Λ) generated by M (X) with X a propersmooth curve. Moreover, if ch(k) = 0, every unipotent group scheme splits as a directeff∗CI
sum of Ga , hence the categroy logDMeff≤1 (k, Λ) is generated by ω DM≤1 and ω Ga [n].
eff

logDM
Moreover, if Λ is a Q-algebra, the functor ω≤1
factors through logDMeff≤1 (k, Q):
indeed the category D(RSC≤1 (k, Q)) is generated by h0 (C)[n] with C = (C, C∞ ) a proper
modulus pair of dimension 1 such that C − |C∞ | is affine, and there is a fiber sequenceh0 (Cred )[n − 1] → G⊕ra [n] → h0 (C)[n].
Let C ∈ SmlSm(k) be the log scheme (C, ∂C) with |∂C| = |C∞ |, then
eff

1

1

logDM
∗ A
∗
ω≤1
h0 (Cred )[m] = ω ∗ hA
0 (C − C∞ )[m] = ω (h0 (C − C∞ ))[m] = ω M (C − C∞ )[m]
eff

logDM
and by construction ω CI Ga [n] = ω≤1
Ga [n]. By repeating this argument backwards
logDM
we conclude that the functor ω≤1eff

is also essentially surjecrive on logDMeff≤1 (k, Λ).

From now on, we consider again ch(k) = 0 and Λ = Q. We have the following generalization of [ABV09, Theorem 2.4.1]:
logDM
Theorem 5.15. The composition L Alblog ◦ω≤1eff

is equivalent to the constant pro-

eff

logDM
object functor. In particular, the functor ω≤1
is fully faithful and induces an equivalence∼
logDMeffω≤1: D(RSC≤1 (k, Q)) −→ logDMeff
L Alblog .
≤1 (k, Q) : lim←−
Proof. By construction, the proof follows from Proposition 5.16 and Lemma 5.18 below.
Proposition 5.16. There is a commutative diagram of stable ∞-cateogries:
L Alb≤1

D(HI≤1 (k, Q))j

Pro-D(RSC≤1 (k, Q))
L Alblog

DMeff≤1 (k, Q)

DMeff (k, Q)ω∗

logDMeff (k, Q).

Proof. The category DMeff≤1 (k, Q) is generated by M (C) with C affine curve, and for such
1
C we have NS (Ck ) = 0. It is then enough to show that
L Alblog (ω ∗ M (C)) ' jL Alb≤1 (M (C)).
Since C is affine, the right hand side is equivalent to the constant pro-object Alb(C)[0].
By [BPØ20, Theorem 8.2.11], we have that ω ∗ M (C) = M (C, ∂C) with C the smoothprojective compactification of C. To conclude, we have to show that(
Alb(C) if i = 0
πi L Alblog (C, ∂C) =0
otherwise.
Notice that in this case, (C, ∂C) is not Alblog -trivial since H 1 (C, OC ) 6= 0 if g(C) 6= 0.
On the other hand, there exist x, y ∈ C such that C − {x}, C − {y} and C − {x, y} are

DERIVED LOG ALBANESE SHEAVES

39

affine. In particular, (C − {x, y}, ∂C), (C − {x}, ∂C) and (C − {y}, ∂C) are Alblog -trivial
and there is a sZar-distinguished square(C − {x, y}, ∂C)(C − {x}, ∂C)(C − {y}, ∂C)(C, ∂C)

which induces a fiber-cofiber sequence in logDMeff (k, Q):
M (C − {x, y}, ∂C) → M (C − {x}, ∂C) ⊕ M (C − {y}, ∂C) → M (C, ∂C)and so a fiber-cofiber sequence in Pro-D(RSC≤1 (k, Q)):
L Alblog (C−{x, y}, ∂C) → L Alblog (C−{x}, ∂C)⊕L Alblog (C−{y}, ∂C) → L Alblog (C, ∂C).
Since (C − {x, y}, ∂C), (C − {x}, ∂C) and (C − {y}, ∂C) are Alblog -trivial, by Proposition
5.10 and Remark A.9 we conclude thatlog
Alb (C, ∂C) if i = 0log
Alb(C−{x},∂C)⊕
πi L Alblog (C, ∂C) = ker Alblog (C − {x, y}, ∂C) → Alblog (C−{y},∂C)if i = 10otherwise.
By construction, we have that Alblog (C, ∂C) is the constant pro-object Alb(C), so toconclude, we need to show that the map in pro- RSC≤1,ét (k, Q)
Alblog (C − {x, y}, ∂C) → Alblog (C − {x}, ∂C) ⊕ Alblog (C − {y}, ∂C)
is a monomorphism. It is an easy consequence of [Isa02, Proposition 3.2] that a map of of
pro-objects is a monomorphism if and only if it is injective levelwise on a coinitial system.
In particular, it is enough to show that for n  0 the map
Alb(C,|∂C|+n({x}+{y}) → Alb(C,|∂C|+n{x}) ⊕ Alb(C,|∂C|+n{y})is an injective map of étale reciprocity sheaves. By Proposition 3.20 and the proof of
Proposition 3.14, we have AlbC = Pic(C, C∞ ) for any C = (C, C∞ ), hence by [Sai20,
Theorem 0.2], it is enough to show that for every function field K over k, the map
Pic(C K , |∂C|K +n({x}K +{y}K )) → Pic(C K , (|∂C|K +n{x}K )⊕Pic(C K , (|∂C|K +n{y}K )is injective, which follows by Lemma 5.17 below.

Lemma 5.17. Let X be a proper smooth curve over a field and D, E be effective Cartierdivisors on X such that |D| =6 ∅ and |D| ∩ |E| =6 ∅. Then there exist an exact sequence×
0 → H 0 (X, OE) → Pic(X, D + E) → Pic(X, D) → 0.

If E = E1 + E2 with |E1 | ∩ |E2 | = ∅, then the map
Pic(X, D + E) → Pic(X, D + E1 ) ⊕ Pic(X, D + E2 )is injective.
Proof. The second assertion follows immediately from the first. To show the first, recall
an isomorphism
Pic(X, D + E) ∼) with O×= Ker(O× → O× ).
= H 1 (X, O×
X|D+E

X|D+E

X

D+E

and a similar isomorphism with D + E replaced by D. By the assumption |D| ∩ |E| 6= ∅,we have an exact sequence×××
0 → OX|D+E→ OX|D→ OE→ 0,×which induces the desired exact sequence noting H 0 (X, OX|D
) = 0 if |D| =
6 ∅.



40

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO
eff

logDM
Lemma 5.18. For all i > 0, Li Alblog (ω≤1
Ga ) = 0.

Proof. By purity and étale descent, it is enough to show the vanishing over every algebraically closed field extension K/k. Let X ∈ SmlSm(K) and (X, D) be a Cartiercompactification of X. By the construction of Log in [Sai21, §6] and [RS21, Cor. 6.8(1)],
we have(5.18.1)

CIωlog
Ga (X) = lim Γ(X, OX (nD)) = Γ(X, OX ).
−→nHence, there is an explicit resolution (the normalized Bar construction, see [Bre69]):
×•^
1 , triv)
Qltr (A(5.18.2)

eff

logDM
→ ω≤1
Ga .
i

0
^
1 , triv) is the complement of the retraction map Q −where Qltr (A→Qltr (A1 , triv) → Q,^
1 , triv) → ω CI G (X) is induced byand the augumentation Qltr (AaZ(A1 ) = Z(HomSchk ( , A1 )) → HomSchk ( , A1 ) = Ga
See [Bre69, §1-2]. The map factors through Qtr (A1 ) and by (5.18.1) gives naturally a
map in Shvltr . The log scheme (A1 , triv) is Alb-admissible by Proposition 5.10 and Albadmissible objects are stable under direct summands, so we get the following explicit
construction:
eff

logDM
^
1 , triv))×• = “ lim ”(Pic0 (P1 , m∞) ⊗ Q)×• .
L Alb(ω≤1
Ga ) = Alb(Qltr (A←−m

Let K as above: we have that
(Pic0 (P1 , m∞) ⊗ Q)(K) = Pic0 (P1K , m∞)where the right hand side is the group of vector bundles of degree 0 with a trivialization
at m∞, which is already a Q-vector space since ch(K) = 0. It is then enough to showthat for m, n ≥ 1, πn (Pic0 (P1K , m∞)×• ) = 0. Let us fix m. We compute the boundary
maps δn . For n ≥ 1 have the following square:dBnZ0 (AnK )Q(5.18.3)

Z0 (An−1
K )Q(Alb(P1 ,m∞) )×nδn

Pic0 (P1K , m∞)×n(Alb(P1 ,m∞) )×n−1

Pic0 (P1K , m∞)×n−1

where Z0 (AnK )Q = Qltr (An , triv)(K) is the group of zero-cycles with rational coefficientsand the map dB
n is the boundary maps for the complex (5.18.2) evaluated on K, definedon generators as follows (see [Bre69]): let ({a1 }, . . . , {an }) ∈ Z0 (AnK )Q be a generator,then(5.18.4) dB
n (K) : ({a1 }, . . . , {an }) 7→ ({a2 }, . . . , {an })
+

n−1
X

(. . . , {ai + ai+1 }, . . .) + (−1)n ({a1 }, . . . , {an−1 }).

i=1

On the other hand, the Albanese map sends ({a1 }, . . . , {an }) to the n-uple


(1)

O,

T1

(1)
− a1 T0 

(1)

T1

(1)

− T0



, . . . , O,

(n)

T1

(n)
− an T0 

(n)

T1

(n)

− T0

DERIVED LOG ALBANESE SHEAVES
(i)

41

(i)

where [T0 : T1 ] is the coordinate of the i-th copy of P1K . Consider zi :=for every i an isomorphism that on generators is defined as follows:
Pic0 (P1K , m∞) ∼= 1 + zi K[zi ]/(zi )m(5.18.5)



O,

(i)
(i)
T1 − ai T0 (i)(i)
T1 − T0

7→

(i)

T1

(i)

T0

, we have

1 − ai zi
1 − ziwhere the operation on 1 + zi K[zi ]/(zi )m is the multiplication of polynomials. To ease
the notation, for a ∈ K we denote by [a] the element 1 − azi in 1 + zi K[zi ]/(zi )m andiby [a] + [b] the element (1 − azi )(1 − bzi ), hence we will denote 1−az
1−zi by [a] − [1]. From
(5.18.3), (5.18.4) and (5.18.5) we conclude that the differential δn is computed as follows:
the i-th component of δn ([a1 ] − [1], . . . , [an ] − [1]) isi−1X(−1)q ([ai+1 ] − [1]) + (−1)i ([ai+1 + ai ] − [1]) +

q=0

n−1
X


(−1)q ([ai ] − [1]) .

q=i+1

On the other hand, we have that ([1], . . . , [1]) = Alb({0}, . . . , {0}), hence(0
if n ≡2 1
δn ([1], . . . , [1]) = Alb(dBn ({0}, . . . , {0})) =([1], . . . , [1]) if n ≡2 0.
Finally we can compute δn ([a1 ], . . . , [an ]) = δn ([a1 ] − [1], . . . , [an ] − [1]) + δn ([1], . . . , [1]).
We let ( )i denote the i-th component, there are four cases:[ai+1 + ai ]if n ≡2 i ≡2 0
[a ] + [a ] − [aii+1i+1 + ai ] if n ≡2 0, i ≡2 1
δn ([ar1 ], . . . , [arn ])i =
if n ≡2 1, i ≡2 0
[ai + ai+1 ] − [ai ][ai+1 ] − [ai + ai+1 ]if n ≡2 i ≡2 1.
The group 1 + zi K[zi ]/(zi )m has a bounded filtration where for 1 ≤ r ≤ m:rm
Fil(i)
r = 1 + zi K[zi ]/(zi ) .

For each r, we let [a]r = 1 + azir . We make the following claim:
(1)

(n)

Claim 5.19. Let r ≥ 1 and let ([ar1 ]r , . . . , [arn ]r ) ∈ Filr ⊕ . . . Filr . Then modulo(i)mod Filr+1 we have:[(ai+1 + ai )r ]r
if n ≡2 i ≡2 0[ar + ar − (ari+1 + ai ) ]r if n ≡2 0, i ≡2 1ii+1δn ([ar1 ]r , . . . , [arn ]r )i =r[(ai + ai+1 ) − ari ]r
if n ≡2 1, i ≡2 0


 r
r
if n ≡2 i ≡2 1.
[ai+1 − (ai + ai+1 ) ]r
Admitting Claim 5.19, since K is algebraically closed we conclude that for each r, wehave that


(n)
(n−1)
δn Fil(1)⊆ Fil(1).
r ⊕ . . . ⊕ Filrr ⊕ . . . ⊕ Filr
In particular, the complex (Pic0 (P1K , m∞)• , δ• ) has a bounded filtration as follows:(Pic0 (P1K , m∞)• , δ• ) =

•
M
i=1

(i)

(Fil1 , δ• ) ⊇ . . .

•
M
i=1

(Fil(i)r , δ• ) ⊇•M(Fil(i)m , δ• ) = 0.

i=1Moreover, for all r we have an evident isomorphism (see also (5.19.2) and (5.19.3) below)
grr Pic0 (P1K , m∞)n ∼= K n ([a1 ]r , . . . , [an ]r ) 7→ (a1 , . . . an )42

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

and the boundary map grr (δn ) : K n → K n−1 is computed as in Claim 5.19. Using thespectral sequence of the filtered complex (notice that the filtration is bounded so it always
abrupts), we are reduced to show that for all n ≥ 1 and 1 ≤ r ≤ m,πn (grr Pic0 (P1K , m∞)• , grr (δ• )).
Let (a1 . . . an ) ∈ K n such that grr (δn )(a1 . . . an ) = 0, we need to find (b1 , . . . , bn+1 ) suchthat ai = grr (δn+1 )(b1 , . . . , bn+1 )i . Since K is algebraically closed we can suppose thatai = xri . By Claim 5.19 we get that
if n ≡2 i ≡2 00
xr + xri
i+1 if n ≡2 0, i ≡2 1
(5.19.1)
grr (δn )(xr1 . . . xrn )i = 0 ⇔ (xi + xi+1 )r =rx
if n ≡2 1, i ≡2 0

i

 r
xi+1
if n ≡2 i ≡2 1.
Let i ≥ 2 and suppose that we have constructed b1 , . . . , bi such that for all ` ≤ i − 1,xr` = grr (δn+1 )(b1 , . . . , bi , 0 . . . , 0)`then we have a polynomial relation on bi+1 by imposing the condition
xri = grr (δn+1 )(b1 , . . . , bi , bi+1 , 0 . . . , 0)iand deduce the existence of bn+1 as root of such polynomial. Hence it remains to constructb1 and b2 in the case i = 1. If n ≡2 0, we choose b1 = (−x1 )r and b2 = (x1 + x2 )r : fromthe second case of(5.19.1) we have (x1 + x2 )r = xr1 + xr2 , so we get thatgrr (δn+1 )(b1 , b2 , 0 . . . , 0)1 = (x1 + x2 )r − (x1 + x2 − x1 )r = xr1 + xr2 − xr2 = xr1 .
If n ≡2 1, we choose b1 = xr1 and b2 = xr2 : from the last case of (5.19.1) we have(x1 + x2 )r = xr2 , so we get thatgrr (δn+1 )(b1 , b2 , 0 . . . , 0)1 = xr1 + xr2 − (x1 + x2 )r = xr1 .
This implies that πn (grr Pic0 (P1K , m∞)• , grr (δ• )) = 0 for all n ≥ 1 and reduces the proof
of Lemma 5.18 to Claim 5.19.
Proof of Claim 5.19. Let δn,i := δn ([ar1 ]r , . . . [arn ]r )1 . Notice that for all b ∈ K:(i)

− [b]r = (1 − bzir )−1 = 1 + bzir + O(zi2r ) = [−b]r mod Filr+1(5.19.2)

and for all b, c ∈ K;(i)

(5.19.3) [b]r + [c]r = (1 − bzir )(1 − czir ) = 1 − (b + c)zir + O(zir+1 ) = [b + c]r mod Filr+1 .
Let ζr be a primitive r-th root of unity in K, then[ari ]r = 1 − ari zir =r−1
Y(1 − ζrq ai zi ) =

q=0

r−1
X

[ζrq ai ],

q=1

q
q
q=1 δn ([ζr a1 ], . . . , [ζr an ])i . We check the four cases:
Pr−1 qP
qr(1) n ≡2 i ≡2 0: δn,i = q=1[ζr ai+1 + ζrq ai ] = r−1
q=1 [ζr (ai + ai+1 )] = [(ai + ai+1 ) ]r
Pr−1 qqqq(2) n ≡2 0, i ≡2 1: δn,i = q=1 [ζr ai+1 ] + [ζr ai ] − [ζr ai+1 + ζr ai ]= [ari+1 ]r + [ari ]r − [(ai + ai+1 )r ]rP(3) n ≡2 1, i ≡2 0: δn,i = r−1[ζrq ai+1 + ζrq ai ] − [ζrq ai ] = [(ai + ai+1 )r ]r − [ari ]rPr−1q=1q(4) n ≡2 i ≡2 1: δn,i = q=1 [ζr ai+1 ] − [ζrq ai+1 + ζrq ai ] = [ari ]r − [(ai + ai+1 )r ]r .

thus δn,i =Pr−1Here all congruences follow from (5.19.2) and (5.19.3).

DERIVED LOG ALBANESE SHEAVES

43

6. Some computations
Recall from [BVK16, Theorem 9.2.3] (see also [Par21, Theorem 1.1] for a statement ina language more similar to ours) that
L Alb(ω(X)) ' Alb(ω(X)) ⊕ NS∗ (ω(X))[1],(6.0.1)

where NS∗ (ω(X)) is the torus dual to the Néron-Severi. The goal of this section is to givean explicit description of L Alblog (X): for X = (X, ∂X), recall that ω(X) is defined as
X − |∂X|. We will prove the following result:
Theorem 6.1. Let X ∈ SmlSm geometrically connected and (X, D) a Cartier compactification of X. Then we have that
i (X, O (nD))∨ ) ⊗ G
for 2 ≤ i ≤ dim(X)“ lim ”(H
a
k

X

←− 


“ lim ” (H 1 (X, O (nD))/H 1 (X, O ))∨ ⊗ G ⊕ NS∗ (ω(X)) for i = 1
a
k
X
X
←−
Li Alblog (X) ∼=log

for i = 0
Alb (X)0otherwise.
dim(X)

Moreover, the canonical map L Alblog (X) → ⊕i=0

Li Alblog (X)[i] is an equivalence.

We start with the following observation:
Proposition 6.2. The inclusion D(iA1 ) : D(HIét,≤1 ) → D(RSCét,≤1 ) has a left adjoint
1L≤1such that πi (L≤1)(F ) = hAi (F ) (the Suslin hyperhomology).
A1
A1
tr
Proof. By Proposition 4.31, the inclusion D(RSCét,≤1 ) → D(Shvét(k, Q)) is fully faithful,then for F ∈ D(RSCét,≤1 ) and H ∈ D(HIét,≤1 ):MapD(RSCét,≤1 ) (F, D(iA1 )(H)) ' MapD(Shvtr (k,Q)) (F, D(iA1 )(H))
ét11' MapDMeff (k,Q)) (C∗A (F ), H) ' MapD(HIét,≤1 ) (L Alb C∗A (F ), H)

which implies that L≤1exists and it coincides with L Alb C∗A . We can write F =A11trhocolimi,n Fi [n] with Fi ∈ RSC?≤1 . Since C∗A : D(Shvét(k, Q)) → DMeff (k, Q)) commutes with all (homotopy) colimits as a left adjoint, we have
1

1

1

(∗)

1

C∗A (F ) ' hocolim C∗A (Fi [n]) ' hocolim(hA
0 (Fi )[n]),1where (∗) follows from Remark 4.25. Hence the homotopy groups of C∗A (F ) are 1-motivic
1
1
1
so that πi L≤1(F ) = πi L Alb C∗A (F ) = πi C∗A (F ) = hA
i (F ).
A1
Lemma 6.3. Let Pro-L≤1: Pro-D(RSCét,≤1 ) → Pro-D(HIét,≤1 ). For X ∈ SmlSm(k)A1geometrically connected, Pro-L≤1
L Alblog (X) is a constant pro-object and
A1
L Alblog (X) ' L Alb(ω(X)).
lim pro-L≤1
A1←−
Proof. Let Qltr (Y• ) → Qltr (X) be a resolution in Shvltr
dét (k, Q) by Alb-trivial objects.
log
log
Then Li Alb (X) = πi (Alb (Y• )). On the other hand, by construction, Qtr (ω(Y• )) →tr
Qtr (ω(X)) is a resolution in Shvét(k, Q) by affine NS1 -local objects in the sense of [ABV09],
so Li Alb(ω(X)) = πi Alb(ω(Y• )). By Proposition 3.19 and (5.2.1), for Y• Cartier compactifications of Y• , we have a fiber-cofiber sequence:(n)“ lim ”U (Y• ) ⊗k Ga → Alblog (Y• ) → c(Alb(ω(Y• ))),←−

(6.3.1)n(n)where U (Y• ) are the k-vector spaces from Definition 3.5. The complex Alb(ω(Y• )) isA1 -local, so we have that
Pro-L≤1c(Alb(ω(Y• ))) ' c(Alb(ω(Y• ))) = c(L Alb(ω(X)))
A1

44

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITOr(n)

and for all i, n, there exist ri,n such that U (Yi ) = Gai,n , so we conclude since:
(n)

(n)

Pro-L≤1(“ lim ”(Y• )) = “ lim ”L≤1
(U (Y• )) = 0.
A1
←−
←− A1
n

n


log

The lemma above gives a natural map L Alb(X) → cL Alb(ω(X)). Let

Fib(X) := hofib(L Alblog (X) → cL Alb(ω(X))).

(6.3.2)

In view of (6.3.1), we have that(n)Fib(X) ' “ lim ”U (Y• ).
←−(6.3.3)nsince for Qltr (Y• ) → Qltr (X) an Alb-trivial resolution and Y• a Cartier compactification(n)(n)
of Y• , the map Alb(Yi ) → Alb(ω(Yi )) is surjective with kernel U (Yi ) for all i.
Definition 6.4. Let hGa i ⊆ D(RSCét,≤1 ) be the stable ∞-subcategory generated bydirect sums of Ga and let hGa iω be the full subcategory of compact objects. In particular,F ∈ hGa iω if and only if there exist s ≤ t ∈ Z and rs , . . . , rt ≥ 0 such that
F '

t
M

Grai [i].

i=sRemark 6.5.
Pro-(hGa iω )Let Pro-(hGa iω ) ⊆ Pro-D(RSCét,≤1 ), then for all X

as above Fib(X) ∈Remark 6.6. Let Vectfd be the category of finite dimensional k-vector spaces. By Proposition 4.31 and [Ser75, VII]:
MapD(RSCét,≤1 ) (Gra , Gsa [i]) ' HomVectfd (k r , k s )[i] ' MapD(Vectfd ) (k r , k s [i]).
In particular, the functor V 7→ Spec(k[V ∨ ]) induces an equivalence of ∞-categories⊗k Ga : D(Vectfd ) → hGa iω
with quasi inverse given by RΓ(Spec(k), ).
Let ( )∨ : Vectfd → Vectop
fd denote the equivalence given by the dual vector space. It
induces an equivalence:( )∨ : Pro-D(Vectfd ) → Ind-D(Vectop
fd ).
Remark 6.7. There is a commutative diagram of stable ∞-category:
Map( ,Ga )Pro-hGa iω

Pro-RΓ(Spec(k), )Pro-D(Vectfd )( )∨Ind-D(Vectop
fd )where Map denotes the mapping space in Pro-D(RSCét,≤1 ). This easily follows from thefact that any map of sheaves f : Gra → Gsa is indeed a map of vector groups, hence since
Ga = k ⊗k Ga , we have a commutative diagram in D(Vectfd ):Map(Gsa , Ga )'f∗

Map(Gra , Ga )RΓ(Spec(k), Gsa )∨f (k)t'RΓ(Spec(k), Gra )∨ .

Proposition 6.8. Let X ∈ SmlSm(k) geometrically connected. Then for any (X, D)
Cartier compactification of X and i ≥ 2, we have that
Li Alblog (X) ' “ lim ”(H i (X, OX (nD))∨ ) ⊗k Ga .
←−In particular, Li Alblog (X) = 0 for i ≥ max(dim(X), 2).

DERIVED LOG ALBANESE SHEAVES

45

Proof. By (6.0.1), for X as above we have that for i ≥ 2
Li Alblog (X) ' πi Fib(X).
By Remark 6.7 we have that
(6.8.1)

Fib(X) = RΓ(Spec(k), Fib(X)) ⊗k Ga ' Map(Fib(X), Ga )∨ ⊗k Ga .

By Theorem 5.1 and [BPØ20, Theorem 9.7.1] and (5.18.1).
(6.8.2)CI
MapPro-D(RSCét,≤1 ) (L Alblog (X), Ga ) ' MaplogDMeff (M (X), ωlog
Ga ) ' RΓ(X, OX ).
Let now (X, D) be a Cartier compactification of X. This gives an isomorphism:
RΓ(X, OX ) ' lim RΓ(X, OX (nD)).
−→n
op
and the right hand side is in Ind-D(Vectfd ), which completes the proof.



Since L Alblog (X) is bounded, there are maps
L Alblog (X) → τ≤i L Alblog (X) ← τ≥i τ≤i L Alblog (X) ' Li Alblog (X)[i]which induces a zig-zag(6.8.3)dim(X)L Alblog (X) → ⊕i=1

dim(X)

τ≤i L Alblog (X) ← ⊕i=1

Li Alblog (X)[i].

Lemma 6.9. For X ∈ SmlSm(k) geometrically connected, we have an equivalence:dim(X)L Alblog (X) ∼= ⊕i=0 Li Alblog (X)[i].

Proof. By Proposition 4.31, we have that for every F ∈ RSCét,≤1 , and i ≥ 2,
Extipro- RSCét,≤1 (πj Fib(X), c(F )) = 0,

resp. Extipro- RSCét,≤1 (πj L Alb(ω(X)), c(F )) = 0,

so by (6.3.2), we get Extipro- RSCét,≤1 (Lj Alblog (X), c(F )) = 0. Then the spectral sequence
E2ij = Extipro- RSCét,≤1 (Lj Alblog (X), c(F )) ⇒ πi+j MapPro-D(RSCét,≤1 ) (L Alblog (X), c(F )[0])degenerates at page 2, giving for every j a short exact sequence(6.9.1)
0 → Ext1pro- RSCét,≤1 (Lj Alblog (X), c(F )) → π1+j MapPro-D(RSCét,≤1 ) (L Alblog (X), c(F ))→ Hompro- RSCét,≤1 (L1+j Alblog (X), c(F )) → 0.
On the other hand, the complex ⊕Li Alblog (X)[i] also fits in short exact sequences
0 → Ext1pro- RSCét,≤1 (Lj Alblog (X), c(F )) → π1+j MapPro-D(RSCét,≤1 ) (⊕Li Alblog [i], c(F ))→ Hompro- RSCét,≤1 (L1+j Alblog (X), c(F )) → 0
so the zig-zag of (6.8.3) gives for all F = “ lim ”Fi and all n an equivalence←−log
MapPro-D(RSCét,≤1 ) (⊕Li Alb (X)[i], F [n]) ' lim MapPro-D(RSCét,≤1 ) (⊕Li Alblog (X)[i], c(Fi )[n])←−' lim Σ−n MapPro-D(RSCét,≤1 ) (⊕Li Alblog (X)[i], c(Fi ))←−' lim Σ−n MapPro-D(RSCét,≤1 ) (L Alblog (X), c(Fi ))←−' lim MapPro-D(RSCét,≤1 ) (L Alblog (X), c(Fi )[n])←−' MapPro-D(RSCét,≤1 ) (L Alblog (X), F [n]).
Then a spectral sequence argument and the fact that L Alblog (X) is bounded implies thatfor all C ∈ Pro-D(RSCét,≤1 ) we have
MapPro-D(RSCét,≤1 ) (⊕Li Alblog (X)[i], C) ' MapPro-D(RSCét,≤1 ) (L Alblog (X), C)
which allows us to conclude.

46

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITOProof of Theorem 6.1. The only case left is i = 1: we consider the long exact sequence of
homotopy groups arising from (6.3.2). The map L1 Alb(ω(X)) → π0 (Fib(X)) is zero sinceby (6.0.1), L1 Alb(ω(X)) is a torus, so we get a short exact sequcend in pro- RSCét,≤1 :(6.0.1)0 → π1 Fib(X) → L1 Alblog (X) → L1 Alb(ω(X)) ∼= c(NS∗ (ω(X))) → 0.
Since π1 Fib(X) is a pro-vector group, this short exact sequence splits and(6.9.2)L1 Alblog (X) ∼= π1 Fib(X) ⊕ NS∗ (ω(X)).
Moreover, we haveMap(π1 Fib(X), Ga ) ' Map(L1 Alblog (X), Ga ) ' Hompro- RSCét,≤1 (L1 Alblog (X), Ga )[0],where Map denotes the mapping space in Pro-D(RSCét,≤1 ). Hence (6.9.2) gives:(6.9.3)π1 L Alblog (X) ∼= (Hom(L1 Alblog (X), Ga )∨ ) ⊗k Ga ⊕ NS∗ (ω(X)).

Let us compute Hom(L1 Alblog (X), Ga ): by (6.8.2) and (6.9.1) we have an exact sequence:(6.9.4)0 → Ext1 (Alblog (X), Ga ) → H 1 (X, OX ) → Hom(L1 Alblog (X), Ga ) → 0.

In particular, we have a similar exact sequence for the log scheme (X, triv), which we now
inversigate. For X proper we have by construction (see Theorem 4.28) that Alblog (X) isthe constant pro-object Alb(X), so there is a surjective map
Alblog (X)  Alblog (X)whose kernel is an extension of the torus T := ker(Alb(ω(X)) → Alb(X)) by the pro-vector
group “ lim ”U (X, nD) ⊗k Ga , where U (X, nD) comes from Definition 3.5. For i ≥ 1, we←−
have that lim Exti (U (X, nD), Ga ) = 0 and Exti (T, Ga ) = 0, so we have a surjective map:−→
(6.9.5)
Ext1 (Alblog (X), Ga )  Ext1 (Alblog (X), Ga ).
Combining (6.9.4), and (6.9.5) we have a commutative diagram with exact rows:0

Ext1 (Alblog (X), Ga )H 1 (X, OX )Hom(L1 Alblog (X), Ga )00

Ext1 (Alblog (X), Ga )H 1 (X, OX )Hom(L1 Alblog (X), Ga )0

so to conclude it is enough to show that L1 Alblog (X) ∼= NS∗ (X), which implies thatlog
Hom(L1 Alb (X), Ga ) = 0, so the diagram above implies
Hom(L1 Alblog (X), Ga ) ' coker(H 1 (X, OX ) → H 1 (X, OX ))and we will conclude by duality and (6.9.3).
By (6.9.2), it is enough to show that π1 Fib(X) = 0. By (6.8.1), we have
π1 Fib(X) ∼= π1 (Map(Fib(X), Ga )∨ ⊗k Ga ) ∼= (π−1 Map(Fib(X), Ga )∨ ⊗k Ga ),so it is enough to show that π−1 Map(Fib(X), Ga ) = 0. By (6.3.2) for X we have afiber-cofiber sequence:(6.9.6)

Map(L Alb(X), Ga ) → Map(L Alblog (X), Ga ) → Map(Fib(X), Ga ).

By (6.0.1), since NS∗ (X) is a torus and Alb(X) is an abelian variety, by [Ser75, VII] wehave
π−1 Map(L Alb(X), Ga ) ∼= π0 Map(NS∗ (X), Ga ) ⊕ π−1 Map(Alb(X), Ga )∼= Ext1 (Alb(X), Ga ) ∼= H 1 (X, OX ),where the last isomorphism is classical, and
π−2 Map(L Alb(X), Ga ) = 0.

DERIVED LOG ALBANESE SHEAVES

47

Finally, π−1 Map(L Alblog (X), Ga ) ∼= H 1 (X, Ga ) by (6.8.2), so the map
π−1 Map(L Alb(X), Ga ) → π−1 Map(L Alblog (X), Ga )in the long exact sequence of homotopy groups of (6.9.6) is an isomorphism, which implies
the desired vanishing.

Remark 6.10. We observe from Theorem 6.1 two extreme cases: if X is affine, we havethat H i (X, OX ) ' limH i (X, OX (nD)) = 0 for i ≥ 1, so
−→
n

log

Alb (X) if i = 0
Li Alblog (X) = NS∗ (X)if i = 10otherwise.
In particular, Li Alblog (X) is constant for i ≥ 1. For X proper, H 1 (X, OX ) ∼= H 1 (X, OX ),so
if i = 0
Alb(X)log
Li Alb (X) = NS∗ (X)if i = 1 i∨(H (X, OX ) ) ⊗k Ga if i ≥ 2.
In this case, L Alblog (X) is a constant pro-object. This shows that Proposition 5.16 cannot
be extended to the whole DMeff (k, Q): in general, if M ∈ DMeff (k, Q), L Alblog (ω ∗ M ) is
not equal to L Alb(M ): the difference is controlled by coherent cohomology of degree ≥ 2.
6.1. Open questions. We end this Section with the following observation. It seems to
be an interesting question to determine under which conditions Li Alblog is a constant
pro-object.
This is related to the following problem: let X 0 → X be a desingularisation of a ddimensional, integral variety over a field k, and let D be an effective Cartier divisor on X 0
covering the exceptional fibre, and assume that codimX (π(D)) ≥ 2. Let rD denote ther-th infinitesimal thickening of D and F d K0 (X 0 , rD) the subgroup of the relative K-group
K0 (X 0 , rD) generated by the cycle classes of closed points of X 0 − |D|, for each r ≥ 1.
Bloch and Srinivas conjectured (see [Sri85, p. 6]) that the pro-object “ limn ”F d K0 (X 0 , nD)←−
is essentially constant and equal to F d K0 (X). The Bloch–Srinivas conjecture was proved
for normal surfaces by Krishna–Srinivas [KS02, Theorem 1.1], and for ch(k) = 0 it waslater extended to higher dimensional projective and affine varieties over an algebraicallyclosed field by Krishna [Kri06, Theorem 1.1] [Kri10, Thorem 1.2] and Morrow [Mor15,
Theorem 0.1, (iii), (iv)]7.
The proof of [Mor15] indeed relies on a natural reformulation of the Bloch–Srinivas
conjecture for the Chow groups with modulus:
Theorem 6.11 (cfr. [Mor15, Theorem 0.3]). Let k be an algebraically closed field of
characteristic zero and π : X 0 → X and D be as above and assume that X is projective.
Then the pro-object “ lim ”n CH0 (X, nD) is constant with stable value equal to the Levine–←−
Weibel Chow group of zero cycles CHLW
0 (Y ) of [LW85] (see also [BK18]).
By the universal property of the Albanese map, we deduce that if in the situationof Theorem 6.11 we assume that D is a simple normal crossing divisor, the pro-object“ lim ”n Alb(X 0 ,nD) is indeed essentially constant, so the pro-object L0 Alblog (X−π(|D|), triv)←−
is essentially constant. Then the following question arises naturally:
Question 6.12. Let X be a proper variety and U ⊂ X be a smooth open subvariety. Whenis the pro-reciprocity sheaf Li Alblog (M (U, triv)) essentially constant?
7The conjecture is indeed true in a more general class of examples: the interested reader can check[Mor15, Theorem 0.1 (i)-(vii)]

48

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

Notice that in general Li Alblog (M (U, triv)) is not essentially constant: let X be a proper
non-singular surface, and U = X − Y for some closed subscheme Y of X. Let (X 0 , D) be
the blow-up of X in Y . As observed in [Har68, p. 407], if some irreducible componentof Y is a point, then dim(H 1 (U, OU )) = ∞. On the other hand, dim(H 1 (X 0 , OX 0 (nD)))
is finite for every n, so L1 Alblog (M (U, triv)) is not constant by Theorem 1.3. At the
moment, we do not know if there is a nice family of pairs U ⊆ X that answers Question
6.12 positively.
7. Laumon 1-motives and compact objects
In this Section, we combine the results of [Ber14] with some arguments of [BVK16]. As
before, let k be a field of characteristic zero.
7.1. Review of Laumon 1-motives. The following definition is adapted from [Ber14,
1].
uDefinition 7.1. An effective Laumon 1-motive is a two-terms complex M = [Γ −→ G],
where Γ is a formal k-group and G is a connected algebraic k-group, both seen as objectsof Shvét (k). We say that M is étale if Γ is a lattice.8. An effective morphismu

M = [Γ −
→ G]

(f,g)

0

u
/ M 0 = [Γ0 −→ G0 ]

is a map of complexes. We denote the category of effective (resp. étale) Laumon 1-motives(resp. Ma,eff
by Ma,eff
11,ét ). An effective Laumon 1-motive is an effective Deligne 1-motive if
for the full subcategory of effective Deligne 1-motives.
G is semi-abelian. We write MD,eff
1
Definition 7.2. An effective morphism (f, g) : M → M 0 is strict if g has (smooth) connected kernel, and a quasi-isomorphism if g is an isogeny, f is surjective and ker(f ) =ker(g) is a finite k-group scheme.
Note that if (f, g) is strict and g is an isogeny, then g is an isomorphism of commutative
algebraic groups. Write Σ for the class of quasi-isomorphisms: it admits a calculus ofright fractions (see [BVK16, C.2.4] or [Ber13, Lemma 1.6]). We can now give the following
Definition (see [Ber14, Definition 2] and [BVB09, Definition 1.4.4]).
Definition 7.3. The category of étale Laumon (resp. Laumon, resp Deligne) 1-motives
Ma1,ét (resp. Ma1 , resp. MD
1 ) is the localization by Σ of the effective category.
Recall [BVK16, Appendix B] for the notion of C ⊗ Q for an additive category C. The
proof of the following proposition is identical to [BVK16, Corollary C.7.3]:aaProposition 7.4. The categories MD
1 ⊗ Q, M1 ⊗ Q and M1,ét ⊗ Q are abelian.
aDefinition 7.5 (see [Ber14]). Let Ma,?
1,ét be the full subcategory of M1,ét whose objectsuare 1-motives M = [Γ −→ G] with ker(u) = 0.
aLemma 7.6. The category Ma,?
1,ét ⊗ Q is a generating subcategory of M1,ét ⊗ Q, and it
is closed under kernels and extensions. Moreover, for every object M ∈ Ma1,ét ⊗ Q there
exists a monomorphism f : M 0 → M 00 in Ma,?
1,ét ⊗ Q such that M = coker(f ).

Proof. This is essentially [Ber14, Lemma 4].

Remark 7.7. The reader might wonder if there are interesting examples of étale Laumon
1-motives which are not Deligne 1-motives. The prototype of such example is given by
the 1-motive M \ = [Γ → G\ ] which is the universal Ga -extension of the Deligne 1-motive[Γ → G]. Starting from the motive M \ it is possible to construct the universal sharp
8 Note that this definition is different from the one given in [BVB09, 1.4], where the authors require in
addition that U (G) = 0

DERIVED LOG ALBANESE SHEAVES

49

extension M ] of M , as discussed in [BVB09]. Note however that the cateogry Ma1,ét is
b a → G2 ],not closed under ]-extensions: as remarked in [BVB09, 3.1.5], [0 → Ga ]] = [Gawhich is clearly not étale.
Remark 7.8. The category of Deligne 1-motives has an interesting self-duality, inducedby the classical Cartier duality for algebraic groups. This extends to Laumon 1-motives,
see [BVB09]. Note that while the Cartier dual of a Deligne 1-motive is again a Deligne
1-motive, the dual of an étale Laumon 1-motive is in general not étale. For example, if
A is an Abelian variety (see as 1-motive [0 → A]), its universal Ga extension is the étale
Laumon 1-motive A\ = [0 → A\ ], which is not a Deligne 1-motive. Its Cartier dual (A\ )∗b0 → A0 ], where A0 is the dual Abelian variety of A and A
b0 is the formal
is the 1-motive [A0\∗completion of A along the identity. Clearly, (A ) is a Laumon 1-motive that is not étale.
Remark 7.9. Consider the functor
ρeff : Ma,eff
1,ét → Shvét (k, Q)u[L −→ G] 7→ coker(u) ⊗Z Q.

u1
u2
If [L1 −→
G1 ] → [L2 −→G2 ] ∈ Σ, then by definition coker(u1 ) ⊗ Q ∼= coker(u2 ) ⊗ Q. This
together with [BVK16, Lemma B.1.2] implies that ρeff induces:ρ : Ma1,ét ⊗ Q → Shvét (k, Q).
Lemma 7.10. The restriction of ρ to Ma,?
1,ét ⊗ Q induces an equivalence (cf. Def. 4.23):∼=?
.
→ RSCét,≤1
ρ? : Ma,?
1,ét ⊗ Q −?
Proof. By definition, for [L → G] ∈ Ma,?
1,ét ⊗ Q, ρ([L → G]) ∈ RSCét,≤1 , and by Proposi?
tion 2.10 for every morphism f we have that ρ(f ) is a map in RSCét,≤1, hence ρ? is welldefined. The presentation of Proposition 4.24 gives then a quasi-inverse of ρ? .
Remark 7.11. For a category C, we write Ind(C) for the Ind-category of C as in e.g. [KS06].
?
By [KS06, Prop. 6.3.4] and Remark 4.26 the functor Ind(RSCét,≤1) → RSCét,≤1 inducedby filtered colimits is fully faithful. It is also essentially surjective by Proposition 4.24,
hence it is an equivalence. Combining this with Lemma 7.6 and 7.10, we have a functor(7.11.1)(∗)

T : Ma1,ét ⊗ Q → Ind(Ma1,ét ⊗ Q) ' Ind(Ma,?
1,ét ⊗ Q) ' RSCét,≤1 (k, Q)

awhere (∗) follows from the fact that Ma,?
1,ét ⊗ Q is a generating subcategory of M1,ét ⊗ Q.
Since Ma1,ét ⊗ Q is abelian by Lemma 7.4, it is idempotent-complete, hence following thesteps of [KS06, Exercise 6.1] the functor (7.11.1) is fully faithful and it identifies Ma1,ét ⊗ Qwith a set of compact generators of RSCét,≤1 . Moreover, by [KS06, Proposition 8.6.11],
the category T (Ma1,ét ⊗ Q) is closed under extensions in RSCét,≤1 .

7.2. The derived category of étale Laumon 1-motives. By [KS06, Proposition
8.6.11] and Remark 7.11, the image of the functor T of (7.11.1) is a Serre subcategory ofb
RSC≤1,ét (k, Q), hence we can consider the triangulated category DM(RSC≤1,ét (k, Q))a1,étof bounded complexes of RSC≤1 (k, Q) such that Hn (C) = T ([Ln → Gn ]) for [Ln → Gn ] ∈Ma1,ét ⊗ Q.
Remark 7.12. The functor T of (7.11.1) induces an equivalence of triangulated categories:ba
Db (Ma1,ét ⊗ Q) ' DM
(RSCét,≤1 (k, Q))
1,ét

where the latter is the triangulated derived category , since by Proposition 4.31, every?
object of RSCét,≤1
is of projective dimension at most 1 in the sense of [Kel99], in particular
the image of T satisfies [Kel99, 1.21 Lemma (c2)], hence the equivalence comes from [Kel99,
1.21 Lemma (c)].

50

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITODefinition 7.13. Let Db (Ma1,ét ⊗ Q) be the full ∞-subcategory of D(RSCét,≤1 ) spannedby bounded complexes C ∈ Db (RSCét,≤1 ) such that πn C = T ([Ln → Gn ]) for [Ln →Gn ] ∈ Ma1,ét ⊗ Q.
Remark 7.14. Notice that since the category Ma1,ét ⊗ Q does not have enough injective
nor projective objects, we cannot use [Lur17, 1.3] to construct Db (Ma1,ét ⊗ Q) directly.
Lemma 7.15. There is an equivalence of ∞-categories:
D(RSCét,≤1 (k, Q))ω ' Db (Ma1,ét ⊗ Q)
where the left hand side denotes the subcategory of compact objects as in [Lur09, Notation
5.3.4.6].
Proof. Since the set of objects of RSCét,≤1 (k, Q) lying in the image of (7.11.1) is a set ofcompact generators, by [Sta16, Lemma 094B] we have an equivalence
D(RSCét,≤1 (k, Q))ω ' Idem(Db (Ma1,ét ⊗ Q))
where the right hand side is the idempotent completion of Db (Ma1,ét ⊗ Q), see [Lur09,
Definition 5.1.4.1]. On the other hand, the category Db (Ma1,ét ⊗Q) is idempotent-complete
since the image of (7.11.1) is idempotent complete (it is an abelian subcategory), hence
every object of D(RSCét,≤1 (k, Q))ω lies in Db (Ma1,ét ⊗Q). The other inclusion is clear. 
logDM
eff
ω
Theorem 7.16. Let logDMeff≤1,gm (k, Q) := logDM≤1 (k, Q) . The functor ω≤1
preserves compact objects and it induces an equivalenceeff∼

Db (Ma1,ét ⊗ Q) −→ logDMeff≤1,gm (k, Q).
Proof. By Lemma 7.15, if C ∈ D(RSC≤1 ) is compact, then it is a bounded complex suchthat πi (C) = T (Mi ) for Mi ∈ Ma1,ét ⊗Q. In particular, there exists n such that C = τ≥n C,
so we get a fiber-cofiber sequence in D(RSCét,≤1 ):
T (Mn )[n] → C → τ≥n−1 C.
If C is compact, then τ≥n−1 C is compact, so by induction on the length of the bounded
eff

logDM
(T (M )[n]) iscomplex it is enough to show that for M ∈ Ma1,ét ⊗ Q, the object ω≤1compact. As observed in Lemma 7.6, there is an exact sequence in Ma1,ét ⊗ Q:

0 → M 0 → M 00 → M → 0
with M 0 , M 00 ∈ Ma,?
1,ét , and since T is exact we have
eff

eff

eff

logDM
logDM
logDM
ω≤1
(T (M )[n]) = cofib(ω≤1(T (M 0 )[n]) → ω≤1(T (M 00 )[n]).
u

eff

logDM
Thus, it is enough to show that ω≤1(T (M )[n]) is compact for M = [L ,→ G]. In thiscase, we have that T (M ) = coker(u), so we have a cofiber sequence
eff

eff

eff

logDM
logDM
logDM
(G[n]) → ω≤1(T (M )[n]).
ω≤1(L[n]) → ω≤1
eff

eff

logDM
logDM
CI (G)[n] are comWe conclude since ω≤1(L)[n] = ω ∗ L[n] and ω≤1(G[n]) = ωlog
pact. The equivalence then follows from Theorem 5.15 and Lemma 7.15.
Appendix A. Pro-left derived functors
In this appendix we generalize to pro-left adjoints the results discussed in [ABV09, 2.1]
and [KS06, 14.3] for left adjoints. We use in an essential way the formalism of (stable)∞-categories of [Lur17] and [Lur09].

DERIVED LOG ALBANESE SHEAVES

A.1.

51

We consider the following commutative square:G

C(A.0.1)

LC

C0

D
LD

G0

D0

where C, C 0 , D and D0 are ∞-categories, LC (resp. LD ) has a fully faithful right adjoint iC(resp. iD ). Let α : LD G → G0 LC be the equivalence that makes the diagram commute.
As observed in [Lur17, Definition 4.7.4.13], α induces a natural transformation (the Beck–
Chevalley map)
BC(α) : GiC → iD G0 .
Definition A.1. In the situation of A.1, an object X of D is BC(α)-admissible if andonly if for any Y ∈ C 0 the Beck-Chevalley map induces an equivalence:(A.1.1)

MapD (X, iD G0 (Y )) ' MapD (X, GiC (Y )).

Assume that G and G0 have left adjoints F and F 0 , then by adjunction α induces a map
β : F 0 LD → LC F.
A.2. Let C be an (arbitrary) ∞-category. We consider as in [Hoy18, Section 2] the ∞category of pro-objects Pro-C together with the “constant pro-object” embedding c : C →Pro-(C) such that MapPro-C ( , c(Y )) commmutes with cofiltered limits. Every element in
Pro-C is corepresented by a diagram I → C for I (the nerve of) a small cofiltered poset. We
will often denote an object of Pro-C as “ limi∈I ”Xi for a diagram I → C. By construction←−
we have that
MapPro-C (“ lim ”Xi , “ lim ”Yj ) ' lim lim MapC (Xi , Yj )
←−
←−
←− −→
i∈I1

j

j∈I2

i

where the limits and colimits are computed in S.
Remark A.2. The functors LC and iC extend levelwise to an adjunction (Pro-LC , Pro-iC ) on
Pro-C and Pro-C 0 with the same properties. The verification is immediate. In particular,
if D and D0 have all limits, G and G0 give the following commutative diagram:
Pro-C

Pro-G

Pro-LC

DLD
0

Pro-C 0 Pro-G D0
which satisfies the hypotheses of Situation A.1 with equivalence
αpro : LD Pro-G → Pro-G0 Pro-LC .
In particular, since iD commutes with all limits being a right adjoint, it is immediate thatX ∈ D is BC(α)-admissible if and only if it is BC(αpro )-admissible.
Remark A.3. If C is an accessible stable ∞-category equipped a t-structure (C≤0 , C≥0 ) withheart C ♥ , the ∞-category Pro-C is also stable (see e.g. [KST19, Lemma 2.5]) and it comesequipped with a t-structure such that (Pro-C)≤0 (resp. (Pro-C)≥0 ) is the full subcategoryof objects which are formal limits of objects in C≤0 (resp C≥0 ).
A.3. Let A be a Grothendieck abelian category. Write Ch(A) for the model category ofchain complexes with the injective model structure. Let W be the class of quasi isomorphisms. We consider the ∞-categories (see [Lur17, 1.3.5]) Chdg (A) = Ndg (Ch(A)) andD(A) = Ndg (Ch(A))[W −1 ]. An exact functor G : A → B between Grothendieck abeliancategories induces a dg-functor Ch(G) : Ch(A) → Ch(B) which preserves W , so by taking
the dg-nerve it induces a functor Chdg (G) : Chdg (A) → Chdg (B) such that Chdg (G)(C) =Ch(G)(C), and by e.g. [Hin20, Proposition 4.3], it induces D(G) : D(A) → D(B) on the

52

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

localizations. Note that both functors are clearly stable (i.e. they commute with shifts).
By construction, we have a commutative square of ∞-categories:
Chdg (A)(A.3.1)

Chdg (G)LA

Chdg (B)
LB

D(G)

D(A)

D(B)

where LA and LB have fully faithful right adjoints iA and iB . We will fix α that makes(A.3.1) commute and just say that an object is BC-admissible.
Remark A.4. Note that, since G is exact, we can identify the Beck-Chevalley transformation Chdg (G)iA → iB D(G) as follows. For any object I ∈ D(A) (i.e. a fibrant complex
in Ch(A) for the injective model structure), the object D(G)(I) is a fibrant replacementof Chdg (G)(iA (I)). The map Chdg (G)(iA I) → iB D(G)(I) in Ch(B) is thus given by
the functorial fibrant replacement. In particular, if Ch(G) is a right Quillen functor, themap Chdg (G)(iA I) → iB D(G)(I) is an equivalence in Chdg (B). On the other hand, thefunctors condidered here are not necessarily right Quillen.
Remark A.5. If the functor G has a left adjoint F , then X is BC-admissible if and only
if it is F -admissible in the sense of [ABV09, Definition 2.1.5].
A.4. Pro-left derived functors. Fix A, B and G as above and we assume that G
and iB commute with filtered colimits. Since G is exact, it preserves finite limits, so
it has a pro-left adjoint F : B → pro-A. The functor D(G) is then an accessible functor
between presentable ∞-categories that preserves finite limits, hence it has a pro-left adjoint
LF : D(B) → Pro-D(A) (see e.g. [Hoy18, Remark 2.2]).
For any chain complex C, let σ≤n C and σ≥n C denote the stupid truncations (see [Sta16,
Tag 0118]) 9. We have an equivalence in Chdg :(A.5.1)C∼(lim σσ C).
= lim−→ ←− ≥−m ≤nnmDefinition A.6. We say that a chain complex C is strictly bounded if there exists m, nsuch that C = σ≤n C = σ≥−m C.
Remark A.7. Notice that if C ∈ Ch(B), the object Ch(F )(C) a priori lives in Chdg (pro-A),
which contains strictly Pro- Chdg (A). If C is a strictly bounded complex, let m, n suchthat C = σ≤n C = σ≥−m C, then for r ∈ [−m, n] let F ♥ (Cr ) = “ lim ”i∈Ir (Xr )i . Then one←−
can find a cofinal set I ⊆ Ir for all r such that
Ch(F )(C) = “ lim ”(. . . → (Xr )i → (Xr−1 )i → . . .).
←−i∈I

In particular, Ch(F )(C) ∈ Pro-(Chdg (A)).
Proposition A.8. For all C ∈ Chdg (B), there is an equivalence in S:
MapPro- Chdg (A) (Ch(F )(C)[0], Y ) ' MapChdg (B) (X[0], Chdg (G)(Y )).
Proof. Let Cn = 0 for n ∈/ [−r, s]. The cofiber of the map σ≤s−1 C → C is equivalent
to Cs [s], hence by induction on r + s we are reduced to the case where C = Cs [s] withCs ∈ B, and clearly Ch(F )(Cs [s]) = F (Cs )[s]. Since Pro- Chdg (A) is pointed, by [Lur17,
Remark 1.1.2.8] it is enough to show that for all m,
π0 MapPro- Chdg (A) (F (X)[m], Y ) ' π0 MapChdg (B) (X[m], Chdg (G)(Y )).
Let F (X) = “ lim ”Ti , then for all m we have an isomorphism of abelian groups:←−
HomCh(B) (X[m], Ch(G)(Y )) = lim HomCh(A) (Ti [m], Y ),−→9Notice that we chose the convention for chain complexes, which is different from [ABV09, Lemma
2.1.10]: there the convention is for cochain complexes

DERIVED LOG ALBANESE SHEAVES

53

and since π0 commutes with filtered colimits in S we have by [Lur17, Remark 1.3.1.5,
Remark 1.3.1.11 and Definition 1.3.2.1]:
π0 MapPro- Chdg (A) (F (X)[n], Y ) ∼π MapChdg (A) (Ti [n], Y )
= lim
−→ 0
∼
HomCh(A) (Ti [n + 1], Y ) → lim HomCh(A) (Ti [n], Y )).
= coker(lim−→−→∼= coker(HomCh(B) (X[n + 1], Ch(G)(Y )) → HomCh(A) (X[n], Ch(G)(Y )))∼= π0 MapChdg (B) (X[n], Chdg (G)(Y )).

Remark A.9. A priori, there is no relation between F and LF , but if X is strictly bounded
and BC-admissible, then Proposition A.8 implies that
LF (LB (X)) ' LPro-A Ch(F )(X)[0].
In particular for X ∈ B such that X[0] is BC-admissible,(
F (X) if n = 0
πn (LF (LB (X[0]))) =0otherwise.
Remark A.9 motivates the following definition:
Definition A.10. In the situation of A.4, LF is said to be a pro-left derived functor of
F if for every X ∈ B
π0 LF (X[0]) ∼= F (X).
A.5. BC-admissible resolution. We will fix the setting of A.4. By abuse of notation,
we will say that P ∈ B is BC-admissible if P [0] ∈ Chdg (B) is.
Proposition A.11. Let P• ∈ Chdg (B) be a strictly bounded complex (see Definition A.6)
such that Pn is BC-admissible for all n. Then P• is BC-admissible.
Proof. Up to shift, we can suppose that P• = σ≥0 P• = σ≤n P• for some n ≥ 0: we proceed
by induction on n. If n = 0, then P• = P0 [0], and it is BC-admissible by assumption. Letn > 0 and consider the fiber-cofiber sequence in Chdg (B):σ≤n−1 P• → P• → Pn [n].
For all I ∈ D(A), we conclude by the following diagram in S where the left and rightvertical maps are equivalences by induction:
Map(Pn [0], Chdg (G)(iA I[−n]))

Map(P• , Chdg (G)(iA I))'

Map(Pn [0], iB D(G)(I[−n]))Map(σ≤n−1 P• , Chdg (G)(iA I))'

Map(P• , iB D(G)(I))

Map(σ≤n−1 P• , iB D(G)(I)).


Recall that B is said to be generated by a set of objects E if and only if E is closed
under direct sums and for every X ∈ B there exists a surjective map(A.11.1)

P0 → X → 0

with P0 ∈ E. Suppose that that B is generated by a set of objects E which are BCadmissible. Then let K be the kernel of (A.11.1), so there exists P1 ∈ E together with asurjective map P1  K, hence we have an exact sequence:(A.11.2)

P1 → P0 → X → 0.

By iterating (A.11.2) one can construct a resolution P• → X[0] where Pn ∈ E and Pn = 0
for n < 0. We will call this a connective BC-admissible resolution.

54

FEDERICO BINDA, ALBERTO MERICI AND SHUJI SAITO

Lemma A.12. Suppose that B is generated by a set of objects which are BC-admissible.
For any X ∈ B and any connective BC-admissible resolution P• → X[0], we have that
LF (LB (X[0]) ' lim LPro-A (Ch(F )(σ≤n P• )).
−→n

Proof. Since P• → X[0] is a resolution, we have LB (P• ) ' LB (X[0]). Moreover, since P•
is connective, we have that P• = lim σ≤n P• , and σ≤n P• are BC-admissible by Proposition−→
A.11. Since LF and LB commute with all colimits, by Remark A.9 we conclude that
LF (LB X[0]) ' lim LF (LB σ≤n P• ) ' lim LPro-A (Ch(F )(σ≤n P• )),
−→
−→
n


We can now prove the main theorem of this appendix:
Theorem A.13. In the situation of A.4, suppose that B is generated by a set objects
which are BC-admissible. Then the functor LF is a pro-left derived functor of F .
Proof. Let X ∈ B and P• → X[0] a BC-admissible resolution, in particular P• ∈ D(B)≥0 .
Since LB X[0] ∈ D(B)≥0 and LF is right t-exact, LF (LB X[0]) ∈ Pro-D(A)≥0 , hence
π0 LF (LB X[0])[0] ' τ≤0 LF (LB X[0]).
Since τ≤0 is a left adjoint, it commutes with colimits, so by Lemma A.12 we haveτ≤0 LF (X[0]) ' τ≤0 lim LPro-A Ch(F )(σ≤n P• ) ' lim τ≤0 LPro-A Ch(F )(σ≤n P• )
−→
−→
n

n

where the last colimit is computed in Pro-D(A)≤0 . On the other hand, by definition of
the t-structure on Pro-D(A) we haveτ≤0 LPro-A Ch(F )(σ≤n P• ) ' LPro-A τ≤0 Ch(F )(σ≤n P• ).
For n ≥ 1, we have that τ≤0 Ch(F )(σ≤n P• ) = coker(F (P1 ) → F (P0 ))[0], and since F is aleft adjoint, it preserves cokernels, so
coker(F (P1 ) → F (P0 )) = F (coker(P1 → P0 )) = F (X).
We conclude that in Pro-D(A)≤0 we havelim τ≤0 LPro-A Ch(F )(σ≤n P• ) ' lim LPro-A F (X)[0] ' LPro-A F (X)[0]
−→
−→
n

n

since LPro-A

F (X)[0] ∈ Pro-D(A)♥ we conclude that
π0 LF (X[0]) ' π0 (lim τ≤0 LPro-A F (σ≤n P• )) ' F (X).
−→
n


A.6.

We end this appendix with a criterion of BC-admissibility.

Lemma A.14 (See [ABV09, Lemma 2.1.10]). In the situation of A.4, let P ∈ B suchthat P [0] is compact in Chdg (B). Then P is BC-admissible if and only if for any injective
object I0 ∈ A, ExtiB (P, G(I0 )) = 0 for i 6= 0.
Proof. Suppose first that P is BC-admissible. If I0 ∈ A is injective, then I0 [0] is fibrant,
so I0 [0] = iA LA (I0 [0]). Let I = LB (I0 [0]), then:
ExtnB (P, G(I0 )) = π0 MapChdg (B) (P [0], iB D(G)(I[−n])).
Since P [0] is BC-admissible, we have:
π0 MapChdg (B) (P [0], iB D(G)(I[−n])) = π0 MapChdg (B) (P [0], Chdg (G)(iA I[−n])))= π0 MapChdg (B) (P [0], G(I0 )[−n])).
The last term is zero for n 6= 0, hence ExtnB (P, G(I0 )) = 0 if n 6= 0.

DERIVED LOG ALBANESE SHEAVES

55

Let us now show the converse implication. We need to show that for every Y ∈ D(A)the Beck-Chevalley map induces an equivalence:
MapChdg (B) (P [0], iB D(G)(Y )) ' MapChdg (B) (P [0], Chdg (G)(iA Y )).
Let I := iA (Y ), so σ≤−n σ≥m I is a strictly bounded complex of injectives; since P [0] iscompact and the colimit in (A.5.1) is filtered, Lemma A.15 below implies that:(A.14.1)
MapChdg (B) (P [0], Chdg (G)(I)) ' lim(lim MapChdg (B) (P [0], σ≤−n σ≥m Chdg (G)(I)))
−→ ←−
n

m

' lim(lim MapChdg (B) (P [0], Chdg (G)(σ≤−n σ≥m I)))
−→ ←−
n

m

' lim(lim MapChdg (B) (P [0], iB D(G)(LA σ≤−n σ≥m I))
−→ ←−
n

m

' MapChdg (B) (P [0], lim lim iB D(G)(LA σ≤−n σ≥m I)).
−→ ←−
n

m

Next, observe that, for any n and m, the map Chdg (G)(σ≤−n σ≥m I) → iB D(G)(LA σ≤−n σ≥m I)
is a fibrant replacement (see Remark A.4) of bounded complexes, which is given by the
total complex of injective resolutions of each G(Ir ) → Jr• . So, we have that
(A.14.2)
lim iB D(G)(LA σ≤−n σ≥m I) ' lim Totr∈[−n,m] (Jr• ) = Totr≥−n (Jr• ) ' iB D(G)(LA σ≤−n I).
←−
←−
m

m

Since iB commutes with filtered colimits by assumption, (A.14.1) and (A.14.2) imply:(A.14.3) MapChdg (B) (P [0], Chdg (G)(iA Y )) ' MapChdg (B) (P [0], iB lim D(G)(LA σ≤−n I)).
−→n

For every q ∈ Z, we have that for n  q:πq D(G)(Y ) ∼= G(πq Y ) ∼= G(πq σ≤−n I) ∼= πq D(G)(LA σ≤−n I),so since homotopy groups commute with filtered colimits:(A.14.4)πq lim D(G)(LA σ≤−n I) ∼π D(G)(LA σ≤−n I) ∼= lim
= πq D(G)(Y )
−→
−→ q
n

n

The proof follows then from (A.14.3) and (A.14.4).

Lemma A.15. Let P ∈ B such that for any injective object I0 ∈ A, ExtiB (P, G(I0 )) = 0
for i 6= 0. Then for any strictly bounded complex I b ∈ Ch(A) of injective objects of A:
MapChdg (B) (P [0], Chdg (G)(I b )) ' MapChdg (B) (P [0], iB D(G)(LA I b )).
Proof. Let Inb = 0 for n ∈/ [−r, s]. The cofiber of σ≤s−1 I b → I b is equivalent to Isb [s]: byinduction on r + s we reduce to I b = Is [s] with Is an injective object of B. We conclude:(0
if n 6= ss−nπn MapChdg (B) (P [0], iB D(G)(LA Is [s])) ∼= ExtB (P, G(Is )) =HomB (P, G(Is )) if n = s.

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Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via
Cesare Saldini 50, 20133 Milano, Italy
E-mail address: federico.binda@unimi.it
Institut für Mathematik, Universität Zurich, Winterthurerstr. 190, CH-8057 Zürich,
Switzerland
E-mail address: alberto.merici@math.uzh.ch
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo
153-8941, Japan
E-mail address: sshuji@msb.biglobe.ne.jp