MOTIVES WITH MODULUS, III:THE CATEGORIES OF MOTIVESBRUNO KAHN, HIROYASU MIYAZAKI, SHUJI SAITO,
AND TAKAO YAMAZAKI
Abstract. We construct and study a triangulated category of
motives with modulus MDMeff
gm over a field k, that extends Voeff
evodsky’s category DMgm in such a way as to encompass nonhomotopy invariant phenomena. In a similar way as DMeff
gm is
eff
constructed out of smooth k-varieties, MDMgm is constructed outof proper modulus pairs, introduced in Part I of this work. To sucha modulus pair we associate its motive in MDMeff
gm . In some cases,
betweenthemotivesof two modulus
the Hom group in MDMeffgmpairs can be described in terms of Bloch’s higher Chow groups.

Contents
Introduction
1. Review of modulus pairs
2. The tensor structure on modulus pairs
3. Motives with modulus
4. Brown-Gersten property
5. The derived Suslin complex
6. Comparisons
7. Some computations
Appendix A. Categorical toolbox, III
Appendix B. Cubical objects and intervals
References

1
6
12
16
21
29
31
39
44
48
59

Date: July 23, 2021.
2010 Mathematics Subject Classification. 19E15 (14F42, 19D45, 19F15).
The first author acknowledges the support of Agence Nationale de la Recherche(ANR) under reference ANR-12-BL01-0005. The second author is supported by
RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program, and by
JSPS KAKENHI Grant (19K23413). The third author is supported by JSPS KAKENHI Grant (15H03606). The fourth author is supported by JSPS KAKENHI
Grant (15K04773).
1

2

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Introduction
In this paper, we construct triangulated categories of “motives withmodulus” over a field k, in parallel with Voevodsky’s construction oftriangulated categories of motives in [44]. Our motivation comes fromthe reciprocity sheaves studied in [14]; the link between the present
theory and [14] is established in [19].
Let Sm be the category of smooth separated k-schemes of finite type.
Voevodsky’s construction starts from an additive category Cor, whose
objects are those of Sm and morphisms are finite correspondences. The
category of effective geometric motives DMeff
gm is then defined to be thepseudo-abelian envelope of the localisation of the homotopy category
K b (Cor) of bounded complexes by two types of “relations”:(HI): [X × A1 ] → [X], X ∈ Cor;(MV): [U ∩ V ] → [U ] ⊕ [V ] → [X], X, U, V ∈ Cor;
where, in the latter, U t V → X ranges over all open covers of X.
VThis makes DMeff
gm a tensor triangulated category. We denote by M
the canonical functor Sm → DMeff
gm . The following fundamental result
computes some Hom groups in concrete terms:
Theorem 1 ([5, 6.7.3] and [45, Cor. 2]). Assume that k is perfect.
For X, Y ∈ Sm, with X proper of dimension d and j ∈ Z, there is a
canonical isomorphism
HomDMeff(M V (Y )[j], M V (X)) ' CH d (Y × X, j)
gm
where the right hand side is Bloch’s higher Chow group. In particular,
this group is 0 for j < 0 and isomorphic to CH d (Y × X) for j = 0.
In the present work, we construct a tensor triangulated category
MDMeffgm in a parallel way. The category Cor is replaced by a category
MCor whose objects are modulus pairs, which only played an auxiliaryrôle in [14]: this category has been studied in [17]. A modulus pairM = (M , M ∞ ) consists of a proper k-variety M and an effective Cartierdivisor M ∞ such that M −|M ∞ | ∈ Sm. A morphism from (M , M ∞ ) to
(N , N ∞ ) is a finite correspondence from M − |M ∞ | to N − |N ∞ | which
satisfies a certain inequality on Cartier divisors (Definition-Proposition
1.1.2).
The category MCor enjoys a symmetric monoidal structure (Definition 2.1.1). The right object replacing A1 in this context turns out
to be its minimal compactification(0.1)

 = (P1 , ∞),MOTIVES WITH MODULUS, III3

the compactification of A1 ' P1 −{∞} with reduced divisor at infinity.
This provides an analogue of (HI):(CI): [M ⊗ ] → [M ], M ∈ MCor.
To introduce an analogue1 of (MV), we use the cd-structure on
MCor which was introduced in [29] and developed in [18]. This yields
a Mayer-Vietoris condition in K b (MCor) (§3.1). We may then define a tensor triangulated category MDMeff
gm in a similar fashion to
Voevodsky (Definition 3.1.1), with a “motive” functor M : MCor →eff
MDMeffgm . We can also compute the Hom-groups of MDMgm :Theorem 2 (see Theorem 5.2.4). For any X , Y ∈ MCor and i ∈ Z,
we have an isomorphism(0.2) HomMDMeff(M (X ), M (Y)[i]) '
gm

lim
−→
fin
0

X ∈Σ

0

HiNis (X , RC∗ (Y)X 0 ).
↓X

Here Σfin ↓ X denotes a certain category of morphisms X 0 → X
which (in particular) induce isomorphisms on the interiors (see Theorem 1.2.3), RC∗ (Y) is the derived Suslin complex of the modulus pair0
Y (see Definition 5.2.3), and RC∗ (Y)X 0 denotes its restriction to X Nis(see Definition 4.2.4). Briefly, RC∗ (Y) is defined like the Suslin complex, with three differences: a) we use  instead of A1 ; b) we use
a cubical version instead of Suslin-Voevodsky’s simplicial version (seeRemarks 5.2.5 and B.2.6 for an important comment on this point); c)
we use derived internal Homs instead of classical internal Homs.
Recall that a key technical tool of Voevodsky for proving Theorem 1
eff
is to embed DMeffof motivic
gm into a larger triangulated category DM
effcomplexes. The situation is similar here: MDMgm is embedded intoa category MDMeff . This is how the derived Suslin complex RC∗ (Y)arises.
On the other hand, there is a canonical “forgetful” functor ω : MCor
→ Cor sending (X, X ∞ ) to X − |X ∞ |, whence a comparison between
our theory and Voevodsky’s. This is summarised in the following diagram, assuming k perfect:
M

(0.3)

ι

MCor −−−→ MDMeff−−−→ MDMeff

 gm




ωeff,gm yωeff y
ωy
Cor

MV

−−−→

DMeff
gm

1This is not quite accurate: see §3.1.

ι−−−→ DMeff ,4

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

in which the functors denoted ι are fully faithful. The yoga of Thomason-Neeman of compactly generated categories [31] shows that ωeff has
a right adjoint ω eff : DMeff → MDMeff , and we have:
Theorem 3 (See Th. 6.3.1 and Cor. 6.3.4). a) Let X be a smoothproper k-variety. Then ω eff M V (X) = M (X, ∅).
b) If p is the exponential characteristic of k, then ω eff (DMeff
gm [1/p]) ⊂eff
eff
MDMgm [1/p]. The functor ω is symmetric monoidal.
Note that ω eff is fully faithful (Propositions 6.1.1 and 6.1.2). As acorollary, we get the following partial analogue of Theorem 1:
Theorem 4 (See Cor. 6.3.6). Suppose k is perfect. Let X be a smoothproper k-variety of dimension d. We have a canonical isomorphism:
HomMDMeff(M (Y)[j], M (X, ∅)) ' CH d ((Y − |Y ∞ |) × X, j)
gmfor any modulus pair Y = (Y, Y ∞ ) and j ∈ Z.
Though we consider proper modulus pairs in the above, we can also
construct a theory of motives with modulus for pairs (X, X ∞ ) with
X not necessarily proper. They come with a reasonable topology (see[17]). By a similar construction to the one above, this leads to trieff
angulated categories MDMeffof motives with modulus for
gm , MDM
non-proper modulus pairs. Even though MDMeff seems to be the central object of interest (e.g. it is the sheaf theory on proper modulus
pairs which has a relationship with reciprocity sheaves in [19]), it is
impossible to avoid developing MDMeff at the same time. This leads
to a regrettably heavy exposition, for which we apologize to the reader.
Besides, the non-proper version is used in [35] as an important technicaltool.
Relationship with previous work. This work completes the revision of [15], which was started in [17] and [18]. For the readers aware ofeff
are the same as
this first version, the categories MDMeff
gm and MDM
in [15], as well as their non-proper versions. (The constructions given
here are different and simpler.) The only difference with the results of[15] is in Theorem 2, where the formula for the Hom group given in[15] missed the direct limit.
Perspectives. The story of motives with modulus does not stop here:
there are many further things to explore, some being explored now. We
quote a few:
• Extend more of Voevodsky’s results to this modulus context.
See already Section 7 (and the references therein), and [35]. In

MOTIVES WITH MODULUS, III

5particular, extend Voevodsky’s theorem on strict A1 -invariance[43, Th. 5.6] to the -invariant context. See Example B.7.5, as
well as [17, Question 1], and [35, Th. 0.6] for a partial result.
• A comparison with the category of log-motives of Binda-ParkØstvær [7]; see already [36].
• Versions for other topologies, notably topologies related to theétale topology on schemes. The theory developed here is intrinsically restricted to the Nisnevich topology, via the theoryof cd-structures. One may think of the model-theoretic techniques of [7] — as soon as one has guessed the right definitionof topologies on modulus pairs!
• A -homotopy theory of modulus pairs similar to that of [30].
It should be easy to develop from the material here.
• Contrary to DMgm , there is evidence that the category MDMgm
obtained by ⊗-inverting the Tate object is not rigid, see Proposition 6.5.4. Another fact is that many cohomology theories,
starting with higher Chow groups with modulus, satisfy, notthe modulus condition, but its opposite. This suggests that one
should construct an even larger category based on “modulus
triples” (two Cartier divisors at infinity with opposite modulus
conditions). Work in this direction has been made by Binda[6] in the context of -homotopy theory (as above), and by
Ivorra-Yamazaki [11, 12] in the additive context.
• Of course, develop the various theories over a base.
Acknowledgements. Part of this work was done while three of theauthors stayed at the University of Regensburg, supported by the SFB
grant “Higher Invariants”. Another part was done in a Research in trioin CIRM, Luminy. Yet another part was done while the fourth author
was visiting IMJ-PRG, supported by Fondation Sciences Mathématiques de Paris. We are grateful to the support and hospitality received
in all places.
Independently of our work, Moritz Kerz conjectured in [24] the existence of a category of motives with modulus and gave a list of expected
properties. His conjectures are closely related to our construction. We
thank him for communicating his ideas, which were very helpful to us.
We thank Joseph Oesterlé for his help in the proof of Lemma B.3.2,
Joseph Ayoub for Remarks 4.2.2 and 5.2.5, and Shane Kelly for useful
discussions on §4.4. Thanks are also due to the referees for their carefulreading and valuable comments.
We would like to dedicate this work to the memories of Michel Raynaud and Laurent Gruson, whose article [33] was essential for developing this theory (as it was previously for that of Voevodsky).

6

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Organisation of this paper. In Section 1, we review part of our previous work on modulus pairs which will be used here. (More reminders
are inserted in the sequel at appropriate places.) In Section 2, we introduce a new ingredient: the tensor structure. In Section 3, we give anelementary construction of the triangulated categories of motives withmodulus in the spirit of [20, §4.2], and prove their basic properties in
Theorem 3.3.1. In Section 4, we describe MDMeff and MDMeff interms of sheaves with transfers: this yields in particular a first computation of Hom groups in terms of Nisnevich cohomology (Corollary
4.2.5). In Section 5, we use the theory of intervals from Appendix B toreformulate this computation in more concrete terms (Theorem 5.2.4);
we also prove that the natural functor MDMeff → MDMeff is fully
faithful (Th. 5.2.2). In Section 6, we compare our categories with
Voevodsky’s categories, as well as with the category of Chow motives.
The most important result there is Theorem 6.3.1. In Section 7, we
prove various results on MDMeff and MDMeff similar to those of Voevodsky in [44], and include for the reader’s convenience some whichwere proven by other authors.
There are two appendices. Appendix A gathers new technical categorical results. Appendix B is central to the results of Section 5: it
generalises Voevodsky’s theory of intervals (in a cubical version) to thecase of symmetric monoidal categories. Its most important results are
Theorems B.4.5 and B.7.2 (the latter is used in the proofs of Theorems
5.2.2 and 6.3.1).
Notation and terminology. Throughout this paper, we fix a basefield k. We denote by Sch the category of separated schemes of finite
type over k, and by Sm the full subcategory of Sch consisting of thoseobjects which are smooth over k. For any Cartier divisor D on a scheme
X, the support of D is denoted by |D|. We write Cor for Voevodsky’s
category of finite correspondences [44].
Let Sq be the square of the category {0 → 1}, depicted as
00

/ 01




/ 11.

10

For any category C, denote by C Sq for the category of functors from
Sq to C. A functor f : C → C 0 induces a functor f Sq : C Sq → C 0 Sq .
A ⊗-category is a unital symmetric monoidal category; a ⊗-functor
F between ⊗-categories is a strong unital symmetric monoidal functor

MOTIVES WITH MODULUS, III

7(this means that the maps F (A) ⊗ F (B) → F (A ⊗ B) are all isomorphisms).
1. Review of modulus pairs
1.1. The categories MCor, MCor, MSm and MSm.
Definition 1.1.1. A modulus pair is a pair M = (M , M ∞ ) such that
M ∈ Sch, M ∞ is an effective Cartier divisor on M , and M o := M −|M ∞ | ∈ Sm. We call M , M ∞ and M o the ambient space, modulus andinterior of M , respectively. A modulus pair M is called proper if its
ambient space M is proper over k.
The ambient space M is reduced for any modulus pair M [17, Rem.
1.1.2 (3)].
As in Voevodsky’s theory [44], we define two types of categories withthe same objects out of modulus pairs. One, analogous to Sm, is used
as a support for Grothendieck topologies. The other, analogous to Cor,is used to define a transfer structure on (pre)sheaves. We start withthe latter:
Definition-Proposition 1.1.2. Let M and N be modulus pairs. An
elementary modulus correspondence α : M → N is an elementary correspondence αo : M o → N o between the interiors which satisfies thefollowing properties:
Properness condition: Let α be the scheme-theoretic closure of
αo in M × N . Then α is proper over M . (This is automatic if
N is a proper modulus pair.)
Modulus condition: Let ν : αN → M × N be the compositionof the normalisation αN → α with the inclusion α ,→ M × N .
Then we have ν ∗ (M ∞ ×N ) ≥ ν ∗ (M ×N ∞ ) (inequality of Cartierdivisors). We say α is admissible if this condition is satisfied.
A modulus correspondence α : M → N is a formal Z-linear sum ofelementary modulus correspondences. By [17, Def. 1.3.1], the composition of modulus correspondences (regarded as finite correspondences[44, §2.1]) is again a modulus correspondence. Therefore, modulus pairs
and modulus correspondences define the category of modulus correspondences, denoted by MCor. The full subcategory of MCor consistingof proper modulus pairs is denoted by MCor.
Definition 1.1.3. We denote by MSm the category whose objects
are modulus pairs and in which a morphism f : M → N is a morphism f o ∈ Sm(M o , N o ) whose graph defines an elementary modulus
correspondence as in Definition 1.1.2; we write MSm for the full subcategory of MSm consisting of proper modulus pairs.

8

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

There is a commutative diagram of natural functors/ MCor

c

MSm
ωs

(1.1)

$

ω

τs



MSm

ωs

Sm
;

cV

/ Cor
e

y
τ
ω

c


/ MCor .

Here, the vertical functors τ and τs are the full embeddings mentioned above, and the horizontal functors c and c are graph functors
similar to Voevodsky’s graph functor cV ; the diagonal functors are all
induced by M 7→ M o (retain the interior). By [17, Lemma 1.5.1, Th.
1.5.2 and Prop. 1.10.4] (see also [29, Cor. 2.2.5]), we have the following
important results:
Theorem 1.1.4. The functors ω, ω and τ have pro-left adjoints ω ! , λand τ ! , given respectively byω ! X = ” lim ”M ∈Σ↓X M, λX = (X, ∅), τ ! M = ” lim ”N ∈Comp(M ) N←−←−(λ is an honest left adjoint), where Σ = {u | ω(u) is an isomorphism}(it admits a calculus of right fractions), and Comp(M ) is the category whose objects are pairs (N, j) consisting of a modulus pair N =(N , N ∞ ) ∈ MSm equipped with a dense open immersion j : M ,→ Nsuch that N ∞ = MN∞ + C for some effective Cartier divisors MN∞ , C
on N satisfying N − |C| = j(M ) and j ∗ N ∞ = M ∞ .
Similarly for ωs , ω s and τs (same formulas).
Theorem 1.1.5. Let f1 : M1 → N and f2 : M2 → N be two morphisms
in MSm (resp. MSm), and assume that M1o ×N o M2o belongs to Sm.
Then the fibre product M1 ×N M2 exists in MSm (resp. in MSm).
(In other terms, ωs and ω s reflect fibre products.) Moreover, MSm
and MSm have the final object 1 := (Spec k, ∅). In particular, finite
products exist in these categories.
1.2. The category MSmfin . There is another category, which plays
a technically important rôle:Definition 1.2.1. We write MSmfin for the subcategory of MSm
with the same objects and such that f ∈ MSm(M, N ) belongs to
MSmfin (M, N ) if and only if the rational map M → N defined by f o
is a morphism. We say that f is ambient.
We write bs : MSmfin → MSm for the inclusion functor.

MOTIVES WITH MODULUS, III

9Note that, for an ambient morphism f , the properness condition istrivial and the modulus condition simplifies to∗ν ∗M ∞ ≥ ν ∗f N ∞.

(1.2)
where ν : M

N→ M is the normalisation.

Definition 1.2.2. We say that f is minimal if there is equality in(1.2).
By [17, Prop. 1.9.2], we have
Theorem 1.2.3. Let Σfin be the class of minimal morphisms f : M →N in MSmfin such that f o : M o → N o is an isomorphism in Sm andf : M → N is proper. Then Σfin admits a calculus of right fractions,any morphism in Σfin becomes invertible in MSm and the induced functor (Σfin )−1 MSmfin → MSm is an equivalence of categories.
Remark 1.2.4. Contrary to Theorem 1.1.5, fibre products (or evencartesian products) are not representable in MSmfin in general.
1.3. Topologies on modulus pairs. The categories MSm and MSm
have Grothendieck topologies which are derived from the Nisnevich
topology on Sm. We recall their definitions from [17] and [29].
First, we consider the “naive” topology on MSmfin .
Definition 1.3.1. An MVfin -cover is an ambient morphism f : U → M
in MSmfin which is minimal and such that the underlying morphism
f : U → M in Sch is a Nisnevich cover. The Grothendieck topology
on MVfin generated by MVfin -covers is called the MVfin topology.
There is another characterisation of this topology from the “cdstructure” point of view (see [46] for the definitions and properties
of cd-structures).
Definition 1.3.2. Let PMVfin be the cd-structure on MSmfin consistingof commutative squares in MSmfin
W

/V





W

/V




/M

/M
U
such that all arrows are minimal and the underlying square of schemesU10

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

is a distinguished Nisnevich square. (We always take U → M to be an
open immersion.) An element of PMVfin is called an MVfin -sqaure.
By “transport of structure” of the Nisnevich cd-structure, which iscomplete, regular and bounded in the sense of [46], we obtain
Proposition 1.3.3. The MVfin topology coincides with the topology associated with the cd-structure PMVfin . Moreover, the latter is complete,regular and bounded.
Next, we introduce topologies on MSm and MSm. The former iseasily defined as follows.
Definition 1.3.4. Let PMV be the “smallest” cd-structure on MSm
which contains the images of all squares in PMVfin under the functor
bs : MSmfin → MSm of Theorem 1.2.3. In other words, a commutative square in MSm belongs to PMV if and only if it is isomorphic
in MSmSq to the image of an MVfin -square (see Notation for thedefinition of Sq). An element of PMV is called an MV-square.
The Grothendieck topology on MSm associated with the cd-structure
PMV is called the MV topology.
Remark 1.3.5. For any MVfin -square or MV-square S, the square S o in
Smo is a distinguished Nisnevich square.
By [17, Prop. 3.2.2], we have
Theorem 1.3.6. The cd-structure PMV is complete and regular.
The definition of the topology on MSm is a bit tricky. It is designed
to satisfy completeness and regularity, and to be compatible with the
MV topology. First we need to recall the “off-diagonal functor” from
[29].
Definition 1.3.7. Let MEt be the category whose objects are morphisms f : U → M in MSm such that f o is étale, a morphism fromf : U → M to g : U 0 → M 0 being given by a pair of morphismss : U → U 0 , t : M → M 0 which are compatible with f and g and suchthat so and to are open immersions. Let MEt be the full subcategoryof MEt consisting of those f : M → N with M, N ∈ MSm.
By [29, Th. 3.1.3], we have
Proposition 1.3.8. There exists a functor OD : MEt → MSm together with a natural isomorphism∼

U t OD(f ) −→ U ×M U,

MOTIVES WITH MODULUS, III11

for each (f : U → M ) ∈ MEt, where t denotes coproduct and the righthand side denotes the fiber product in MSm. This functor restricts toa functor MEt → MSm.
Definition 1.3.9. Let PMV be the cd-structure on MSm consisting of
those commutative squares T of the form
T (00)(1.3)/ T (01)pqT (10)

/ T (11)which satisfy the following properties:(1) T is cartesian in MSm.
(2) There exists an MV-square S and a morphism ι : S → T in
MSmSq such that ι(11) : S(11) → T (11) is an isomorphism in
MSm, and ι(ij)o : S(ij)o → T (ij)o is an isomorphism in Sm
for any (ij) ∈ Sq. In particular, the square S o ∼= T o in SmSq
is a distinguished Nisnevich square.
(3) The morphism OD(q) → OD(p), which is induced by the funtoriality of OD, is an isomorphism.
An element of PMV is called an MV-square. The topology associatedwith the cd-structure PMV is called the MV topology.
Example 1.3.10. Let X be proper and let D, D1 , D2 , D0 be effective
Cartier divisors on X such thatX − D is smoothD ≤ Di ≤ D0
|D1 − D| ∩ |D2 − D| = ∅D0 − D2 = D1 − D.
Then

(X, D0 ) −−−→ (X, D1 )




T =
y
y

(X, D2 ) −−−→ (X, D)
is an MV-square. Indeed, (1) holds by [17, Lemma 1.10.1 and Prop.
1.10.4]. Let S(01) = X − |D1 − D|, S(10) = X − |D2 − D|, S(00) =X − |D0 − D| and S(ij)∞ = j(ij)∗ T (ij)∞ where j(ij) is the inclusion
S(ij) ,→ X. This yields a square S as in (2), and (3) is trivial since T o
is a Zariski square.
By [29, Th. 4.3.1, 4.4.1], we have

12

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Theorem 1.3.11. The cd-structure PMV is complete and regular.
(Condition (3) in Definition 1.3.9 is crucial for the proof of regularity.)
We now recall the main result of [16], its Theorem 1.5.6. It will
be used in the proof of Theorem 4.1.1 (see (iv) in §4.5). Recall from
[16, Def. 1.5.3] that the category Comp(M ) of Theorem 1.1.4 can beextended to squares of modulus pairs.
Theorem 1.3.12. Let S ∈ MSmSq , and let CompMV (S) denote thefull subcategory of Comp(S) consisting of those T which are MVsquares. Then CompMV (S) is cofinal in Comp(S).
We shall need the following lemma in the proof of Theorem 3.3.1 (4)below.
Lemma 1.3.13. Let T ∈ MSmSq , verifying Conditions (1) and (3) of
Definition 1.3.9. Assume that p has a section sp . Then q has a section
sq , one can write T (00) = sq (T (10)) t T 0 (00), T (01) = sp (T (11)) t
T 0 (11), and the morphism T (00) → T (01) induces an isomorphism∼
u : T 0 (00) −→ T 0 (01).
Proof. The section sq is obtained from sp because T is cartesian (property (1)). The decompositions exist because po and q o are étale (cf.
[29, proof of Th. 3.1.3]). The morphism u exists by construction; itremains to see that it is an isomorphism. But an easy computation
provides decompositionsaa
OD(q) ' T 0 (00)T 0 (00)
OD(T 0 (00) → T (10))aaOD(p) ' T 0 (01)T 0 (01)
OD(T 0 (01) → T (11))∼

respected by the isomorphism OD(q) −→ OD(p) (property (3)). This
yields the conclusion.

2. The tensor structure on modulus pairs
2.1. Definition. Recall that the tensor structure on Voevodsky’s category DMeff
gm comes from the product of smooth varieties. However, itturns out that the product of Theorem 1.1.5 cannot be used to coneff
struct categories MDMeff
with good properties. The
gm and MDM
basic reason is that the product morphism A1 × A1 → A1 used by
Voevodsky to define an interval structure on A1 does not define a morphism  ×  → , where  = (P1 , ∞) (see Remark 5.1.2). This andother considerations lead us to introduce another tensor structure:

MOTIVES WITH MODULUS, III

13

Definition 2.1.1. For two modulus pairs M and N , we define theirtensor product M ⊗ N to be
M ⊗ N := (M × N , M ∞ × N + M × N ∞ ).
Remark 2.1.2. If we pull back the modulus of Definition 2.1.1 by the
projection p : BlM ∞ ×N ∞ (M × N ) → M × N , we get an isomorphic
modulus pair with modulus p∗ (M ∞ × N + M × N ∞ ) = p# (M ∞ ×
N + M × N ∞ ) + 2E, where p# denotes proper transform and E isthe exceptional divisor. By contrast, the cartesian product M × N isrepresented by (BlM ∞ ×N ∞ (M × N ), p# (M ∞ × N + M × N ∞ ) + E):
this is a special case of [17, Prop. 1.10.4 (3)] and its proof. See also
Remark 2.1.4 below.
Definition 2.1.1 provides the categories MSm, MSm, MCor, MCorand MSmfin of Definitions 1.1.2, 1.1.3 and 1.2.1 with symmetric monoidal structures with unit 1 = (Spec k, ∅), for which the various functors
between them are ⊗-functors. To see this, we have to check:
Lemma 2.1.3. Let f ∈ MCor(M1 , N1 ) and g ∈ MCor(M2 , N2 ).
Consider the tensor product correspondence f ⊗g ∈ Cor(M1o ×M2o , N1o ×
N2o ). Then f ⊗ g ∈ MCor(M1 ⊗ M2 , N1 ⊗ N2 ).
Proof. We may assume that f and g are given by integral cycles Z ⊂M1o × N1o and T ⊂ M2o × N2o . Then f ⊗ g is given by the product cycleN
Z × T . Let Z → Z be the normalizations of the closures Z of Z, andN
similarly for T → T . By hypothesis, we have(pZ1 )∗ M1∞ ≥ (pZ2 )∗ N1∞ ,where pZ1 is the composition Z
for pZ2 , pT1 , pT2 . Hence:N(pT1 )∗ M2∞ ≥ (pT2 )∗ N2∞ ,→ Z ⊂ M 1 × N 1 → M 1 , and likewiseN

N(pZ1 ×pT1 )∗ (M1∞ ×M 2 +M 1 ×M2∞ ) = (pZ1 )∗ M1∞ ×T +Z ×(pT1 )∗ M2∞
N

N≥ (pZ2 )∗ N1∞ ×T +Z ×(pT2 )∗ N2∞ = (pZ2 ×pT2 )∗ (N1∞ ×N 2 +N 1 ×N2∞ ).
We conclude that Z × T is admissible, because the projection (Z ×N
NT )N → Z × T factors through Z × T . Finally, Z × T is obviously
proper over M 1 × M 2 .

To conclude checking that we have indeed defined symmetric monoidal structures, we need to verify identities of the form f ⊗ (g ◦ h) =(f ⊗ g) ◦ (f ⊗ h), and to define associativity, commutativity and unit
constraints. The first point holds because it holds in Cor and Sm; forthe second one, we leave to the reader the pleasure to check that the14

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

constraints of the symmetric monoidal structure on Cor are proper andadmissible, hence induce similar ones on MCor, etc., which enjoy thecorrect identities. The (symmetric) monoidality of the various functors
is tautological.
Remark 2.1.4 (see also Remark 2.1.2). Given two modulus pairs M, N ,the canonical morphisms M → 1, N → 1 give morphisms M ⊗ N →M ⊗ 1 = M , M ⊗ N → 1 ⊗ N = N . The universal property of the
product then yields a natural transformation(2.1)

M ⊗ N → M × N.

Take M = N . The right hand side comes with a diagonal morphism
M → M ×M , corresponding to the diagonal morphism ∆ : M o → M o ×
M o (indeed, products commute with ω s and ωs since they have pro-left
adjoints). If (2.1) were an isomorphism, ∆ would define a morphism
M → M ⊗ M ; but Definition 2.1.1 and the modulus condition showthat this happens if and only if M ∞ = 0. Conversely, it can easily beshown that (2.1) is an isomorphism if M ∞ = 0 or N ∞ = 0.
Proposition 2.1.5. All functors in Diagram (1.1) are symmetric monoidal. This also applies to the functors of Theorem 1.1.4, and to bs in
Definition 1.2.1.
Proof. For Diagram (1.1) and bs , it is easily deduced from the construction of the functors. For the functors ω ! and ωs! , recall from [17, Def.
1.4.1] the functors (−)(n) given by M (n) = (M , nM ∞ ). Clearly, (−)(n)is monoidal. On the other hand, by [17, Lemma 1.7.4 b)], an inverse
system defining ω ! X for X ∈ Sm is given by (M (n) )n≥1 for any M suchthat ω(M ) = X; this proves the claim in this case, and similarly forωs! .
We now show the monoidality of τ ! , arguing as in the case of ω !
(although we cannot quite use the functors (−)(n) here). Let M ∈MCor. We use the category Comp(M ) of Theorem 1.1.4. TakeN ∈ Comp(M ) and write N ∞ = MN∞ + C as in loc. cit. DefineComp(N, M ) as the full subcategory of Comp(M ) consisting of thoseP such that P = N (compatibly with the open immersions M ,→ N ,M ,→ P ) and P ∞ = MN∞ + nC for some n > 0. (Strictly speaking,Comp(N, M ) depends on the choice of the decomposition N ∞ = MN∞ +C.) The proof of [17, Claim 1.8.4] shows that Comp(N, M ) is cofinal
in Comp(M ). If M 0 ∈ MCor is another object and N 0 ∈ Comp(M 0 )000with a decomposition N 0 ∞ = M 0 ∞N 0 +C , then N ⊗N ∈ Comp(M ⊗M )00
as (N ⊗ N 0 )∞ = (MN∞ × N 0 + N × M 0 ∞N 0 ) + (C × N + N × C ), and itMOTIVES WITH MODULUS, III15

is easy to see that the obvious functor
Comp(N, M ) × Comp(N 0 , M 0 ) → Comp(N ⊗ N 0 , M ⊗ M 0 )is cofinal. The same proof applies to τs! .



2.2. Tensor product and cd-structures. The following lemma will
be used later to define tensor structures for motives with modulus (seeTheorem 3.3.1 (3)).
Lemma 2.2.1.
(1) If S ∈ MSmSq is cartesian, so is S ⊗ M for any M ∈ MSm.
(2) If T ∈ MSmSq is cartesian, so is S ⊗ M for any M ∈ MSm.
(3) If S is an MVfin -square, then so is S ⊗M for any M ∈ MSmfin .
(4) If S is an MV-square, then so is S ⊗ M for any M ∈ MSm.
(5) If T is an MV-square, then so is T ⊗ M for any M ∈ MSm.
Proof. (1): By the construction of fiber products in [29, §2.2], we may
assume that the square S is of the form
L
S:

p1

p2



M1

/ M2
f2



f1

/N

where all arrows are ambient, E := p∗1 M1∞ ×L p∗2 M2∞ is an effective
Cartier divisor on L, and L∞ = p∗1 M1∞ + p∗2 M2∞ − E.
Set S 0 := S ⊗ M and write
L0
0

S :

p01

p02

2



M10

/ M0


f10

f20

/ N 0,

where the arrows are obviously ambient, and there is a natural morphism S 0 → S in (MSmfin )Sq . In particular, we have an ambientmorphism π : L0 → L.
0∗
0∞
0∞
0
(1): Set E 0 := p0∗
1 M1 ×L0 p2 M2 . Then E is an effective Cartier0
divisor on L . Indeed:
0

Claim 2.2.2. E 0 = π ∗ E + L × M ∞ .
0∗∞
Proof. Let I1 , I2 , I be the ideals of definition of p0∗
1 (M1 ×M ), p2 (M 1 ×0
M ∞ ), L × M ∞ , respectively. Then the ideal of definition of E 0 is given
by I · I1 + I · I2 = I · (I1 + I2 ). Since E = p∗1 M1∞ ×L p∗2 M2∞ by definition,
I1 + I2 is the ideal of definition of π ∗ E, hence the claim.
16

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

This also shows the following: by definition we have L0∞ = L∞ ×0
M + L × M ∞ . Thus we obtain:0L0∞ = L∞ × M + L × M ∞0= (p∗1 M1∞ + p∗2 M2∞ − E) × M + L × M ∞
0

0∞
0∗
0∞
∗
∞
= p0∗
1 M1 + p2 M2 − π E − L × M
0
0∞
0∞
0∗
= p0∗
1 M1 + p2 M2 − E ,which implies that S 0 is cartesian.
(2): This is a direct consequence of (1).
(3): This is obvious by the definition of MVfin -squares.
(4): Let S be an MV-square. By definition, there exists an MVfin square S 0 which is isomorphic to S in MSmSq . Then we have M ⊗ S ∼
=
Sq
fin
0
0
M ⊗ S in MSm , and M ⊗ S is an MV -square by (3).
(5): Since T is an MV-square, it is cartesian, there exist an MVsquare S and a morphism S → T in MSmSq such that S(11) = T (11)and, finally, OD(q) ∼= OD(p) (see (1.3) for the notation). Set S 0 :=S ⊗ M and T 0 := T ⊗ M . Then T 0 is cartesian by (1), S 0 is an MVsquare by (4), and S 0 (11) = T 0 (11).
It remains to show that OD(q 0 ) ∼= OD(p0 ), where p0 := p ⊗ M , q 0 :=q ⊗ M . For this, it suffices to show that for any (f : U → N ) ∈ MEt,
we have OD(f ) ⊗ M ∼= OD(f ⊗ M ).
We use a similar argument to the one in the proof of [29, Prop. 3.1.4].
Set f 0 := f ⊗ M, U 0 := U ⊗ M, N 0 := N × N . By construction of OD,
we have canonical isomorphisms
U 0 t OD(f 0 ) ∼= U 0 ×N 0 U 0 ,U t OD(f ) ∼= U ×N U.
Therefore, we obtainU 0 t (OD(f ) ⊗ M ) ∼= U 0 t OD(f 0 ),= (U ×N U ) ⊗ M ∼= † U 0 ×N 0 U 0 ∼where ∼=† follows from (1). It is easy to see that this isomorphism
restricts to each component (indeed, it is the identity on the interiors).
Therefore, we conclude OD(f ) ⊗ M ∼= OD(f 0 ) = OD(f ⊗ M ), finishing
the proof.

3. Motives with modulus
In this section, we construct the categories of motives with modulus
eff
eff
MDMeffgm and MDMgm and their sheaf-theoretic versions MDM ,MDMeff , and prove their fundamental properties.

MOTIVES WITH MODULUS, III

17

In the sequel, we write K b (A) (resp. K(A)) for the bounded (resp.
unbounded) category of complexes on an additive category A, andD(A) for its unbounded derived category when A is abelian. We also(−)
write (−)\ for pseudo-abelianisation. We also write (−)
for the (Verdier)
localisation of a triangulated category. If X is a subset of a triangulated
category T , we write hXi (resp. < X >) for the thick (resp. localising)subcategory generated by X, i.e. the smallest triangulated subcategoryof T containing X and closed under direct summands (resp. direct
summands and infinite direct sums).
3.1. Geometric motives. Recall that Voevodsky’s category DMeffgm
is defined as\
K b (Cor)eff
DMgm =,hHIV , MVV i
where HIV and MVV are the objects of the form
1 ×p

X(HIV ): [X × A1 ] −−−→ [X],(MVV ): [U ∩ V ] → [U ] ⊕ [V ] → [X],
where U tV → X runs over all elementary Zariski covers of all X ∈ Sm.
The category which compares naturally with our constructions is avariant of this one (cf. [20, Definition 4.3.3]):

DMeff
gm,Nis =K b (Cor)V
hHIV , MVNis
i

\
,

where
V
(MVNis) : [U ×X V ] → [U ] ⊕ [V ] → [X],X ∈ Sm,in which U t V → X runs over all elementary Nisnevich covers of X,
i.e., covers associated with distinguished Nisnevich squares.
It is a highly nontrivial theorem of Voevodsky that the obvious funceff
tor DMeff
gm → DMgm,Nis is an equivalence of categories when k is perfect; we shall not enter the corresponding issue for modulus pairs inthis paper.
eff
Our definitions of MDMeff
gm and MDMgm faithfully mimic that of
DMeff
gm,Nis :Definition 3.1.1. We define b\K (MCor)eff
MDMgm =,
hCI, MVi

MDMeffgm =

K b (MCor)
hCI, MVi

where CI, MV, CI, MV are the objects of the form
1 ⊗pX(CI): [X ⊗ ] −−−→ [X],

\,

18

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI(MV): [U ×X V ] → [U ] ⊕ [V ] → [X],in which U t V → X runs over all elementary MV-covers of all X ∈MSm, i.e. those covers associated with MV-squares (Definition 1.3.4),
and
1X ⊗p
(CI): [X ⊗ ] −−−→ [X],(MV): [U ×X V ] → [U ] ⊕ [V ] → [X],in which U t V → X runs over all elementary MV-covers of all X ∈MSm, i.e. those covers associated with MV-squares (Definition 1.3.9),
eff
and we write M gm : MCor → MDMeff
gm , Mgm : MCor → MDMgm
for the corresponding canonical functors. Moreover, if f : X → Y isa morphism in MCor, we write M gm [f ] for the image in MDMeff
gm offthe complex [f ] = [X ] →− [Y] ∈ K b (MCor) with [Y] placed in degree
0, so that we have a distinguished trianglef∗+1→ M gm (Y) → M gm [f ] −→ .
M gm (X ) −(This notation will only be used in the proof of Theorem 7.3.2.)
3.2. Sheaf-theoretic motives. The sheaf-theoretic category of motives DMeff (unbounded version) is defined to be
DMeff =

D(NST),< ZVtr (HIV ) >where NST is the category of Nisnevich sheaves with transfers and ZVtr
is the additive Yoneda functor.
In the following, we shall replace NST with the categories of modulus
sheaves with transfers which were studied in [17] and [18]. Let us recall
them:(1) MPST (resp. MPST) is the category of left modules (additive
contravariant functors) on MCor (resp. MCor).
(2) MNST (resp. MNST) is the full subcategory of MPST (resp.
MPST) of functors whose restriction to MSm (resp. MSm)
via the graph functor is a sheaf for the MV (resp. MV) topology.
All these categories are abelian Grothendieck: the ones of (1) as
categories of left modules [17, Th. A.10.2], and the ones of (2) by [17,Th. 4.5.5] and [18, Th. 4.2.4].
Recall from [17] the following notion:
Definition 3.2.1. A functor between additive categories is strongly
additive if it preserves all direct sums.
By [17, Prop. 2.4.1 and Th. 4.5.5] and [18, Lemma-Def. 4.2.1, Th.
4.2.4 and Th. 5.1.1] (see also [17, Prop. A.4.1 b)]), we have

MOTIVES WITH MODULUS, III

19

Proposition 3.2.2.
(1) The inclusion functors iNis : MNST ,→ MPST and iNis :MNST ,→ MPST have exact left adjoints aNis and aNis .
(2) The inclusion functor τ : MCor ,→ MCor of (1.1) induces
fully faithful, exact, strongly additive functors
τ! : MPST → MPST,

τNis : MNST → MNST

and we have an isomorphism τNis aNis ' aNis τ! . Moreover, τ! andτNis have exact right adjoints τ ∗ and τ Nis .
The additive Yoneda functors
Ztr : MCor → MPST,

Ztr : MCor → MPST

induce triangulated functors(3.1)

K b (MCor) → D(MPST),K b (MCor) → D(MPST).

Lemma 3.2.3. The functors D(τ! ) : D(MPST) → D(MPST) andD(τNis ) : D(MNST) → D(MNST) are fully faithful.
Proof. This follows from Propositions 3.2.2 and A.4.2.



We now slightly diverge from Voevodsky to define MDMeff and
MDMeff . We will get back to the analogues of his definiton in Theorem
4.1.1 below.
Definition 3.2.4. We define
D(MPST)
D(MPST)
MDMeff =, MDMeff =,< CI, MV >< CI, MV >where < CI, MV > and < CI, MV > are the localising subcategories of
D(MPST) and D(MPST) generated by the images of hCI, MVi and
hCI, MVi by the functors (3.1).
3.3. Full embeddings and tensor structures.
Theorem 3.3.1.
(1) The functors (3.1) induce triangulated functors
(3.2)

eff
ιeff : MDMeff
gm → MDM ,eff
ιeff : MDMeff
gm → MDM .

We write M = ιeff ◦ M gm and M = ιeff ◦ Mgm (cf. Definition
3.1.1).
(2) The functors ιeff and ιeff are fully faithful with dense images;
their essential images consist of the compact objects of the target
categories. In particular, MDMeff and MDMeff are compactly
generated.

20

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI(3) The tensor structure on MCor induces tensor structures on
D(MPST), D(MPST) and all categories of (1). The functorsof (3.2) are ⊗-triangulated.
(4) The functors τ and τ! of (1.1) and Proposition 3.2.2 induce⊗-triangulated functors
effτeff,gm : MDMeff
gm → MDMgm ,τeff : MDMeff → MDMeff .

(5) The functor τeff is strongly additive and has a right adjoint τ eff .
Proof. (1) is obvious by construction. For (2), apply Theorem A.3.9
and Example A.3.6. Let us prove (3). To start with, Theorem A.4.1
provides tensor structures on MPST, MPST, K b (MCor), K b (MCor),D(MPST) and D(MPST) such that the functors of (3.1) are tensor
functors. Then Lemma 2.2.1 implies that hCI, MVi and hCI, MVi are⊗-ideals in K b (MCor) and K b (MCor), thus so are < CI, MV > and< CI, MV > in D(MPST) and D(MPST).
Let us prove (4), which is the most difficult point. Since D(τ! ) isstrongly additive, it suffices to show that K b (τ )(hCIi) ⊆ hCIi andK b (τ )(hMVi) ⊆ hMVi. The first inclusion is obvious. The second
one is a consequence of the continuity of τs [16, Th. 1], but we provide
a direct and concrete proof. Let T ∈ MSmSq be an MV-square, and
let S → τs (T ) be an associated MV-square (property (2) of Definition
1.3.9). Consider the Sq × Sq-object of MSm
X = S ×τs (T (11)) τs (T ).
We can compute Tot(X) in K b (MSm) in two different ways (see [41,I.2.2] for the totalisation of multicomplexes, and loc. cit., (2.2.4.1) fora “Fubini theorem”): we drop τs to lighten the notation.
(i) For every (kl), S ×T (11) T (kl) is an MV-square. Hence Tot(X) ∈hMVi.
(ii) For every (ij) 6= (11), Tij = S(ij) ×T (11) T is a cartesian squarein which the map S(ij) → T (ij) over T (11) provides a sectionof the projection S(ij) ×T (11) T (ij) → S(ij). For (ij) = (10),this projection is an isomorphism on the open parts by property(2) of Definition 1.3.9, so is a monomorphism, hence an isomorphism. In other words, T10 (10) → T10 (11) is an isomorphism,
and the same holds for its pull-back T00 . Hence Tot(T10 ) andTot(T00 ) are contractible. For (ij) = (01), we apply Lemma
1.3.13: the hypotheses of this lemma are satisfied. We check
Condition (3): Suppose T is of the form (1.3). Since T is
an MV-square by definition, we have OD(q) ∼= OD(p). Moreover, by [29, Prop. 3.1.4], the off-diagonal OD is stable underMOTIVES WITH MODULUS, III

21

pullbacks. More precisely, if we set p0 := p ×T (11) S(01) andq 0 := q ×T (11) S(01), we have OD(p0 ) = OD(p) ×T (11) S(01) andOD(q 0 ) = OD(q) ×T (11) S(01). Therefore, we have OD(q 0 ) ∼=OD(p0 ). It follows that Tot(T01 ) is also contractible. In fine,Tot(Tij ) is contractible except for the square T11 = T .
This shows that Tot τs (T ) ∈ hMVi (more precisely, Tot τs (T ) belongs
to the triangulated ⊗-ideal generated by Tot(S)).
Finally, we prove (5). Let π : D(MPST) → MDMeff and π :D(MPST) → MDMeff be the projection functors. By Corollary
A.3.10, they have right adjoints i and i which themselves have rightadjoints. In particular, all these functors are strongly additive. This,and the strong additivity of D(τ! ) easily implies that τeff is strongly additive; the existence of τ eff then follows from (2) (compact generationof MDMeff ), Theorem A.3.8 and Lemma A.3.2.
Remarks 3.3.2.
(1) Similar to the proof of Theorem 3.3.1 (2), Theorem A.3.9 andExample A.3.6 imply that the canonical functor ιVeff : DMeff
gm,Niseff→ DM is fully faithful over any k (compare [20, p. 6801]).
(2) We shall see in Theorem 5.2.2 that τeff and τeff,gm are fully faithful. This seems to require the theory of intervals from Appendix
B.
4. Brown-Gersten property
4.1. Main result. The adjunctions of Proposition 3.2.2 (1) induce
adjunctions
D(aNis ) = RaNis : D(MPST)  D(MNST) : RiNis ,D(aNis ) = RaNis : D(MPST)  D(MNST) : RiNis ,and D(aNis ), D(aNis ) are localisations by Proposition A.4.2.
Theorem 4.1.1.
(1) The kernel of the localisation functor
D(aNis ) : D(MPST) → D(MNST)
equals < MV >.
(2) The kernel of the localisation functor
D(aNis ) : D(MPST) → D(MNST)
equals < MV >.

22

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI(3) The localisation functor D(MPST) → MDMeff induces anequivalence of triangulated categories
D(MNST) ∼= MDMeff .
< CI >(4) The localisation functor D(MPST) → MDMeff induces anequivalence of triangulated categories
D(MNST) ∼= MDMeff .
< CI >(5) The categories D(MNST) and D(MNST) are compactly generated and inherit tensor structures from those of D(MPST)
and D(MPST) obtained in Theorem 3.3.1 (3). The functor
D(τNis ) : D(MNST) → D(MNST) is ⊗-triangulated.
The proof is given in Subsection 4.5.
4.2. Corollaries.
Corollary 4.2.1 (cf. Hypothesis B.6.1 (i)). Via aNis and aNis , the tensor structures on MPST and MPST induce right exact tensor structures on MNST and MNST.
Proof. Let F, G ∈ MNST. We define
F ⊗MNST G = H0 (F [0] ⊗D(MNST) G[0]).
for the tensor structure of Theorem 4.1.1 (5). For its right exactness,
one sees that Hi (F [0] ⊗D(MNST) G[0]) = 0 for i < 0 by reducing to
F = Ztr (M ), G = Ztr (N ). By Theorem 4.1.1 (5), the functor D(aNis )is monoidal, hence we have aNis F0 ⊗MNST aNis G0 = aNis (F0 ⊗MPST G0 )for F0 , G0 ∈ MPST. Same argument with MNST.
Remark 4.2.2. Proceeding as in [3, Proof of Prop. 4.1.22], it canbe shown that ⊗D(MNST) is actually the total left derived functor of⊗MNST . Similarly for MNST.
Corollary 4.2.3. The localisation functors L : D(MNST) → MDMeff ,L : D(MNST) → MDMeff are strongly additive, symmetric monoidal
and have right adjoints j  , j  , which themselves have right adjoints.
We have a natural isomorphism of functors(4.1)

τeff L ' L D(τNis ).

Proof. Via Theorem 4.1.1, strong additivity and symmetric monoidality follow from those of the localisation functors D(MPST) → MDMeffand D(MPST) → MDMeff . The sequel then follows from Corollary

MOTIVES WITH MODULUS, III

23

A.3.10. The isomorphism (4.1) follows from the construction of τeff in
Theorem 3.3.1 (4), and the fact that D(aNis ) and D(aNis ) are localisations.

For the next corollary, we recall an important notation from [17].
Definition 4.2.4. For F ∈ MPST and for X ∈ MSm, we write FX
for the presheaf on the small étale site of X given by
FX (U → X ) := F (U, X ∞ ×X U).
For F ∈ MPST and X ∈ MSm, we set
FX := (τ! F )X .
We extend this notation to complexes of (pre)sheaves in the obvious
way.
Corollary 4.2.5. For any X ∈ MCor and K ∈ MDMeff , we have an
isomorphism
MDMeff (M (X ), K) '

lim
−→
fin
0

X ∈Σ

0

H0Nis (X , (j  K)X 0 ).
↓X

The same formula holds in MDMeff if X ∈ MCor and K ∈ MDMeff .
Proof. This is obvious by adjunction from Corollary 4.2.3 and [18, Prop.
7.4.2 and Th. 7.5.1].

Since D(MNST) and MDMeff are compactly generated (Theorem
4.1.1), Brown’s representability theorem applied to their tensor structures provides them with internal Homs; similarly for D(MNST) and
MDMeff . The following is an application of Lemma A.1.1:
Proposition 4.2.6. Let K ∈ D(MNST) and L ∈ MDMeff . Then we
have a natural isomorphismj  HomMDMeff (L (K), L) ' HomD(MNST) (K, j  L)
hence, for K 0 , L ∈ MDMeff , a natural isomorphismj  HomMDMeff (K 0 , L) ' HomD(MNST) (j  K 0 , j  L).
Same formulas in D(MNST) and MDMeff , with L and j  .

24

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

4.3. Sheaves on MSmfin and MSm. To prove Theorem 4.1.1, we
have to go back to these categories. As in [17], we write MPSfin and
MPS for the categories of presheaves of abelian groups on MSmfin
and MSm, and MNSfin and MNS for the corresponding categoriesof sheaves (for the MVfin and MV topology, respectively). For genfin
eral reasons, the inclusion functors ifin,→ MPSfin , is,Nis :s,Nis : MNS
MNS ,→ MPS have exact left adjoint sheafification functors afins,Nis ,
as,Nis . Moreover, the adjoint functors
bs,! : MPSfin  MPS : b∗sassociated to the functor bs of Definition 1.2.1 are both exact, and theypreserve sheaves [17, Lemma 4.2.3 and Prop. 4.3.3]. For further reference, we record the corresponding tautological identity for the rightadjoints:Nisb∗s is,Nis = ifins,Nis bs .

(4.2)

Nis
In the adjoint pair (bs,Nis , bNiss ), bs,Nis is exact (but not bs ) .

Definition 4.3.1. Let Zp : MSmfin → MPSfin , Zp : MSm → MPS
denote the “free presheaf” functors (cf. [17, Proposition 2.6.1]). Thatis, for M, N ∈ MSmfin (resp. MSm), the section Zp (M )(N ) is given by
the free abelian group Z HomMSmfin (N, M ) (resp. Z HomMSm (N, M )).
We write < MVfins > (resp. < MVs >) for the localizing subcategoryfinof D(MPS ) (resp. D(MPS)) generated by the objects of the form
Zp (U ×M V ) → Zp (U ) ⊕ Zp (V ) → Zp (M )where M ∈ MSmfin (resp. M ∈ MSm) and U t V → M runs over
all elementary MVfin -covers (resp. MV-covers). Note that < MVs >=D(bs,! )(< MVfins >).
4.4. Technical results.
Proposition 4.4.1.
(1) For any MV-square T ∈ MSmSq , the sequence
0 → Ztr (T (00)) → Ztr (T (01)) ⊕ Ztr (T (10)) → Ztr (T (11)) → 0
is exact in MNST.
(2) For any MV-square T ∈ MSmSq , the sequence
0 → Ztr (T (00)) → Ztr (T (01)) ⊕ Ztr (T (10)) → Ztr (T (11)) → 0
is exact in MNST.

MOTIVES WITH MODULUS, III

25

Proof. (1) is [17, Th. 4.5.7]. For (2), by [29, Cor. 5.2.7] we have thedesired exactness if we consider the terms as sheaves on MSm; equivalently, the sequence becomes exact after applying τNis . The conclusion
then follows from Proposition 3.2.2 (2).

Proposition 4.4.2. We have a naturally commutative diagram:< MV >/K< MV >

/K

(4.3)


< MVs >O/KOs

/ D(MPST)
D(τ! )/ D(MPS)O/ D(MNST)

D(aNis )D(as,Nis )/ D(MNST)/ Kfins

D(cNis )/ D(MNS)O

D(bs,! )< MVfins >D(τNis )/ D(MPST)
D(c∗ )

D(aNis )D(bs,Nis )D(afins,Nis )/ D(MPSfin )/ D(MNSfin ),where K, K, Ks and Kfins are the kernels of D(aNis ), D(aNis ), D(as,Nis )fin
and D(as,Nis ), respectively.
Proof. Note that F 7→ D(F ) is functorial in exact functors F by [18,Lemma A.2.4]. The commutativity of the upper (resp. middle, lower)right square in the diagram follows from Proposition 3.2.2 (2) (resp.
from as,Nis c∗ = cNis aNis , which is proven in [17, Prop. 4.5.6], resp. fromthe isomorphism as,Nis bs,! ' bs,Nis afins,Nis which we obtain by taking left
adjoints of both sides of (4.2)).
A fortiori, this provides the vertical functors in the second column.
fin
We have the inclusions < MVs >⊂ Ks and < MVfins >⊂ Ks by [46,Lemma 2.18] and the regularity of the cd-structures PMV and PMVfin .
We also have < MV >⊂ K and < MV >⊂ K by Proposition 4.4.1.
Finally, the arrows in the left column follow tautologically from thedefinitions of < MV >, < MVs > and < MVfins >, except for the top
one which follows from the proof of Theorem 3.3.1 (4) (this arrow will
not be used in the proof of Theorem 4.1.1).
inRemark 4.4.3. It would be more natural to use D(b∗s ) and RbNiss
Diagram (4.3). Unfortunately, the commutation of the corresponding
square would imply the exactness of bNiss , which is false. This will force
us to use a more indirect argument for the proof of (ii) in Subsection
4.5 below.
Lemma 4.4.4. The functor D(c∗ ) : D(MPST) → D(MPS) is conservative, and < MV >→< MVs > is essentially surjective.

26

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Proof. Let C ∈ D(MPST) be such that D(c∗ )C = 0. For any M ∈MSm and any i ∈ Z, we have c! Zp (M ) = Ztr (M ) by [17, Proposition 2.6.1], where Zp is as in Definition 4.3.1. Moreover, the presheaf
Zp (M ) ∈ MPS is a projective object, since HomMPS (Zp (M ), F ) =F (M ) by definition of Zp . This implies Lc! Zp (M ) = c! Zp (M ). Therefore,
0 = D(MPS)(Zp (M ), D(c∗ )C[i]) = D(MPST)(Ztr (M ), C[i])by adjunction. This shows that C = 0. The second statement istrivial.

Proposition 4.4.5. We have the following isomorphism of functors:finD(bs,! )Rifins,Nis D(as,Nis ) ' Ris,Nis D(as,Nis )D(bs,! ).

Proof. In view of the commutativity of the right lower square in (4.3),
it suffices to show the isomorphisms∼∼− R(bs,! ifin
→ R(is,Nis )D(bs,Nis ).
D(bs,! )R(ifins,Nis ) ←s,Nis ) ' R(is,Nis bs,Nis ) −Indeed, the first isomorphism follows from the exactness of bs,! by [18,Lemma A.2.4], and the second one is tautological.
For the third one, by [18, Lemma A.2.7], it suffices to show that bs,Nis
sends injectives to is,Nis -acyclic sheaves, which holds by [17, Lemma
4.4.3], and that D(bs,Nis ), R(is,Nis ) and R(is,Nis bs,Nis ) are strongly additive. Since bs,Nis is exact and strongly additive as a left adjoint, D(bs,Nis )is strongly additive by [18, Prop. A.2.8 a)]. For R(is,Nis ), we invoke[18, Prop. A.2.8 c)]: by [18, Prop. A.2.5], Ris,Nis has the left adjoint
D(as,Nis ), which sends Zp (M )[n] (M ∈ MSm, n ∈ Z) to Z(M )[n]. The
first are compact generators of D(MPS) by Example A.3.6, and thesecond are compact in D(MNSfin ) by [18, Prop. 7.1.1].
Finally, we prove the strong additivity of R(is,Nis bs,Nis ) ' R(bs,! ifins,Nis ).
For this, we use [18, Prop. A.2.8 b)]. We must check that
• Rp (bs,! ifins,Nis ) is strongly additive for all p ≥ 0;
• there is a set E of compact projective generators of MPS andintegers cd(E) for E ∈ E such that MPS(E, Rp (bs,! ifins,Nis )(A)) =fin
0 for any p > cd(E) and for any A ∈ MNS .
p fin
Noting that Rp (bs,! ifins,Nis ) ' bs,! R is,Nis , the first point follows from thecommutation of Nisnevich cohomology with filtering direct limits. For
the second one, we take for E the collection of Zp (M ) for M ∈ MSm,

MOTIVES WITH MODULUS, III27

and claim that cd(Zp (M )) = dim M o works. Indeed:pp fin
MPS(Zp (M ), Rp (bs,! ifins,Nis )(A)) = MPS(Z (M ), bs,! R is,Nis (A))

=

lim
−→
fin

N ∈Σ

MPSfin (Zp (N ), Rp ifins,Nis (A)) =

lim
−→
fin

N ∈Σ

↓M

p
HNis
(N , AN )↓M= 0 for p > dim M o ,where we used [17, (2.5.1)] for the second equality. We are done.



4.5. Proof of Theorem 4.1.1. Assertions (3) and (4) follow from (1)and (2). Assertion (5) is a consequence of Theorem A.3.9 and the factthat D(MPST) and D(MPST) are compactly generated (see Example
A.3.6). The assertion on tensor structures holds since < MV > and< MV > are ⊗-ideals by Lemma 2.2.1 (cf. the proof of Theorem 3.3.1(3)).
It remains to prove (1) and (2). Let < MV >⊥ (resp. < MV >⊥ )denote the right orthogonal of < MV > in K (resp. of < MV > in K),⊥
and define < MVs >⊥ , < MVfins > similarly. By Theorem A.3.7, we⊥have < MV >= K ⇐⇒ < MV > = 0, etc. We shall play with theseequivalences. More precisely, the layout is:⊥(i) < MVfins > = 0.
(ii) < MVs >= Ks .
(iii) < MV >⊥ = 0 (i.e. (1)).
(iv) < MV >⊥ = 0 (i.e. (2)).
Proof of (i). It follows from [46, Th. 3.2], since the cd-structure MVfin
is complete and bounded (Proposition 1.3.3).
Proof of (ii). Let x ∈ Ks . Since bs,! is exact, the functor Rb∗s is rightadjoint to D(bs,! ) by Proposition A.4.2. Consider the distinguishedtriangle in D(MPSfin ):
(4.4)

f

+1

∗
fin
Rb∗s x →− Rifins,Nis D(as,Nis )Rbs x → z −→,finwhere f is the unit of the adjunction (D(afins,Nis ), R(is,Nis )) and z is a coneof f . Applying D(bs,! ) to (4.4), we obtain the following distinguishedtriangle in D(MPS):

(4.5)

f

+1

∗
fin
D(bs,! )Rb∗s x →− D(bs,! )Rifins,Nis D(as,Nis )Rbs x → D(bs,! )z −→ .

By Proposition 4.4.5, the second term of (4.5) is isomorphic to
Ris,Nis D(as,Nis )D(bs,! )Rb∗s x, and we have D(bs,! )Rb∗s = Id by Proposition A.4.2 and by the full faithfulness of b∗s [17, Prop. 2.5.1]. Hencethe first term is isomorphic to x, and the second term is 0 by x ∈ Ks .
We thus get an isomorphism x ' D(bs,! )z[−1]. Moreover, z ∈ Kfin
s as

28

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

one sees by applying D(afins,Nis ) to (4.4). By (i) and Proposition 4.4.2,

this implies that x ∈< MVs > as requested.
Proof of (iii). Let x ∈< MV >⊥ : we must prove that x = 0. Since
D(c∗ ) is conservative by Lemma 4.4.4, it suffices to show D(c∗ )x = 0.
Since < MVs >⊥ = 0 by (ii), it is enough to prove that D(c∗ )x ∈<MVs >⊥ . By Definition 4.3.1, < MVs > is generated by complexes ofthe form Tot Zp (S) for MV-squares S. Therefore, it suffices to provethat HomD(MPS) (Tot Zp (S), D(c∗ )x) = 0 for any such S. We compute:
HomD(MPS) (Tot Zp (S), D(c∗ )x) ∼= HomD(MPST) (L(c! ) Tot Zp (S), x)= HomD(MPST) (Tot Ztr (S), x)= 0,
where the first isomorphism follows from Lemma A.4.3, since the leftderived functor L(c! ) is defined at the bounded complex Tot Zp (S)of projective objects.2 The second equality follows from the equality L(c! )Zp (M ) = Ztr (M ) for any modulus pair M (this was alreadyused in the proof of Lemma 4.4.4), and the third equality follows fromTot Ztr (S) ∈< MV >. This finishes the proof.
Proof of (iv). Since D(τ! ) is fully faithful by Lemma 3.2.3, we are reduced by (iii) to proving
D(τ! )(< MV >⊥ ) ⊂< MV >⊥ .

(4.6)

Take any x ∈< MV >⊥ . It suffices to prove that the abelian group
K(Tot Ztr (S), D(τ! )(x)[i]) = D(MPST)(Tot Ztr (S), D(τ! )(x)[i])is 0 for any MVfin -square S and any i ∈ Z, where Tot denotes totaliN
sation. For each (ij) ∈ Sq, set S N (ij) := (S(ij) , πij∗ S(ij)∞ ), whereN

πij : S(ij) → S(ij) is normalisation. Then the edges S(ij) → S(i0 j 0 )in S uniquely lift to S N (ij) → S N (i0 j 0 ), and we obtain a new square
S N . The maps πij induce a morphism S N → S in MSmSq , which is
an isomorphism since normalisation is proper. Moreover, the étalenessof the edges in S implies that S N (ij) = S(ij) ×S(11) S N (11), and therefore that S N is again an MVfin -square. In the following, replacing S
with S N , we may assume that the ambient space S(ij) is normal forall (ij) ∈ Sq.
2In fact, Lc! is everywhere defined by [22, Th. 14.4.3].

MOTIVES WITH MODULUS, III

29

Now, we compute for i ∈ Z:
D(MPST)(Tot Ztr (S), D(τ! )(x)[i])∼=1 K(MPST)(Tot Ztr (S), τ! (x)[i])= H 0 Hom•Ch(MPST) (Tot Ztr (S), τ! (x)[i])∼lim
H 0 Hom•Ch(MPST) (Tot Ztr (T ), x[i])=2−→T ∈Comp(S)

=

lim
−→

K(MPST)(Tot Ztr (T ), x[i])lim−→D(MPST)(Tot Ztr (T ), x[i])T ∈Comp(S)∼=3T ∈Comp(S)

∼
=4

lim
−→MV

T ∈Comp

D(MPST)(Tot Ztr (T ), x[i])
(S)

5

= 0,
where Ch(−) denotes the category of chain complexes, and Hom• denotes the Hom complex. Here Comp(S) and CompMV (S) are as in
Theorem 1.3.12.
The isomorphisms ∼=3 hold because each component of Ztr (S)=1 and ∼and Ztr (T ) is projective, and ∼=2 follows from the formula in Theorem 1.1.4 for the pro-left adjoint of τ ! of τ! . Moreover, ∼=4 follows
from Theorem 1.3.12. Finally, the assumptions x ∈< MV >⊥ and
Tot Ztr (T ) ∈< MV > imply =5 .
This completes the proof of Theorem 4.1.1.
Remark 4.5.1. This proof rests fundamentally on the fact that the cdstructure PMVfin on MSmfin is bounded; an easier but similar proof
shows that the kernel of the localisation functor D(aVNis ) : D(PST) →D(NST) equals < MVNis >, cf. [5, Proposition in §4.2.1]. The main
reason why the boundedness of PMVfin is sufficient here seems to be thatsheaves in MNST and MNST also have cohomological dimensions
bounded by the dimension of the total space of a modulus pair [18,Cor. 2.2.10 and 5.1.4].
We do not know whether the cd-structures PMV and PMV are themselves bounded.
5. The derived Suslin complex
5.1. -invariance. We start with:

30

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Lemma 5.1.1. Let  = (P1 , ∞) ∈ MSm. The interval structure of
A1 ' P1 − {∞} ∈ Sm from [42] induces an interval structure on 
for the ⊗-structure of Definition 2.1.1.
Proof. We need to check that the structure maps p, i0 , i1 , µ are morphisms in MCor. The unit object is (Spec k, ∅), so i0 , i1 and p areclearly admissible. As for µ, its points of indeterminacy in P1 × P1
are (0, ∞) and (∞, 0); the closure Γ of its graph in P1 × P1 × P1 isisomorphic to Bl(0,∞),(∞,0) (P1 ×P1 ), where the two exceptional divisors
are given by 0 × ∞ × P1 and ∞ × 0 × P1 . In particular, Γ is smooth.
Thenp∗2 ∞ = P1 × ∞ × ∞ + ∞ × P1 × ∞whilep∗1 (P1 ×∞+∞×P1 ) = P1 ×∞×∞+0×∞×P1 +∞×P1 ×∞+∞×0×P1
which completes the proof.

Remark 5.1.2. Lemma 5.1.1 is false if we replace the ⊗-structure ofDefinition 2.1.1 by the cartesian product structure: indeed, µ does notfactor through the morphism  ⊗  →  ×  of (2.1). Conversely,
the diagonal A1 → A1 × A1 obviously yields a diagonal morphism
 →  × , but the latter does not factor through (2.1) either.
The following definition will not be used in the sequel, except in
Theorem 6.4.1, but is key to [19].
Definition 5.1.3. We say F ∈ MPST (resp. F ∈ MPST) is invariant if the projection map p : M ⊗ → M induces an isomorphism∼p∗ : F (M ) −→ F (M ⊗ ) for any M ∈ MSm (resp. M ∈ MSm).
∼
Equivalently, F −→ Hom(Ztr (), F ).
5.2. The derived Suslin complex. We shall need:
Proposition 5.2.1. Consider the tensor structures on D(MNST)
and D(MNST) given by Theorem 4.1.1 (5). Then the interval structure on  ∈ MSm from Lemma 5.1.1 yields categories with interval (D(MNST), Ztr ()), (D(MNST), Ztr ()) which verify hypotheses
B.4.1 and B.6.1.
Proof. We do the proof for D(MNST), the case of D(MNST) being
identical.
Since ⊗D(MPST) is the total derived functor of ⊗MPST by Theorem
A.4.1 d), there is a canonical natural transformation
λP C ⊗D(MPST) λP D ⇒ λP (C ⊗K(MPST) D)MOTIVES WITH MODULUS, III

31

for (C, D) ∈ K(MPST) × K(MPST), where λP : K(MPST) →D(MPST) is the localisation functor. Applying D(aNis ) to it, we geta natural transformation
λN K(aNis )C ⊗D(MNST) λN K(aNis )D' D(aNis )λP C ⊗D(MNST) D(aNis )λP D ' D(aNis )(λP C ⊗D(MPST) λP D)⇒ D(aNis )λP (C ⊗K(MPST) D) ' λN K(aNis )(C ⊗K(MPST) D)' λN (K(aNis )C ⊗K(MNST) K(aNis )D)
where λN : K(MNST) → D(MNST) is the localisation functor.
Since K(aNis ) is a localisation, this yields by [17, Lemma A.3.3] thedesired natural transformation(5.1)

λN C 0 ⊗D(MNST) λN D0 ⇒ λN (C 0 ⊗K(MNST) D0 )for (C 0 , D0 ) ∈ K(MNST) × K(MNST).
It remains to check properties (iii) and (iv) of Hypothesis B.6.1: (iii)
is obvious by construction, and (iv) is true because it is already truein D(MPST) by the representability of Ztr (), and aNis is exact. In the next theorem, we use the functors L , L from Corollary 4.2.3.
Theorem 5.2.2. The base change morphism(5.2)

L ◦ D(τ Nis ) ⇒ τ eff L

as in (B.11) is an isomorphism; the functors τeff,gm and τeff of Theorem
3.3.1 (4) are fully faithful.
Proof. The first claim follows from Theorem B.7.2. By Lemma 3.2.3,
D(τNis ) is fully faithful, hence3 the unit map
Id ⇒ D(τ Nis )D(τNis )is an isomorphism. Applying L , we obtain a natural isomorphism∼L −→ L D(τ Nis )D(τNis ).
On the other hand, (4.1) and (5.2) yield natural isomorphisms∼∼

L D(τ Nis )D(τNis ) −→ τ eff L D(τNis ) −→ τ eff τeff L
and one checks that their composition L ⇒ τ eff τeff L is induced by theunit of the adjunction (τeff , τ eff ). Since L is a localisation, we conclude
that this unit is an isomorphism. This implies the full faithfulness ofτeff , which in turn implies that of τeff,gm by Theorem 3.3.1 (2).
3This is actually part of the proof, see Proposition A.4.2.

32

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Definition 5.2.3 (cf. Definition B.6.2). For any K ∈ D(MNST), weset•RC∗ (K) = Hom(Ztr (ν ), K) ∈ D(MNST);
this is the derived Suslin complex of K. Similarly for K ∈ D(MNST).
For X ∈ MCor or MCor, we abbreviate RC∗ (Ztr (X )[0]) to RC∗ (X ).
Recall from Corollary 4.2.3 that L and L have right adjoints j  , j  .
As a consequence of Theorem B.6.3, we have:
Theorem 5.2.4. For any K ∈ D(MNST), we have an isomorphismj  L (K) ' RC∗ (K).
Similarly, we have an isomorphism for any K ∈ D(MNST)
j  L (K) ' RC∗ (K).
In particular, the isomorphisms of Corollary 4.2.5 translate as
MDMeff (M (X ), L K) '

lim
−→
fin
0

X ∈Σ

MDMeff (M (X ), L K) '

lim
−→
fin
0

X ∈Σ

0

H0Nis (X , (RC∗ (K))X 0 )↓X0

H0Nis (X , (RC∗ (K))X 0 )↓X

for (X , K) ∈ MCor ×D(MNST) (resp. (X , K) ∈ MCor ×D(MNST)).
Remark 5.2.5. Theorem B.6.3 also yields a version of Voevodsky’s results for DMeff and D(NST) [44, 26], where he uses simplicial objects
rather than cubical objects. Comparing the two, we get an a posteriori
1
proof that for any K ∈ D(NST) the two “Suslin” complexes RC∗A (K)based on simplicial or cubical sets are quasi-isomorphic. Hopefully thiscan be proven by an explicit chain computation.
On the other hand, the theory of intervals does not yield a simplicial
theory in the case of MCor and MCor, see Remark B.2.6.
6. Comparisons6.1. Relationship with Voevodsky’s categories. We start by comparing MDMeff with DMeff . As usual, the functor ω : MCor → Corfrom (1.1) defines an adjunction(6.1)

ω ! : MPST  PST : ω ∗ .

Thus ω ! is right exact and strongly additive; it is even exact, as theright adjoint of λ! (see Theorem 1.1.4 for λ).
Since λ is fully faithful, so is ω ∗ = λ∗ [17, Prop. 2.3.1].

MOTIVES WITH MODULUS, III

33

Recall the category DMeff
gm,Nis from Subsection 3.1. Since ω obviouslyVV), we get functors ω eff,gm , ω eff
sends (CI) to (HI ) and (MV) to (MVNisin the following commutative diagrams:
K b (ω)K b (MCor) −−−→ K b (Cor)




(6.2)
y
y
ω eff,gm

MDMeff−−−→ DMeff
gm −gm,Nis ,D(ω )!
D(MPST) −−−→
D(PST)
yyMDMeffω

eff−−−→ DMeff ,where the vertical arrows are localisation functors.
As in the proof of Theorem 3.3.1 (5), we deduce from the strong
additivity of ω ! that ω eff is strongly additive and has a right adjoint
ω eff . Recall from [18, Prop. 6.2.1 d)] that the adjunction (6.1) induces
an adjunction
ω Nis : MNST  NST : ω Nis(6.3)

where ω Nis , ω Nis are both exact and ω Nis is fully faithful. The same
picture holds for ωNis = ω Nis ◦ τNis and its right adjoint ω Nis [18, Prop.
6.2.1 c)].
Proposition 6.1.1. The functors ω eff and ω eff fit in commutative diagrams
D(ω

)

Nis
→ D(NST)
D(MNST) −−−−



 A1

L y
yL
(6.4)

MDMeff−−−→ω eff

DMeff ,

D(ω Nis )D(MNST) ←−−−− D(NST)
x
x

 A1

j 
j
MDMeff←−−−ω eff

DMeff .

Moreover, ω eff is strongly additive and fully faithful, while ω eff is alocalisation and is symmetric monoidal.
Proof. The second diagram of (6.2) factors through the first diagramof (6.4), thanks to Theorem 4.1.1 and its analogue for NST (Remark
4.5.1). This yields the second diagram of (6.4) by adjunction. By
Proposition A.4.2, the adjunction (6.3) implies that D(ω Nis ) is rightadjoint to D(ω Nis ), and fully faithful. Hence ω eff is fully faithful, sothat its left adjoint ω eff is a localisation. Since ω Nis is strongly additive[18, Prop. 6.2.1 d)], so is D(ω Nis ), hence ω eff by the diagram.
The symmetric monoidality of ω eff will follow from that of the threeother functors in the diagram. This is already known for the verticalones (see Corollary 4.2.3 for L ), so we are left to show the monoidalityof D(ω Nis ). By the same trick, the latter is reduced to the monoidalityof D(ω ! ), which in turn follows from that of ω (Proposition 2.1.5). 34

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Using Corollary 4.2.3 and the exactness of τ Nis (Proposition 3.2.2(2)), we now get commutative diagrams
D(ω Nis )D(ωNis )D(MNST) −−−−→ D(NST) D(MNST) ←−−−− D(NST)
x
x



 A1

 A1


(6.5)
j 
j
yL
L y
MDMeff−−−→ωeff

DMeff ,

MDMeff←−−−ω eff

DMeff ,

where ω eff is right adjoint to ωeff and ωeff is symmetric monoidal. Moreover,
Proposition 6.1.2. The functor ω eff is strongly additive and fully
faithful, hence ωeff is a localisation.
Proof. Same as for Proposition 6.1.1, using the full faithfulness andstrong additivity of ω Nis [18, Prop. 6.2.1 c)].

We finally have the following commutative diagramι

MDMeff−−eff−→ MDMeff
 gm


τ
τeff,gm y
y eff
(6.6)

ι

eff
eff
MDMeffgm −−−→ MDMωω eff,gm 
y
y eff

ιV

eff
DMeff
gm,Nis −−−→

DMeff

in which ιeff and ιeff are from (3.2) and ιVeff is given in the same way(see Remark 3.3.2 (1)). All rows are fully faithful by Theorem 3.3.1 (2)and [20, (4.5)].
6.2. Relationship with Chow motives. In [44], Voevodsky coneff
structs a ⊗-functor Choweff → DMeff
is the categm , where Chow
gory of effective (covariant) Chow motives. (We refer to [37] or [1]for Chow motives.) This functor sends the Chow motive h(X) of asmooth projective variety X to M V (X), where M V : Cor → DMeffgm
is the canonical functor, and is shown to be fully faithful when k isperfect in [5, 6.7.3] (see also [20, Th. 4.4.1 (3)]).
In fact, Voevodsky’s construction lifts to a ⊗-functor
(6.7)

eff
Φeff
→ DMeff
V : Chowgm,Nis .

Indeed, this construction is as follows: let H(Cor) be the homotopy category of Cor; its Hom groups are h(X, Y ) = Coker(Cor(X ×MOTIVES WITH MODULUS, III

35

A1 , Y ) → Cor(X, Y )). Obviously, the natural functor Cor → DMeff
gm,Nisfactors through H(Cor). There is also a map(6.8)

h(X, Y ) → CHdim X (X × Y )which sends a finite correspondence to the corresponding cycle class.
This map is an isomorphism when X and Y are projective [9, Th. 7.1],
hence the functor (6.7).
Theorem 6.2.1. Write ωeff,gm = ω eff,gm ◦ τeff,gm (see (6.6)). There is aunique functor Φeff : Choweff → MDMeff
gm sending h(X) to M (X, ∅),whose composition with ωeff,gm is (6.7). It is symmetric monoidal.
(We shall see in Corollary 6.3.7 that Φeff is fully faithful when k isperfect.)
Proof. For X, Y as above, the inclusions
MCor((X, ∅), (Y, ∅)) ⊆ Cor(X, Y )MCor((X, ∅) ⊗ , (Y, ∅)) ⊆ Cor(X × A1 , Y )are equalities since the modulus conditions become empty (by definitionof the left hand sides: 1.1.2). Hence we get the refined functor fromthe definition of MDMeff in Definition 3.1.1. Any other such functor
agrees with Φeff on objects, but also on morphisms by (6.8), hence theuniqueness. Its symmetric monoidality is obvious.

6.3. Empty modulus.
Theorem 6.3.1. Let X be a smooth proper k-variety. Then we havenatural isomorphisms∼

M (X, ∅) −→ ω eff M V (X)∼M (X, ∅) −→ ω eff M V (X)
where ω eff and ω eff are the functors from (6.4) and (6.5).
Proof. For any M ∈ MCor, the inclusion
MCor(M, (X, ∅)) ⊆ Cor(M o , X)
is an equality by 1.1.2. Equivalently, ω Nis ZVtr (X) = Ztr (X, ∅) andω Nis ZVtr (X) = Ztr (X, ∅). The result now follows from Theorem B.7.2,
which yields natural isomorphisms
∼

1

∼

1

L ◦ D(ω Nis ) −→ ω eff ◦ LA

L ◦ D(ω Nis ) −→ ω eff ◦ LA

that we apply to ZVtr (X)[0] (recall that ω Nis and ω Nis are exact, see§6.1.)


36

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

eff
Definition 6.3.2. We let DMeff
gm,prop (resp. DMprop ) be the thickeff(resp. localising) subcategory of DMeff
gm,Nis (resp. DM ) generated by
the M (X), where X runs through the smooth proper k-varieties: it isclosed under tensor product.

The following facts are well-known:∼

Lemma 6.3.3. Suppose k is perfect. Then DMeff→ DMeff
gm,Nis −gm . Uneff
eff
der resolution of singularities, we have DMgm,prop = DMgm . In geneff
eral, we have DMeff
gm,prop [1/p] = DMgm [1/p], where p is the exponential
characteristic of k.
Proof. The first fact was recalled in Subsection 3.1 (cf. [43, Th. 5.7]).
The second one follows from the Gysin distinguished triangles of [44,Prop. 3.5.4]. The last one is proven similarly, by resolution of singularities à la de Jong–Gabber plus a transfer argument which refines theone in [10, beg. of App. B].
Corollary 6.3.4. The restriction of ω eff to DMeff
prop is symmetric monoidal and induces a fully faithful symmetric monoidal functoreff
ω eff,gm : DMeff
gm,prop → MDMgm
effwhich is “right adjoint” to the functor ω eff,gm : MDMeff
gm → DMgm of(6.6): namely, this right adjoint is defined on DMeff
gm,prop , with value
eff
eff,gmeff. The same holds when replacing MDMgm by MDMeffω
gm and ω
effby ω eff , yielding ω eff,gm : DMeff
gm,prop → MDMgm . In particular, if k isperfect we have adjoint pairs

(6.9)

eff
eff,gmωeff,gm : MDMeff
gm [1/p]  DMgm [1/p] : ω(6.10)eff
eff,gmω eff,gm : MDMeff,
gm [1/p]  DMgm [1/p] : ω

where p is the exponential characteristic of k, and the same without
inverting p under Hironaka resolution of singularities (in particular, if
char k = 0).
Proof. Everything follows from Theorem 6.3.1 and Lemma 6.3.3, except
for the full faithfulness of ω eff,gm , ω eff,gm and the monoidality assertions.
The first follow from Propositions 6.1.1, 6.1.2 and the full embedding
eff
DMeff
of Remark 3.3.2 (1). Next, the monoidality of ω eff
gm,Nis ,→ DM
yields a natural transformation on DMeffω eff M ⊗ ω eff N → ω eff (M ⊗ N )and similarly for ω eff . By Theorem 6.3.1, this is an isomorphism when
M and N are of the form M (X) and M (Y ) for X, Y smooth proper,

MOTIVES WITH MODULUS, III37

hence in general by the strong additivity of ω eff and ω eff (Propositions
6.1.1 and 6.1.2 again).
Definition 6.3.5. Let Z(1) := Φeff (L)[−2], where L ∈ Choweff is the⊗iLefschetz motive. For i ≥ 0 and M ∈ MDMeff
gm , we put Z(i) = Z(1)and M (i) = M ⊗ Z(i).
Corollary 6.3.6. Assume k is perfect. Let X be a smooth proper kvariety of dimension d, Y ∈ MCor a modulus pair, and i, j integers
with i ≥ 0. Then we have a canonical isomorphism:
HomMDMeff(M gm (Y), M gm (X, ∅)(i)[j]) ' H 2d+j (Y o × X, Z(d + i))gm
where the right hand side is Voevodsky’s motivic cohomology. In particular, this group is isomorphic to the higher Chow group CH d+i (Y o ×
X, 2i − j) and vanishes if j > 2i, by [45, Cor. 2].
The same formula holds in MDMeff
gm if Y ∈ MCor (with Mgm insteadof M gm ).
Proof. By Theorem 3.3.1 (2), we may compute the Hom on the left
hand side using MDMeff instead of MDMeff
gm . By monoidality (Coroleffefflary 6.3.4), ω sends Z(i) ∈ DMgm to Z(i) ∈ MDMeff
gm . By adjunction
and Theorem 6.3.1, we then have an isomorphism
HomMDMeff (M (Y), M (X, ∅)(i)[j]) ' HomDMeff (M V (Y o ), M V (X)(i)[j]).
The result now follows from Poincaré duality for X [5, Prop. 6.7.1].
The case of MDMeff
gm is identical.
Corollary 6.3.7. Assume k is perfect. The functor Φeff : Choweff →MDMeff
gm from Theorem 6.2.1 is fully faithful.
6.4. π0 -invariance. For any modulus pair Y ∈ MCor, we write π0 (Y):= (π0 (Y o ), ∅), where π0 (Y o ) (“scheme of constants”) is the universal
étale k-scheme such that the projection Y o → Spec k factors through
π0 (Y o ). This factorisation induces a morphism pY : Y → π0 (Y). In
contrast to Theorem 6.3.1, we have
Theorem 6.4.1. Let X be smooth and quasi-affine. Then Ztr (X, ∅)is -invariant (Definition 5.1.3) and, more strongly, “properly π0 invariant”: for any proper modulus pair Y ∈ MCor, we have an isomorphism in MNST∼

Hom(Ztr (π0 (Y)), Ztr (X, ∅)) −→ Hom(Ztr (Y), Ztr (X, ∅))induced by pY .

38

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Proof. We may reduce to Y o connected and (up to extending k) evengeometrically connected. Take Z ∈ MCor. It suffices to show thatthe map(6.11)p∗Y : MCor(Z, (X, ∅)) → MCor(Z ⊗ Y, (X, ∅))induced by pY is an isomorphism. For any closed point y ∈ Y o , we findthat MCor(Z, (X, ∅)) → MCor(Z ⊗ (y, ∅), (X, ∅)) is injective, hence(6.11) is injective as well.
To show its surjectivity, let us take an elementary modulus correspondence V ∈ MCor(Z ⊗ Y, (X, ∅)). Let V be the closure of V in
Z × Y × X. We claim that the image V 0 of V in Z × X is closed andfinite surjective over Z. To prove this claim, consider the commutative
diagramVπ0

/ Z ×Y ×X

i

π



V 0

%


i0



a

/ Z ×Yb/ Z ×X



d

c

/ Z.

Since V ∈ MCor(Z ⊗ Y, (X, ∅)), ai is proper and surjective. Since
the same is true of c, we find that cai = dπ is proper surjective. Thisimplies that V 0 is closed and, combined with the surjectivity of π 0 , thatdi0 is proper [EGA2, Cor. 5.4.3]. But di0 is also quasi-affine (since sois d), hence finite. This proves the claim.
Now V 0 := V 0 ∩ (Z o × X) is an element of MCor(Z, (X, ∅)). We
clearly have V ⊂ V 0 × Y o , and V 0 × Y o is irreducible because Y isgeometrically irreducible. By comparing dimensions, we get V = V 0 ×
Y o = p∗Y (V 0 ). This proves the surjectivity of (6.11).
6.5. Inverting the Tate object. In this subsection, we shall abundantly use the multiplicative localisations introduced by Grothendieckfor pure motives (inverting the Lefschetz motive); one may refer to [13,A.2.4, A.2.5] for a detailed discussion, see also §A.2.
Definition 6.5.1. We write MDMgm (resp. MDMgm ) for the cateffegory obtained from MDMeff
gm (resp. from MDMgm ) by ⊗-inverting
Z(1) (resp. τeff Z(1)), see Definition 6.3.5. Similarly for DMgm andeff
DMgm,prop from DMeff
gm and DMgm,prop .
The ⊗-functor Φeff of Theorem 6.2.1 extends canonically to a ⊗functor(6.12)

Φ : Chow → MDMgm

where Chow is the category of (all) Chow motives.

MOTIVES WITH MODULUS, III

39

Proposition 6.5.2.
(1) The categories MDMgm and MDMgm are Karoubian ⊗-triangulated categories.
(2) The functor τgm : MDMgm → MDMgm induced by τeff,gm is⊗-triangulated and fully faithful.
(3) The functor Φ is symmetric monoidal, and fully faithful if k
is perfect. For any smooth projective variety X, the motive
M (X, ∅) is strongly dualisable in MDMgm and in MDMgm .
Proof. For (1), we use Voevodsky’s sufficient condition [13, Prop. A.31]
that the switch endomorphism of Z(1)⊗2 is the identity in MDMeff
gm andeff
MDMgm , which holds because this is true for the Lefschetz motive L
in the category Choweff . The karoubian assertion follows from Lemma
A.2.1 (1). For (2), we apply Lemma A.2.1 (2) together with Theorem
5.2.2. Similarly for (3), with Corollary 6.3.7; the strong dualisability
statement holds because Chow is rigid.
Proposition 6.5.3. The ⊗-functors ω eff,gm , ωeff,gm , ω eff,gm and ω eff,gmω gm

of Corollary 6.3.4 induce ⊗-functors MDMgm −−→ DMgm , MDMgm
ωgm

ω gm

ω gm

−−→ DMgm , DMgm,prop −−→ MDMgm and DMgm,prop −−→ MDMgm .
The functors ω gm and ω gm are fully faithful; when k is perfect, theadjoint pairs (6.9) and (6.10) of Corollary 6.3.4 induce adjoint pairs(6.13)

ωgm : MDMgm [1/p]  DMgm : ω gm [1/p](6.14)

ω gm : MDMgm [1/p]  DMgm : ω gm [1/p].

Under resolution of singularities in characteristic p, we can drop the
affixes [1/p].
Proof. The functors of Corollary 6.3.4 induce the said functors becauseof their monoidality, which also implies the monoidality of these functors. The full faithfulness of ω gm and ω gm is shown as in the previous
proof. The adjunction identities of (6.9) and (6.10) are preserved by⊗-inverting Z(1), which yields corresponding adjunction identities for(6.13) and (6.14).

As is well-known, the ⊗-category DMgm [1/p] (more generally, thecategory DMgm,prop ) is rigid. By contrast, there is evidence that thisdoes not hold for MDMgm . Unfortunately, we have to make two
assumptions: one is the analogue of Voevodsky’s cancellation theorem [47] and the other is that “the derived Suslin complex is quasiisomorphic to the naı̈ve Suslin complex”.

40

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Proposition 6.5.4. If the functor MDMeff
gm → MDMgm is fully faithful and if the base change morphism (B.12) is an isomorphism as in
the end of Example B.7.5, the ⊗-category MDMgm is not rigid. More(2)(2)
precisely, M ( ) is not dualisable, with  = (P1 , 2∞).
Proof. Let M ∈ MDMgm , having a dual M ∗ . Suppose that ωgm (M ) =0. By the monoidality of ωgm , we also have ωgm (M ∗ ) ' ωgm (M )∗ = 0.
Equivalently,
MDMgm (M ∗ , ω gm N ) = 0 ∀N ∈ DMgm .
Suppose that N also has a dual N ∗ . Applying the above to N ∗instead of N , and using this time the monoidality of ω gm , we find
0 = MDMgm (M ∗ , (ω gm N )∗ ) = MDMgm (ω gm N, M ).
Take in particular N = Z; by the assumption of full faithfulness andby Theorems 5.2.4 and 6.3.1, we get
H0Nis (k, RC∗ (M )k ) = 0.
(2)

Take for example M = fibre(M ( ) → Z): clearly, ωgm (M ) = 0.
Under the assumption on the base change morphism (B.12), we canreplace RC∗ (M ) by the naı̈ve Suslin complex C∗ (M ) used in [34].
Applying its Theorem 1.1 with S = Spec k, C = P1 , D = 2∞, we find
(2)

(2)

H0Nis (k, RC∗ ( )k ) = H0Nis (k, C∗ ( )k )= H0S (P1 /k, 2∞) = Pic(P1 , 2∞) = Z ⊕ kwhere the last term is the relative Picard group. Thus we get an
isomorphism H0Nis (k, RC∗ (M )k ) ' k, a contradiction. (Note that the(2)(2)
morphism  → 1 is split by the 0-section 1 →  .)

7. Some computations
For simplicity, we write M and M for Mgm and M gm in this section.
7.1. The tautological isomorphisms and distinguished triangles. These are those which come from the definitions of MDMeffgm
eff
and MDM :Mayer-Vietoris: one has a distinguished triangle in MDMeffgm(resp. MDMeff
):
gm
+1

M (T (00)) → M (T (01)) ⊕ M (T (10)) → M (T (11)) −→for any MV-square (resp. MV-square) T .
Tensor product: one has canonical isomorphisms M (X ⊗ Y) 'M (X ) ⊗ M (Y) for any X , Y in MCor (resp. MCor).

MOTIVES WITH MODULUS, III

41

M (p)

-invariance: the morphism M () −−−→ Z =: M (1) is invertible, where p :  → 1 is the structural map.
7.2. An elementary computation. As a special case, in the situation of Example 1.3.10, one has a distinguished triangle in MDMeff
gm
(7.1)

+1

M (X, D0 ) → M (X, D1 ) ⊕ M (X, D2 ) → M (X, D) −→Let us use this example to reduce the computation of the motive of(P1 , D) to its essential parts, where D is an effective divisor. Generally,
for a modulus pair X ∈ MCor, let us write Z̃tr (X ) = Ker(Ztr (X ) →Ztr (1) = Z): in the presence of a 0-cycle of degree 1 on X o , this is adirect summand of Ztr (X ). We define M̃ (X ) as the class of Z̃tr (X ) in
MDMeff
gm , so that we have a distinguished triangle+1

M̃ (X ) → M (X ) → Z −→split in the presence of a 0-cycle of degree 1.
If X = (P1 , D), write M̃ (X ) =: m(D) for simplicity. Then m(∅) =Z(1)[2] and m(∞) = 0. Let D1 , D2 have disjoint supports. Choose
0-cycle of degree 1 on X o . (We can take a rational point unless k isfinite.) It splits off a distinguished triangle+1m(D1 + D2 ) → m(D1 ) ⊕ m(D2 ) → Z(1)[2] −→from the distinguished triangle (7.1) with D = ∅. But the morphism m(Di ) → m(∅) is 0 if k is perfect by Corollary 6.3.6 (if Di
contains a rational point p, an elementary proof is that it factors
through m(p) ' m(∞) = 0). Thus this triangle splits and yields a
non-canonical isomorphismm(D1 + D2 ) ' m(D1 ) ⊕ m(D2 ) ⊕ Z(1)[1].
P
Suppose k algebraically closed, for simplicity. If D = ri=1 ni pi withthe pi distinct points, we get inductively an isomorphismm(D) 'rMm(ni ∞) ⊕ (r − 1)Z(1)[1].

i=17.3. Motives of vector bundles and projective bundles. Let Y ∈MCor be a modulus pair, and let E be a vector bundle of rank n > 0
on Y, with associated projective bundle P(E). We define modulus
pairs E and P with total spaces E and P(E) by pulling back Y ∞ :the resulting morphisms E → Y, P → Y are minimal in the sense of
Definition 1.2.2.

42

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI(There may be more general notions of vector and projective bundles,
but we do not consider them here.)
Remark 7.3.1. By applying Corollary 6.3.6 with X = Spec(k) andj = 2i, we get CH i (Y o ) ' HomMDMeff(M (Y), Z(i)[2i]). In particular,gm
if P (t1 , . . . , tn ) ∈ Z[t1 , . . . , tn ] is a homogeneous polynomial of weight i(the weight of ts being s), then the Chern classes of E yield a morphism
in MDMeffgm
P (c1 (E), . . . , cn (E)) : M (Y) → Z(i)[2i].
Theorem 7.3.2. Assume k is perfect. Suppose Y smooth. The projection p̄ : P → Y yields a canonical isomorphism in MDMeff
gm
(7.2)

∼

ρY : M (P) −
→

n−1
M

M (Y)(i)[2i].

i=0

The same holds in MDMeff
gm if Y ∈ MCor (with M instead of M ).
Remark 7.3.3. If char k = 0 or dim Y ≤ 3, the assumption on Y is
innocent in view of resolution of singularities.
Proof. We follow the method of Voevodsky in [44, proof of Prop. 3.5.1],
with a simplification and a complication. The complication is that
Voevodsky’s construction of the corresponding morphism in DMeff
gm
uses diagonal maps, which cause a problem here (see Remark 2.1.4).
We bypass this problem by using the morphism(7.3)

˜ : P → P ⊗ (P(E), ∅)∆induced by the diagonal inclusion
P o ,→ P o × P(E).

(7.4)

Here, the modulus condition is obviously verified. Using the morphisms M (P(E), ∅) → Z(i)[2i] induced by the powers of c1 (OP(E) (1))(see Remark 7.3.1), we get morphisms
(7.5)

˜
M (∆)

ρiY : M (P) −−−→ M (P) ⊗ M (P(E), ∅) → M (Y) ⊗ Z(i)[2i],whence ρY . To prove that it is an isomorphism, we first consider thecase where the vector bundle E is trivial. We then have an isomorphismof modulus pairs
P ' Y ⊗ (Pn−1 , ∅)hence a corresponding isomorphism of motives. By using either Theorem 6.3.1 or, more directly, the functor Φeff of Theorem 6.2.1 and theMOTIVES WITH MODULUS, III

43

computation of the Chow motive of Pn−1 , one has a canonical isomorphism
θ:

n−1
M

∼

Z(i)[2i] −→ M (Pn−1 , ∅).

i=0

Tensoring it with M (Y) and composing with ρY , we get a morphismn−1
M

M (Y)(i)[2i] →

i=0

n−1
M

M (Y)(i)[2i]i=0which is seen to be the identity by definition of θ and ρY .
In general, we argue by induction on the number m of terms of anopen cover of Y trivialising E. For notational simplicity, write Ξ(Y)for the right hand side of (7.2). Write Y = Y 0 ∪ U , where E is trivial
over U and Y 0 has an (m − 1)-fold trivialising open cover. Provide
Y 0 , U and Y 0 ∩ U with the induced modulus structures Y 0 , U , Y 0 ∩ U ,and pull P back similarly. We claim that the diagram of distinguishedtriangles (with obvious notation)(7.6)

M (P|Y 0 ∩U )

/ M (P|Y 0 ) ⊕ M (P|U )/ M (P)/ M (P|Y 0 ∩U )[1]
Ξ(Y 0 ∩ U )
/ Ξ(Y 0 ) ⊕ Ξ(U )/ Ξ(Y)/ Ξ(Y 0 ∩ U )[1]

commutes, which will conclude the proof. The commutations of theleft and middle square follow from the naturality of ρ. For the right
one4, consider the morphisms f : Y 0 ∩ U → Y 0 ⊕ U and fP : P|Y 0 ∩U →P|Y 0 ⊕ P|U and their associated motives M [f ], M [fP ] (see Definition
3.1.1). We also have an obviously defined motive Ξ[f ], which is a
canonical cone of the left bottom map. Observe now that (7.4) inducesmorphisms
o
o
o
o
o
o
P|Y
0 ,→ P|Y 0 × P(E), P|U ,→ P|U × P(E), P|Y 0 ∩U ,→ P|Y 0 ∩U × P(E)which in turn induce morphisms analogous to (7.3), a morphism in
K b (MCor)[fP ] → [f ] ⊗ [(P(E), ∅)]compatible with (7.3), and finally a morphism ρf : M [fP ] → Ξ[f ]analogous to ρY (see (7.5)). The Mayer-Vietoris property says that
4We thank one of the referees for stressing this issue.

44

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

there are canonical horizontal isomorphisms in the diagram:
M [fP ]


∼

ρf

Ξ[f ]

∼

/ M (P)ρY/ Ξ(Y).

Since the Chern class c1 (OP(E) (1)) restricts to those of c1 (OP(E|V ) (1))
00for V = Y , U, Y ∩ U , this diagram commutes. Therefore we may
replace M (P) and Ξ(Y) by M [fP ] and Ξ(f ) in (7.6). But then thecommutation is obvious.

Question 7.3.4. When E is trivial, the isomorphism E ' Y ⊗ (An , ∅)∼→ M (Y) ⊗ M (An , ∅). Can one extend
yields an isomorphism M (E) −this isomorphism to the general case?
7.4. Further results. In this subsection, we present results whichwere obtained (in anticipation to the release of this paper!) in different
works.
Proposition 7.4.1 (Toric invariance [23, Lemma 10]). For any positive
integer n ≥ 1, consider the standard closed embedding Pn−1 → Pn
(setting P0 := {∗}), and the proper modulus pair (Pn , Pn−1 ). Then the
projection M (Pn , Pn−1 ) → Z is an isomorphism in MDMeff
gm .
In [23], Kelly and Saito also provide a very concise proof of a modulus
version of [44, Prop. 3.5.2]. Recall from [35] the following definition:
Definition 7.4.2. A modulus pair (X , X ∞ ) is said to be log smooth (inshort: ls) if X is smooth and |X ∞ | is a simple normal crossing divisor.
Theorem 7.4.3 (Smooth blowup triangle with modulus). Let X ∈MCor be an ls modulus pair. Let i : Z → X be an ambient minimal
morphism with i : Z → X a closed immersion such that Z is smooth.
Assume moreover that i is transversal (see [23, Def. 7] for the definitionof transversality). Let π : BlZ (X) → X be the blowup of X along Z,
0and i : E → BlZ (X) the exceptional divisor. Set
BlZ (X) := (BlZ (X), π ∗ X ∞ ),0∗

E := (E, i BlZ (X)∞ ).

Note that the resulting morphisms π : BlZ (X) → X, i0 : E → BlZ (X)and π|E : E → Z are minimal, where π|E is the restriction of thenatural morphism E → X.
Then there exists a distinguished triangle in MDMeff (hence in MDMeffgm )of the form
i0 ⊕−π|E

π⊕i+1

M (E) −−−−−→ M (BlZ (X)) ⊕ M (Z) −−→ M (X) −→ .

MOTIVES WITH MODULUS, III

45

Matsumoto established in [28] the following interesting distinguishedtriangle in MDMeff , which lifts the classical Gysin triangle when theclosed subset is of codimension 1 (see Remark 7.4.5).
Theorem 7.4.4. Let (X, X ∞ ) be an ls modulus pair, and let Z ⊂ X bean effective Cartier divisor which is integral and smooth. Assume that
Z is not contained in X ∞ , and that the support of the divisor X ∞ + Z
is a strict normal crossing divisor on X. Set Z ∞ := X ∞ ×X Z.
Then one has the following distinguished triangle in MDMeff :
(7.7)

+1

M (X, X ∞ + Z) → M (X, X ∞ ) → M (Z, Z ∞ )(1)[2] −→,where [−](1) denotes the Tate twist from Definition 6.3.5.
Remark 7.4.5. Applying the triangulated functor ω eff to the distinguished triangle (7.7), we recover the Gysin triangle in DMeff :
(7.8)

+1

M V (X o − |Z o |) → M V (X o ) → M V (Z o )(1)[2] −→ .

Remark 7.4.6. In [28], under the same assumption as in Theorem 7.4.4,
a second lifting of (7.8) is constructed in MDMeff :+M (X − |Z ∞ |) → M (X) → Th(NZ X, op) −→,where Th(NZ X, op) is a suitable “Thom space” in the modulus setting,whose definition we do not recall here.
Appendix A. Categorical toolbox, III
A.1. Monoidal categories [27, VII.1]. Recall that a monoidal category (C, ⊗) is closed if ⊗ has a right adjoint Hom. We shall use thefollowing lemma several times:
Lemma A.1.1. Let C and D be two closed monoidal categories, andlet u : C → D be a lax ⊗-functor: this means that we have a naturaltransformation(A.1)

uX ⊗ uY → u(X ⊗ Y ).

Assume that u has a right adjoint v. Then, for any (X, Y ) ∈ C × D,
there is a canonical morphism
HomC (X, vY ) → v HomD (uX, Y )bivariant in (X, Y ), which is an isomorphism if (A.1) is a natural
isomorphism.

46

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Proof. Applying u to the evaluation morphism HomC (X, vY )⊗X → vY
and using the counit of the adjunction, we get a composite morphismu HomC (X, vY ) ⊗ uX → uvY → Y , hence a morphismu HomC (X, vY ) → HomD (uX, Y )and finally a morphism
HomC (X, vY ) → v HomD (uX, Y )which is checked by Yoneda’s lemma to be an isomorphism when (A.1)
is.

A.2. Categories with suspension ([22, Ex. 11.1], [13, A.2.4]). A
category A provided with an endofunctor L of A is called a category
with suspension. They form a 2-category as in [13, Def. A.25]: a1-morphism is a functor with a natural isomorphism of commutation
with the suspensions, and a 2-morphism is the “obvious” notion (it will
not be used in this paper). We say that L is invertible if it is a selfequivalence. By [13, Lemma A.26], the full embedding of the 2-categoryof categories with invertible suspension into that of all categories withsuspension has a 2-left adjoint, which sends (A, L) to (A[L−1 ], L̃) whereA[L−1 ] has objects (A, n) for A ∈ A, n ∈ Z, morphisms(A.2)

A[L−1 ]((A, m), (B, n)) =lim−→

A(Lk+m A, Lk+n B)

k+m≥0,k+n≥0

and L̃(A; n) = (A, n + 1). This yields:
Lemma A.2.1. Let (A, L), (A0 , L0 ) be two categories with suspension.
(1) If A is Karoubian, so is A[L−1 ].
(2) Let F : (A, L) → (A0 , L0 ) be a 1-morphism of categories withsuspension. If F is full (resp. faithful), so is the induced 1morphism F̃ : A[L−1 ] → A0 [L0 −1 ].
Proof. (2) is obvious in view of Formula (A.2). For (1), let e = e2 be an
endomorphism of (A, n) ∈ A[L−1 ]. By (A.2) again, there exists k  0such that n + k ≥ 0 and some ek = e2k ∈ End(Ln+k A) mapping to e via
the canonical functor ρ : A → A[L−1 ]. Let B = Im ek ; then (B, −k) is
an image of e.

A.3. Brown representability and compact generation. Recall
the following definitions and results of Neeman:
Definition A.3.1. A triangulated category T has the Brown representability property if(1) It is cocomplete.

MOTIVES WITH MODULUS, III

47(2) Any homological functor H : T op → Ab which converts infinite
direct sums into products is representable.
Lemma A.3.2 ([22, Cor. 10.5.3]). If T has the Brown representability
property, it is complete; a triangulated functor F : T → T 0 has a rightadjoint G if and only if it is strongly additive (Definition 3.2.1), andG is triangulated.

Example A.3.3. Suppose T is cocomplete and let R ⊂ T be a localising
subcategory: R is triangulated and closed under direct sums. Then the
inclusion functor R ,→ T and the localisation functor T → T /R are
strongly additive [8, Lemma 1.5].
Definition A.3.4. Let T be a triangulated category.
a) An object X ∈ T is compact if the functor Y 7→ T (X, Y ) is strongly
additive. We write T c for the thick subcategory of T consisting ofcompact objects.
b) A subset X of Ob(T ) generates T if its right orthogonal is 0.
c) T is compactly generated if it is cocomplete and generated by a(small) set of compact objects.
d) Given a subset X of Ob(T ), the thick hull of X in T is the smallest
triangulated subcategory of T which contains X and is closed under
direct summands.
Remark A.3.5. Suppose that T is cocomplete. Then a set X ⊂ Ob(T )of compact objects generates T in the sense of Definition A.3.4 b) if
and only if the smallest localising subcategory of T containing X isequal to T [38, Lemma 2.2.1].
Example A.3.6. Let A be an essentially small additive category and∼
B = Mod –A. Then T = D(B) is compactly generated and K b (A) −→T c [20, Prop. A.4.1].
We have the following very useful result of Beilinson-Vologodsky [5,Proposition in §1.4.2] (see also §1.2 of op. cit.):
Theorem A.3.7. Let T be a cocomplete triangulated category and let
S ⊆ T be a localising subcategory which is generated by a set of compact objects of T . Then the localisation functor T → T /S has a rightadjoint whose essential image is the right orthogonal S ⊥ of S. In particular, S = T ⇐⇒ S ⊥ = 0.
The two main results on compactly generated triangulated categories
are:
Theorem A.3.8 ([32, Th. 4.1]). Any compactly generated triangulated
category has the Brown representability property. In particular [22,48

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Th. 14.3.1], this is the case for the unbounded derived category of aGrothendieck abelian category.
Theorem A.3.9 ([31, Th. 2.1]). Let T be a compactly generated triangulated category. Let S ⊂ T be a localising subcategory generated
by a set of compact objects of T . Then T /S is compactly generated
and compact objects of T remain compact in T /S; the induced functor
T c /S c → (T /S)c is fully faithful and (T /S)c is the thick hull of T c /S cin T /S.
Corollary A.3.10 (cf. [20, Th. A.2.6]). In the situation of Theorem
A.3.9, the localisation functor T → T /S has a right adjoint, which
also has a right adjoint.
We shall also use the following lemma of Neeman, a special case of[20, Lemma 4.4.5]:
Lemma A.3.11. Let T be a cocomplete triangulated category and let
X ⊂ Ob(T ) be a set of compact objects. If X generates T (see Definition A.3.4 and Remark A.3.5), then the thick hull of X is T c .
A.4. Unbounded derived categories: complements.
Theorem A.4.1. Let A be an additive category.
a) Mod –A is a Grothendieck category with a set of projective generators.
b) If A is monoidal, its tensor structure canonically extends to Mod –A
through the additive Yoneda functor, and provides Mod –A with thestructure of a closed additive monoidal category.
c) The ⊗-structure of A extends uniquely to a ⊗-triangulated structureon the homotopy category K b (A).
d) The ⊗-structure of Mod –A has a total left derived functor, which isstrongly additive and provides D(Mod –A) with a closed ⊗-triangulated
structure.
e) If u : A → B is a monoidal functor, u! : Mod –A → Mod –B
is monoidal, and so are the functors K b (u) : K b (A) → K b (B) andLu! : D(Mod –A) → D(Mod –B).
Proof. a) See e.g. [2, Prop. 1.3.6] for the first statement; the projective
generators are given by E = {y(A) | A ∈ A}. b) is [26, Def. 8.2]
or [21, A.8]. c) is easy (define ⊗ termwise). d) This applies to any
right exact covariant bifunctor T : Mod –A × Mod –A → C, where C
is abelian and cocomplete: by a) and [22, Th. 14.4.3], K(Mod –A)
has enough homotopically projective objects (K-projective in the senseof Spaltenstein [39]), which means that the localisation functor λ :MOTIVES WITH MODULUS, III

49

K(Mod –A) → D(Mod –A) has a left adjoint γ. Then the formula
LT (C, D) := λT (γC, γD)
provides the desired total left derived functor. By Example A.3.3, λand γ are strongly additive; thus if T is strongly additive, so is LT .
Similarly, a left exact contravariant/covariant bifunctor S has a totalright derived functor RS given by the formula
RS(C, D) = λS(γC, ρD)
where ρ is right adjoint to λ (also apply a) and [18, Th. A.2.1 b)]).
In the case T = ⊗Mod –A , S = HomMod –A , these formulas immediately
imply that LT is left adjoint to RS, which gives a second justificationof the strong biadditiveness of ⊗Mod –A .
e) is [21, A.12] for the first statement; the second one is easy andthe third follows from the universal property of left derived functors as
Kan extensions.

Proposition A.4.2. Let G : A  B : F be a pair of adjoint functors
between Grothendieck abelian categories, with G exact. Then RF isright adjoint to RG = D(G). If further F is fully faithful, then so
is RF , and D(G) is a localisation. If, on the other hand, G is fully
faithful and F is exact, then D(G) is fully faithful.
Proof. The first statement is a special case of [22, Th. 14.4.5]. By [17,Lemma A.3.1], the next ones are equivalent to saying that the counit
morphism D(G)RF ⇒ IdA is an isomorphism, which follows from [18,Lemma A.2.4]. In the last case, the full faithfulness of D(G) is provendually since D(F G) ⇒ D(F )D(G) is an isomorphism, again by [18,Lemma A.2.4].
We shall also need the following elementary lemma, that we borrowfrom the Stacks project:
Lemma A.4.3 ([40, Lemma 13.30.2]). Let F : A → B and G : B → A
be functors of abelian categories such that F is a right adjoint to G.
Let K ∈ D(A) and let M ∈ D(B). If RF is defined at K and LG isdefined at M , then there is a canonical isomorphism
D(B)(M, RF (K)) ' D(A)(LG(M ), K).
Appendix B. Cubical objects and intervals
B.1. Cubical objects and associated complexes. We follow [25]but we omit the use of permutations and involutions. Let Cube be
the subcategory of Sets which has as objects n = {0, 1}n for n ∈ Z≥0

50

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI(with 0 = ∗ the terminal object of Sets) and whose morphisms are
generated bypni : n → n − 1(n ∈ Z>0 , i ∈ {1, . . . , n}),nδi,ε : n → n + 1(n ∈ Z≥0 , i ∈ {1, . . . , n + 1}, ε ∈ {0, 1}),nnwhere pi omits the i-th component and δi,εinserts ε at the i-th component.
Definition B.1.1. Let A be a category. A covariant (resp. contravariant) functor A : Cube → A is called a co-cubical (resp. cubical) object
in A;
Remark B.1.2. The definition of Cube in [25] is different from ours.
(It also contains other morphisms called permutations and involutions.)
However, concerning the following lemma, the same proof as in loc. cit.
works in our more basic setting.
Lemma B.1.3. Let A : Cubeop → A be a cubical object in a pseudoabelian category A. Put An := A(n).
(1) We have well-defined objects
n
n




M
M
∼
(n−1)∗degn∗: An →
An−1 ,
An := Im ⊕ pi :An−1 → An −→ Im ⊕ δi,1
i=1

i=1



(n−1)∗

Aνn := ker ⊕ δi,1

: An →

n
M





∼

An−1 −→ Coker ⊕ pn∗
i :

n
M

An−1 → An



i=1
i=1
∼
in A, and PAνn ⊕ Adeg
→ An holds.
n −
n+1
n∗
i n∗
) : An+1 → An . This makes A•(2) Let dn := i=1 (−1) (δi,1 − δi,0
a complex, of which Aν• and Adeg
are subcomplexes. The two•degν
complexes A• /A• and A• are isomorphic.

Proof. See [25, Lemmas 1.3, 1.6].



Remark B.1.4. We have obvious dual statements of Lemma B.1.3 for cocubical objects. We state them here for later use. Let A : Cube → A
be a co-cubical object in a pseudo-abelian category A. Put An := A(n).
(1) We have well-defined objects
n
n




M
M
∼
(n−1)
n−1
n
n
n
n
Adeg := Im ⊕ δi,1∗ :A→A −→ Im ⊕ pi∗ : A →An−1 ,i=1Anν := Ker ⊕ pni∗ : An →

i=1

n
M



∼

(n−1)

An−1 −→ Coker ⊕ δi,1∗

i=1
n ∼



in A, and A −→ Anν ⊕ Andeg holds.

:

n
M
i=1


An−1 → An ,

MOTIVES WITH MODULUS, III

51

P
n
n+1
n
i n
. This makes A•(2) Let dn := n+1i=1 (−1) (δi,1∗ − δi,0∗ ) : A → A
a complex, of which A•ν and A•deg are subcomplexes. The twocomplexes A• /A•deg and A•ν are isomorphic.
Remark B.1.5. Let A be pseudo-abelian and provided with an additive
unital symmetric monoidal structure ⊗. Let A : Cube → A be a cocubical object, and suppose that A is strict monoidal (i.e. A(m × n) =A(m) ⊗ A(n)). Then:
1) A0 = A0ν = 1 is the unit object of A, and A0deg = 0. For n >0, combining A1 = A1ν ⊕ A1deg and An = A1 ⊗ · · · ⊗ A1 , we get adecompositionAnν = A1ν ⊗ · · · ⊗ A1ν ,M
Andeg =A1σ(1) ⊗ · · · ⊗ A1σ(n) ,σ6≡νwhere σ ranges over all maps {1, . . . , n} → {ν, deg} except for the constant map with value ν.
2) A• has a canonical comonoid structure where the counit and comultiplication are respectively given by(B.1)

π • : A• → A0 [0] = 1,

π n = 0 (n > 0) and π 0 = IdA0 ,∆• : A• → Tot(A• ⊗ A• )P=
where ∆n = p+q=n ∆p,q with ∆p,q : Ap+q −→ Ap ⊗ Aq . In view of 1),
we see that A•ν inherits the same structure:(B.2)πν• : A•ν → 1,∆•ν : A•ν → Tot(A•ν ⊗ A•ν ).

B.2. Interval structure. Let A be a unital symmetric monoidal category. Recall from Voevodsky [42] the notion of interval:
Definition B.2.1. Let 1 be the unit object of A. An interval in A
is a quintuple (I, p, i0 , i1 , µ), with I ∈ A, p : I → 1, i0 , i1 : 1 → I,
µ : I ⊗ I → I, verifying the identities
pi0 = pi1 = 11 ,µ ◦ (1I ⊗ i0 ) = µ ◦ (i0 ⊗ 1I ) = i0 p,µ ◦ (1I ⊗ i1 ) = µ ◦ (i1 ⊗ 1I ) = 1I .
Definition B.2.2. Given an interval (I, p, i0 , i1 , µ) in A, we define astrict monoidal co-cubical object A : Cube → A by⊗(i−1)

An = I ⊗n , pni∗ = 1I⊗(n−i)⊗ p ⊗ 1I⊗(i−1)n, δiε∗= 1I⊗(n−i)⊗ iε ⊗ 1I(this does not use the morphism µ). When A is pseudo-abelian, we•
write I • , Iν• , Ideg
for the associated complexes introduced in Remark
B.1.4.

52

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

By definition and Remark B.1.5, we havepIνn = Iν ⊗ · · · ⊗ Iν with Iν = Ker(I −→ 1).
Remark B.2.3. Conversely, Levine introduced in [25] a notion of extended co-cubical object A : ECube → A, where ECube is the smallest symmetric monoidal subcategory of Sets that contains Cube andthe morphismµ̃ : 2 → 1; (a, b) 7→ ab.
Given such a (strict monoidal) extended co-cubical object A, we may
define an interval (I, p, i0 , i1 , µ) in A by
00
I = A(1), p = p11∗ , i0 = δ1,0∗, i1 = δ1,1∗, µ = µ̃∗ .

Such intervals are not arbitrary, as µ makes I a commutative monoid(because so does µ̃ with 1). However, all intervals encountered in practice are commutative monoids, including in [42, 44] and here (Lemma
5.1.1).
Definition B.2.4. a) An object X ∈ A is I-local at Y ∈ A5 if p∼
induces an isomorphism A(Y, X) −→ A(Y ⊗ I, X); X is I-local if it isI-local at Y for any Y ∈ A. If A is closed, it is equivalent to ask forthe morphismp∗

X−→ Hom(I, X)
to be an isomorphism.
f∗b) A morphism f : Y → Z in A is called an I-equivalence if A(Z, X) −→A(Y, X) is an isomorphism for any I-local X.
Lemma B.2.5. Let X, Y ∈ A. Then
(1) If X is I-local at Y , the maps 1Y ⊗ i∗0 , 1Y ⊗ i∗1 : A(Y ⊗ I, X) →A(Y, X) are equal.
(2) If the maps 1Y ⊗I ⊗i∗0 , 1Y ⊗I ⊗i∗1 : A(Y ⊗I ⊗I, X) → A(Y ⊗I, X)
are equal, then X is I-local at Y .
(3) X is I-local if and only if the maps i∗0 , i∗1 : A(Y ⊗ I, X) →A(Y, X) are equal for all Y ∈ A (equivalently when A is closed:
if and only if the maps i∗0 , i∗1 : Hom(I, X) → X are equal).
Proof. For (2), the last two identities of Definition B.2.1 imply thatp∗ i∗0 : A(Y ⊗ I, X) → A(Y ⊗ I, X) is the identity, hence the claim sincei∗0 p∗ is also the identity. (3) now follows from (1) and (2).
5This notion is useful in [35].

MOTIVES WITH MODULUS, III

53

Remark B.2.6. Actually, Definition B.2.1 is more general than Voevodsky’s definition in [42, 2.2] or (with Morel) [30, 2.2.3]. There, the ⊗category A is a site with products (in [42]) or the category of sheaves
on a site (in [30]), and the tensor structure is the one given by products of objects or of sheaves. Voevodsky constructs in [42, loc. cit.] auniversal cosimplicial object, whose general term is I n . Unfortunately,
the formulas of loc. cit. implicitly use diagonal morphisms which are
not available in general ⊗-categories, in particular in the ones we use
here (see Remark 2.1.4). So, while one can develop a cubical theory
out of Definition B.2.1, we do not know if this definition is sufficient todevelop a simplicial theory.
B.3. Homotopy equivalences.
Proposition B.3.1. Let A be a pseudo-abelian ⊗-category, provided
with an interval I. Let I • be as in Definition B.2.2. Then the morphisms(B.3)

1 ⊗ p11∗ : I • ⊗ I 1 [0] → I • ,(B.4)

1 ⊗ p11∗ : Iν• ⊗ I 1 [0] → Iν• ,(B.5)∆•ν : Iν• → Tot(Iν• ⊗ Iν• )are homotopy equivalences.
00
Proof. For (B.3), since p11 δ1,0
= 10 , the composition (1 ⊗ p11∗ )(1 ⊗ δ1,0∗
):
n
1
•
•
n
n+1 ∼
−
→ I ⊗ I be the tautological
I → I is the identity. Let s : I
isomorphism. The identities(n−1((δj,ε∗⊗ 1)sn−1 if j < n + 1
nsn δj,ε∗=1I n ⊗ iε
if j = n + 1

yield
sn dn − (dn−1 ⊗ 1)sn−1 = 1 ⊗ i1 − 1 ⊗ i0 .
Then the composition
sn ⊗1

1⊗µ

σ n+1 : I n+1 ⊗ I 1 −−−→ I n ⊗ I 1 ⊗ I 1 −−→ I n ⊗ I 1
0yields a chain homotopy from 1⊗(δ1,0∗p11∗ ) to 1⊗1, which concludes theproof. Now (B.4) is also homotopy equivalence as a direct summandof (B.3).
Consider (B.5). By induction and the homotopy equivalence (B.3),
we find that for any q > 0(B.6)

Id ⊗ (p11 p21 . . . pq1 )∗ : I • ⊗ I q [0] → I •is a homotopy equivalence. Since Iνq is a direct summand of I q contained in Ker((p11 p21 . . . pq1 )∗ ) by Remark B.1.4, we find that I • ⊗ Iνq [0]54

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

is contractible for q > 0. The same is true of Iν• ⊗ Iνq [0] because it is adirect summand of I • ⊗ Iνq [0]. Lemma B.3.2 (2) below then shows thatTot(1 ⊗ π • ) : Tot(Iν• ⊗ Iν• ) → Iν• is a homotopy equivalence, where π •is as in (B.1). Since Tot(1 ⊗ π • ) is left inverse to ∆•ν , this shows that∆•ν is a homotopy equivalence.

Lemma B.3.2. Let A be an additive category. Let us call a double
complex S •,• in A locally finite if {p ∈ Z | S p,n−p 6= 0} is a finite set
for each n ∈ Z.
(1) Let S •,• be a locally finite double complex in A. Suppose thatthe single complex S •,q is contractible for each q ∈ Z. ThenTot(S •,• ) is contractible.
(2) Let f •,• : S •,• → T •,• be a morphism of locally finite double
complexes in A. If f •,q is a homotopy equivalence for each q ∈Z, then so is Tot(f •,• ) : S •,• → T •,• .
Proof. (1)6 Let us write dS1 : S •,• → S •+1,• , dS2 : S •,• → S •,•+1 for thedifferentials of S •,• , and set dS = dS1 +dS2 . By assumption we have a map
s : S •,• → S •,• of bidegree (−1, 0) such that dS1 s + sdS1 = IdS •,• . ThusdS s + sdS − IdS •,• is an endomorphism of S •,• of bidegree (−1, 1), which
defines an endomorphism u of Tot(S •,• ) of degree 0. By assumption, u
restricted to each degree is nilpotent. Hence Id + u is an isomorphism,
which implies that Tot(S) is contractible.
(2) We shall use the following fact:(*) A morphism g of (simple) complexes is a homotopy
equivalence if and only if Cone(g) is contractible.
•,•
p,q
Let U be a cone of f , that is, U p,q = T
⊕ S p+1,q equipped with
T
T
d2 0
d1 f
: U p,q → U p,q+1 .
: U p,q → U p+1,q and dU2 =
dU1 =
0 dS2
0 dS1
For each q ∈ Z, we have U •,q = Cone(f •,q ), as (single) complexes. By
assumption and (*), they are contractible. Then (1) shows that Tot(U )is contractible. Since we have Cone(Tot(f )) = Tot(U ) by definition,
this implies that Tot(f ) is contractible by (*).

B.4. An adjunction. Let T be a tensor triangulated category, compactly generated (Definition A.3.4) and equipped with an interval(I, p, i0 , i1 , µ). We assume:
Hypothesis B.4.1. The tensor structure of T is strongly biadditive(i.e., − ⊗ − is strongly additive in each entry), and − ⊗ I preserves thefull subcategory T c of compact objects.
6 We learned this proof from J. Oesterlé. We thank him.

MOTIVES WITH MODULUS, III

55

By Theorem A.3.8, T enjoys the Brown representability propertyof Definition A.3.1. By Lemma A.3.2, ⊗ therefore has a right adjoint
Hom.
Definition B.4.2. Let RI ⊂ T be the localising subcategory gener1⊗pated by objects of the form Cone(X ⊗ I −−→ X) for X ∈ T . We write
TI for the Verdier quotient T /RI .
Proposition B.4.3.
(1) The functor HomT (I, −) is strongly additive.
(2) The category TI is compactly generated, hence has the Brown
representability property.
(3) The localisation functor LI : T → TI has a (fully faithful) rightadjoint j I , which also has a right adjoint RI .
(4) The essential image of j I consists of the I-local objects (Definition B.2.4 a)).
(5) The tensor structure on T induces a tensor structure on TI .
Proof. For (Xj )j∈J a family of objects of T , the invertibility of the map
M
M
HomT (I, Xj ) → HomT (I,Xj )can be tested on a set of compact generators; it then follows from
Hypothesis B.4.1. This also implies that RI is generated by a set ofcompact objects of T , hence (2) follows from Theorem A.3.9. Then (3)follows from Corollary A.3.10. (4) is obvious by adjunction, and (5)follows from the fact that if A ∈ RI and B ∈ T , then A ⊗ B ∈ RI . 
Remark B.4.4. The functor j I RI can be described by a double adjunction as follows: for X, Y ∈ T , we have
T (X, j I RI Y ) = T (LI X, RI Y ) = T (j I LI X, Y ).
Our main result in this appendix, Theorem B.4.5, is a computationof the localisation functor j I LI in terms of Iν• (see Definition B.2.2).
Ideally it should be expressed in the above framework. Unfortunately,
we do not know how to totalise Iν• into an object of T in general(compare [8, §3]). A nice setup would be to assume that T is provided
with a t-structure with heart A for which ⊗ is t-exact and such thatI ∈ A; unfortunately, the inclusion A ,→ T does not extend to D(A)(or even K(A)) in this generality. So, for simplicity, we take refuge inthe situation where T is of the form D(A) (and where I ∈ A).
The proof of the following theorem will occupy the next two subsections (see Theorem B.6.3).

56

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

Theorem B.4.5. Under suitable additional hypotheses (B.6.1 below),
there is a canonical isomorphism
j I LI (K) ∼(I • , K)
= Hom
D(A)

ν

for any K ∈ D(A).
B.5. Monadic intermezzo. Let C be a category and (C, η, µ) be amonad in C in the sense of [27, Ch. VI]. Recall what this means:• C is an endofunctor of C.
• η : Id → C is a natural transformation (unit).
• µ : C 2 → C is a natural transformation (multiplication).
• For any X ∈ C, we have the identities(B.7)µX ◦ C(µX ) = µX ◦ µC(X)(B.8)µX ◦ C(ηX ) = µX ◦ ηC(X) = 1C(X) .

We shall not use (B.7) in the sequel.
Let C(C) be the strictly full subcategory of C generated by the imageof C: an object of C is in C(C) if and only if it is isomorphic to C(X)for some X ∈ C; the morphisms of C(C) are the morphisms of C.
Proposition B.5.1. a) If µ is a natural isomorphism, the full embedding j : C(C) ,→ C has the left adjoint C.
b) Let C∗ be a second monad in C. Assume that the condition of a)holds for C and C∗ , and that(i) C∗ (C) ⊆ C(C).
(ii) For any X ∈ C(C), the unit map X → C∗ (X) is an isomorphism.
Then there is a natural isomorphism C ∼= C∗ .
∼

Proof. a) Let Y ∈ C(C) and choose an isomorphism u : Y −→ C(X)
with X ∈ C. By assumption, ηY : Y → C(Y ) is an isomorphism, thus
the second equality of (B.8) and the naturality of η imply that thecomposite
C(u)

µ

u−1

X
εY : C(Y ) −→ C 2 (X) −→C(X) −→ Y
is the inverse of ηY , hence does not depend on the choice of u, X. One
then easily checks that εY for Y ∈ C(C) defines a natural transformation ε : Cj → Id and that (η, ε) provides the unit and counit of thedesired adjunction.
In b), (i) implies that for any X ∈ C, the unit X → C∗ (X) factors
through the unit X → C(X) (use a)). On the other hand, (ii) implies
that C(C) ⊆ C∗ (C), so the same reasoning shows that, conversely, theunit X → C(X) factors through the unit X → C∗ (X).


MOTIVES WITH MODULUS, III

57

Remark B.5.2. The converse of a) is certainly false in general. The
point is that a given endofunctor C on C might have two completely
different monad structures. However, if (η, µ) yields an adjunction
between j and C, then µ must be a natural isomorphism because j isfully faithful. In particular, if we start from an adjunction (j, C) withj fully faithful, then the multiplication of the monad jC is a natural
isomorphism.
B.6. A formula for j I LI . Let T be as in Subsection B.4. We use thenotation introduced in Definition B.4.2. We assume here that T is of
the form D(A) for some Grothendieck abelian category A, whence acanonical t-structure. To Hypothesis B.4.1, we add:
Hypothesis B.6.1.
(i) ⊗D(A) is right t-exact, hence induces a right exact tensor structure on A denoted by ⊗A [4, Prop. 1.3.17 (i)]. (A ⊗A B :=H0 (A[0] ⊗D(A) B[0]).)(ii) Let ⊗K(A) be the canonical extension of ⊗A to K(A). Thenthe localisation functor λ : K(A) → D(A) is lax monoidal, i.e.,there is a collection of morphisms
λC ⊗D(A) λD → λ(C ⊗K(A) D)
binatural in (C, D) ∈ K(A) × K(A) and commuting with theassociativity and commutativity constraints.
(iii) 1D(A) , I ∈ A (hence I = λI[0]).
(iv) The map (λI[0])⊗D(A) n → λ(I ⊗A n [0]) induced by (ii) is an isomorphism for all n ≥ 0.
By adjunction, the composed functor j I LI has a canonical monadstructure. Note that its multiplication is an isomorphism because j I isfully faithful (compare Remark B.5.2).
Definition B.6.2. For K ∈ D(A), we let
RC∗I (K) = HomD(A) (Iν• , K) ∈ D(A).
Here we view the complex Iν• as an object of D(A). We call RC∗I (K)the derived cubical Suslin complex of K (relative to I).
The comonoidal structure on Iν• :π • : Iν• → 1,∆• : Iν• → Tot(Iν• ⊗ Iν• )58

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

given by (B.1), (B.2) induces a monad structure on RC∗I . For example
the multiplication is given by
RC∗I (RC∗I (K)) = HomD(A) (Iν• , HomD(A) (Iν• , K))∼(I • ⊗ I • , K)
= Hom
D(A)

ν

ν

(∆• )∗−→ HomD(A) (Iν• , K) = RC∗I (K)Note that the last map is an isomorphism by Proposition B.3.1. The
following theorem completes the proof of Theorem B.4.5.
Theorem B.6.3. The two monads j I LI and RC∗I are naturally isomorphic.
For any K ∈ D(A), the monad structure on RC∗I provides us with anatural morphism in D(A):(B.9)

ηK : K → RC∗I (K).

We prove the following result together with Theorem B.6.3.
Theorem B.6.4. Let K ∈ D(A).
a) The complex RC∗I (K) is I-local (Definition B.2.4 a)).
b) The morphism (B.9) is an isomorphism if and only if K is I-local.
c) The morphism (B.9) is an I-equivalence (Definition B.2.4 b)).
Proof of Theorems B.6.3 and B.6.4. (Compare [44, proof of Lemma 3.2.2]
or [26, proof of Lemma 9.14].) We first prove Theorem B.6.4 a) andb). In view of Definition B.2.4 and Hypothesis B.6.1 (iv), a) follows
from Proposition B.3.1 by adjunction. In b), if K is I-local, we have
Hom(Iν , K) = 0 and hence
HomD(A) (Iνn , K) ∼= HomD(A) (Iνn−1 ⊗ Iν , K)∼
(I n−1 , Hom(Iν , K)) = 0 for n > 0,
= Hom
D(A)

ν

D(A)

which implies that (B.9) is an isomorphism. Conversely, if (B.9) is an
isomorphism, then K is I-local by a).
Next we prove Theorem B.6.3. As mentioned before Definition B.6.2,
the multiplication of the monad j I LI is an isomorphism, and the sameis true for RC∗I as proven above. Theorem B.6.3 now follows from
Theorem B.6.4 a), b) and Proposition B.5.1 b).
Finally, Theorem B.6.4 c) follows from Theorem B.6.3.
Corollary B.6.5. a) For any K ∈ RI , RC∗I (K) = 0 in D(A).
b) The functor RC∗I is strongly additive.
c) The localising subcategory RI ⊂ D(A) is generated by the cones of

MOTIVES WITH MODULUS, III

59

the X → RC∗I (X) for X ∈ D(A). In particular, K ∈ D(A) is I-local
if and only if the natural map
HomD(A) (RC∗I (X), K[i]) → HomD(A) (X, K[i])is an isomorphism for any X ∈ D(A) and any i ∈ Z.
Proof. a) This is obvious from Theorem B.6.3 since RI ∩ j I D(A)I = 0,
the two categories being mutually orthogonal.
b) This follows from Theorem B.6.3 and the strong additivity of j I
and LI (Example A.3.3).
c) By Theorem B.6.4 c), for any X ∈ D(A) the cone of X → RC∗I (X)vanishes in D(A)I , hence it is in RI . Conversely, let R0I ⊂ D(A) be thelocalising subcategory generated by these cones. In the commutative
diagramp

I[0] ⊗D(A) X


y

−−−→

X


y

p0

RC∗I (I[0] ⊗D(A) X) −−−→ RC∗I (X)
p0 is an isomorphism by a), hence the cone of p belongs to R0I . The
last statement follows.

B.7. Comparison of intervals. Let (A, I), (A0 , I 0 ) be as in §B.6. Wegive ourselves a right exact cocontinuous monoidal functor T : A → A0
sending I to I 0 and respecting the constants of structure of I and
I 0 . By [17, Th. A.10.1 b)], T has a right adjoint S. We assume
that T has a total left derived functor LT : D(A) → D(A0 ), whichis strongly additive, a monoidal functor and sends I[0] to I 0 [0] (thisis automatic if T is exact). By Brown representability (Lemma A.3.2
and [18, Th. A.2.1 a)]), LT has a right adjoint RS, which is the totalright derived functor of S. Then LT induces a triangulated monoidal0functor LT : D(A)I → D(A0 )I 0 via LI and LI .
The following lemma is obvious:
0

Lemma B.7.1. Let j I and j I be the right adjoints of the localisa0tion functors LI : D(A) → D(A)I and LI : D(A0 ) → D(A0 )I 0 .
0Then RS sends j I D(A0 )I 0 into j I D(A)I , and the induced functor RS :D(A0 )I 0 → D(A)I is right adjoint to LT .
By construction, we have a natural isomorphism(B.10)

0

RSj I ' j I RS

60

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI

from which we deduce two “base change morphisms”
0(B.11)LI ◦ RS ⇒ RS ◦ LI

(B.12)LT ◦ j I ⇒ j I ◦ LT .

0

Theorem B.7.2. (B.11) is an isomorphism.
Proof. The monoidality of LT yields the following identity, for (X, K) ∈D(A) × D(A0 ) (Lemma A.1.1):(B.13)Hom(X, RSK) ∼= RS Hom 0 (LT X, K).
D(A)

D(A )

Apply (B.13) to X = Iν• : we get an isomorphism
RC∗I (RSK) ∼= RSRC∗0 (K).
In view of Theorem B.6.3, this converts to an isomorphism
00
j I LI RS(K) ∼= RSj I LI (K)0hence to an isomorphism LI RS(K) ∼= RSLI (K) in view of (B.10) andthe full faithfulness of j I . One checks that this isomorphism coincides
with (B.11).


Definition B.7.3. We say that T verifies Condition (V) if (B.12) is
an isomorphism.
Lemma B.7.4. T verifies Condition (V) if and only if LT (j I D(A)I ) ⊆0
j I D(A0 )I 0 .
Proof. “Only if” is obvious. Conversely, let X ∈ D(A)I be such that00
LT j I (X) ∼= j I Y for some Y ∈ D(A0 )I 0 . Applying LI , we get
0
0
0
Y ∼
= LI j I Y ∼= LT (X).
= LI LT j I (X) ∼= LT LI j I (X) ∼0

Applying j I , it gives an isomorphism00LT j I (X) ∼= jI Y ∼= j I LT (X)and one checks that this is induced by (B.12).



Example B.7.5 (see also [5, Remark (c) in 4.4]). Applying LI to theright of (B.12) and using Theorem B.6.3, one gets a natural transformation(B.14)

0LT ◦ RC∗I ⇒ RC∗I ◦ LT.

Take A = PST, A0 = NST, T = aVNis , I = I 0 = ZVtr (A1 ). Then thecondition of Lemma A.3.11 translates as: the sheafification of an A1 invariant complex of presheaves with transfers is A1 -invariant. Whenk is perfect, this is a theorem of Voevodsky [43, Th. 5.6], which canthen be used to prove the equivalence of categories mentioned in §3.1.

MOTIVES WITH MODULUS, III

61

Moreover, RC∗I yields the naı̈ve Suslin complex, because ZVtr ((A1 )n ) is
projective in PST for any n ≥ 0. Thus, the invertibility of (B.14)
means in this case that the derived Suslin complex is quasi-isomorphic
to the sheafification of the naı̈ve Suslin complex.
So, while the invertibility of (B.11) is a formal and general fact,
this is far from being the case for (B.12). If we take A = MPST,
A0 = MNST, T = aNis and I = I 0 = Ztr (), results in this direction
have been obtained by the third author in [35, Th. 0.4 and 0.6].
References[1] Y. André, Une introduction aux motifs: motifs purs, motifs mixtes,
périodes, SMF, 2004.
[2] Y. André, B. Kahn, Nilpotence, radicaux et structures monoı̈dales (with anappendix by P. O’Sullivan), Rend. Sem. Math. Univ. Padova 108 (2002),
107–291.
[3] J. Ayoub Les six opérations de Grothendieck et le formalisme des cycles
évanescents dans le monde motivique, II, Astérisque 315 (2007), vi+364
pp., 2008.
[4] A. Beilinson, J. Bernstein, P. Deligne, O. Gabber, Faisceaux pervers,
Astérisque 100, 1984/2018.
[5] A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, GAFA
(Geom. Funct. Anal.) 17 (2008), 1709–1787.
[6] F. Binda, A motivic homotopy theory without A1 -invariance, Math. Zeit.
295 (2020), 1475–1519.
[7] F. Binda, D. Park, P. A. Østvær, Triangulated categories of logarithmic motives over a field, preprint, https://arxiv.org/abs/2004.12298(2020).
[8] M. Bökstedt, A. Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234.
[9] E. Friedlander, V. Voevodsky Bivariant cycle cohomology, in : Cycles,
transfers and motivic cohomology theories, by E. Friedlander, A. Suslin, V.
Voevodsky, Ann. Math. Studies 143, Princeton University Press, 2000,138–
187.
[10] A. Huber, B. Kahn, The slice filtration and mixed Tate motives, Compositio Math. 142 (2006), 907–936.
[11] F. Ivorra, T. Yamazaki, Nori motives of curves with modulus and Laumon
1-motives, Canad. J. Math. 70 (2018), 868–897.
[12] F. Ivorra, T. Yamazaki, Mixed Hodge structures with modulus, to appear
in Journal of the IMJ (2020).
[13] B. Kahn, Zeta and L-functions of varieties and motives, LMS Lect. Notes
Ser. 462, Cambridge Univ. Press, 2019.
[14] B. Kahn, S. Saito, T. Yamazaki, Reciprocity sheaves (with two appendicesby Kay Rülling), Compositio Math. 152 (2016), 1851–1898.
[15] B. Kahn, S. Saito, T. Yamazaki, Motives with modulus, https://arxiv.
org/abs/1511.07124, withdrawn.
[16] B. Kahn, H. Miyazaki, Topologies on schemes and modulus pairs, Nagoya
Math. J. online: http://dx.doi.org/10.1017/nmj.2020.15.

62

B. KAHN, H. MIYAZAKI, S. SAITO, AND T. YAMAZAKI[17] B. Kahn, H. Miyazaki, S. Saito, T. Yamazaki, Motives with modulus, I:
Modulus sheaves with transfers for non-proper modulus pairs, ÉpijournalGéom. Algébrique 5 (2021), Art. 1, 46.
[18] B. Kahn, H. Miyazaki, S. Saito, T. Yamazaki, Motives with modulus, II:
Modulus sheaves with transfers for proper modulus pairs, Épijournal Géom.
Algébrique 5 (2021), Art. 2, 31.
[19] B. Kahn, S. Saito, T. Yamazaki, Reciprocity sheaves, II, Preprint 2019,https://arxiv.org/abs/1707.07398, to appear in Homology HomotopyAppl.
[20] B. Kahn, R. Sujatha, Birational motives, II: triangulated birational motives, IMRN 2017 (22), 6778–6831.
[21] B. Kahn, T. Yamazaki, Voevodsky’s motives and Weil reciprocity, Duke
Math. J. 62 (2013), 2751–2796.
[22] M. Kashiwara, P. Schapira, Categories and sheaves, Springer, 2006.
[23] S. Kelly, S. Saito, Smooth blowup square for motives with modulus, Bulletin
Polish Acad. Sci. Math (2020).
[24] M. Kerz, letter to Shuji Saito, 27-4-2014.
[25] M. Levine, Smooth motives, in : Motives and algebraic cycles, Fields Inst.
Commun. 56, 175–231. Amer. Math. Soc., Providence, RI, 2009.
[26] C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology,
Clay Math. Monographs 2, AMS, 2006.
[27] S. Mac Lane, Categories for the working mathematician, Grad. Texts in
Math. 5, Springer (2nd ed.), 1998.
[28] K. Matsumoto, Gysin triangles in the category of motifs with modulus,
preprint (2020), https://arxiv.org/abs/1812.10890[29] H. Miyazaki, Nisnevich topology with modulus, Annals of K-theory, 5(2020), No. 3, 581-604.
[30] F. Morel, V. Voevodsky, A1 -homotopy theory of schemes, Publ. Math.
IHÉS 90 (1999), 45–143.
[31] A. Neeman, The connection between the K-theory localisation theorem ofThomason-Trobaugh and Yao and the smashing subcategories of Bousfieldand Ravenel, Ann. Sci. Éc. Norm. Sup. 25 (1992), 547–566.
[32] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniquesand Brown representability, JAMS 9 (1996), 205–236.
[33] M. Raynaud, L. Gruson, Critères de platitude et de projectivité, Invent.
Math. 13 (1971), 1–89.
[34] K. Rülling,, T. Yamazaki, Suslin homology of relative curves with modulus,
J. London Math. Soc. 93 (2016) 567–589.
[35] S. Saito, Purity of reciprocity sheaves, Adv. in Math. 366 (2020), 107067,
70 pp.
[36] S. Saito, Reciprocity sheaves and logarithmic motives, preprint, 2020.
[37] A. Scholl, Classical motives, Motives, Proc. Symp. pure Math. 55 (I), 1994,
163–187.
[38] S. Schwede, B. Shipley, Stable model categories are categories of modules,
Topology 42 (2003), 103–153.
[39] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math.
65 (1988), 121–154.
[40] The Stacks project, https://stacks.math.columbia.edu.

MOTIVES WITH MODULUS, III

63[41] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque239, 1996.
[42] V. Voevodsky, Homology of schemes, Selecta Math. 2 (1996), 111-153.
[43] V. Voevodsky Cohomological theory of presheaves with transfers, in : Cycles, transfers and motivic cohomology theories, by E. Friedlander, A.
Suslin, V. Voevodsky, Ann. Math. Studies 143, Princeton University Press,
2000, 87–137.
[44] V. Voevodsky, Triangulated categories of motives over a field, in : Cycles,
transfers and motivic cohomology theories, by E. Friedlander, A. Suslin,
V. Voevodsky, Ann. Math. Studies 143, Princeton University Press, 2000,
188–238.
[45] V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chowgroups in any characteristic. Int. Math. Res. Not. 2002, no. 7, 351–355.
[46] V. Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies, J. pure appl. Algebra 214 (2010) 1384–1398.
[47] V. Voevodsky, Cancellation theorem, Doc. Math. Extra Volume Suslin(2010), 671–685.
[EGA2] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, II:étude globale élémentaire de quelques classes de morphismes, Publ. Math.
IHÉS 8 (1961), 5–122.
IMJ-PRG, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France
Email address: bruno.kahn@imj-prg.fr
RIKEN iTHEMS, Wako, Saitama 351-0198, Japan
Email address: hiroyasu.miyazaki@riken.jp
Graduate School of Mathematical Sciences, University of Tokyo,
3-8-1 Komaba,, Tokyo 153-8941, Japan
Email address: sshuji@msb.biglobe.ne.jp
Institute of Mathematics, Tohoku University, Aoba, Sendai 9808578, Japan
Email address: ytakao@math.tohoku.ac.jp