Higher Kac–Moody algebras and moduli
spaces of G-bundles
Giovanni Faonte, Benjamin Hennion, Mikhail Kapranov
January 8, 2017
Abstract
We provide a generalization to the higher dimensional case of theconstruction of the current algebra gppzqq, its Kac-Moody extension rgand of the classical results relating them to the theory of G-bundles
over a curve. For a reductive algebraic group G with Lie algebra g,
we define a dg-Lie algebra gn of n-dimensional currents in g. For any
symmetric G-invariant polynomial P on g of degree n ` 1, we get ahigher Kac–Moody algebra r
gn,P as a central extension of gn by thebase field k. Further, for a smooth, projective variety X of dimension
n ě 2, we show that gn acts infinitesimally on the derived moduli space
RBunrig
G pX, xq of G-bundles over X trivialized at the neighborhood ofa point x P X. Finally, for a representation φ : G Ñ GLr we construct
an associated determinantal line bundle on RBunrig
G pX, xq and provethat the action of gn extends to an action of r
gn,Pφ on such bundle forPφ the pn ` 1qth Chern character of φ.

1

Contents
0 Introduction

3

1 Derived analogs of functions and series
1.1 Derived adelic formalism . . . . . . . . . . . . . . . . . . . . .
1.2 Explicit models . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Comparison with the Dolbeault and B b -complexes. . . . . . . .
1.4 Representation-theoretic analysis . . . . . . . . . . . . . . . .
1.5 Residues and duality . . . . . . . . . . . . . . . . . . . . . . .

8
8
11
15
18
22

2 The residue class in cyclic cohomology
2.1 Cyclic homology of dg-algebras and categories . . . . . . . . .
2.2 Cyclic homology of schemes . . . . . . . . . . . . . . . . . . .
2.3 The residue class ρ P HC 1 pA‚n q . . . . . . . . . . . . . . . . . .

26
26
29
32

3 Higher current algebras and their central extensions
34
3.1 Cyclic homology and Lie algebra homology . . . . . . . . . . . 34
3.2 Higher current algebras and their central extensions . . . . . . 35
4 The Tate extension and local Riemann-Roch
4.1 Background on Tate complexes . . . . . . . . . . . . . . . . .
4.2 Tate objects in dg-categories . . . . . . . . . . . . . . . . . . .
4.3 The Tate class, the residue class and the local Riemann–Roch

38
38
45
49

5 Action on derived moduli spaces
5.1 Background on derived geometry . . . . . . . . . . . . . . . .
5.2 The Kodaira–Spencer homomorphism . . . . . . . . . . . . . .
5.3 Derived current groups and moduli of G-bundles . . . . . . . .
5.4 Central extensions associated to Tate complexes . . . . . . . .
5.5 Action on determinantal torsors . . . . . . . . . . . . . . . . .

53
54
56
63
67
75

A Model categories of dg-algebras and dg-categories

80

2

0

Introduction(0.1) Let G be a reductive algebraic group over C with Lie algebra g.
The formal current algebra gppzqq “ g bC Cppzqq and its central extension rg(the Kac-Moody algebra) play a fundamental role in many fields. It can beconsidered as the algebraic completion of the loop algebra MappS 1 , gq, see[PreS].
In particular, gppzqq is fundamental in the study of BunG pXq, the moduli
stack of principal G-bundles on a smooth projective curve X over C. Moreprecisely, let x P X be a point. We then have the scheme (of infinite type)p,
Bunrig
G pX, xq parametrizing bundles P together with a trivialization on xp
p is OX,x » Crrzss,
the formal neighborhood of x. The ring of functions on x
the completed local ring and its field of fractions Kx » Cppzqq corresponds
to the punctured formal neighborhood xp˝ .
The key result [KNTY] is that the Lie algebra gx “ g b Kx acts on thescheme Bunrig
G pX, xq by vector fields. Moreover, any representation φ of G
gives rise to the determinantal line bundle detφ on Bunrig
G pX, xq; the action
φof gx extends to the action, on det , of the central extension r
gx with central
charge given by a local version of the Riemann-Roch theorem for curves.
(0.2) Our goal in this paper is to generalize these results from curves ton-dimensional varieties X over C, n ě 1 (one can replace C by any fieldof characteristic 0). The first question in this direction is what should play
the role of gppzqq. In the analytic (as opposed to the formal series) theory,
natural generalizations of MappS 1 , gq are provided by the current Lie algebras
MappΣ, gq “ g bC C 8 pΣq where Σ is a compact C 8 -manifold of dimension
ą 1. Our approach can be seen as extending this idea to the derived category.
More precisely, the role of gppzqq will be played by the dg-Lie algebra g‚n “
g bC A‚n , where A‚n “ RΓpDn˝ , Oq is the commutative dg-algebra of derived
global sectionsof the sheaf`
˘ O on the n-dimensional punctured formal disk˝Dn “ Spec Crrz1 , . . . , zn ss ´ t0u. More invariantly, we have the puncturedp˝ » Dn˝ associated to a point x P X and the corresponding
formal disk x
current algebra g‚x » g‚n . For n ą 1, passing from the non-punctured formalp, to xp˝ does not increase the ring of functions (Hartogs’ theorem) but
disk xone gets new elements in the higher cohomology of the sheaf O, so A‚n canbe regarded as a “higher” generalization of the Laurent series field Cppzqq,
3

to which it reduces for n “ 1.
Principal bundles on X form an Artin stack BunG pXq and we can still
form a scheme Bunrig
G pX, xq as above. However these objects are, for n ą 1,
highly singular because deformation theory can be obstructed. The correct
object to consider is the derived moduli stack RBunG pXq obtained, informally, by taking the non-abelian derived functor of BunG , i.e., extending
the moduli functor to test rings which are commutative dg-algebras [TV].
When X is a curve, RBunG pXq » BunG pXq, but for n ą 1 there is a difference. Most importantly, the tangent complex of RBunG pXq is perfect(a smoothness property). We can similarly construct the derived scheme
RBunrig
G pX, xq, (an object which locally looks like the spectrum of a commutative dg-algebra) which should also be intuitively considered as being
smooth.
We show, first of all, (Theorem 5.3.8) that g‚n acts on RBunrig pX, xq by
vector fields, in the derived sense. At the level of cohomology, the action
gives, in particular, a map`
˘
HBn´1 pg‚n q ÝÑ Hn´1 RBunrig
G pX, xq, T(here T is the tangent complex and B is the differential of g‚n ). When n “ 1,it is the action by vector fields in the usual sense. In the first new casen “ 2, after restricting to the non-obstructed smooth part of the moduli
space, on which T is the usual tangent sheaf, the target of this map becomes
the space of deformations of the (part of the) moduli space. Deforming themoduli space can be understood as changing the cocycle condition defining
G-bundles (Remark 5.3.9).
Further, each invariant polynomial P on g of degree pn ` 1q gives rise toa central extension rg‚n,P (the higher Kac-Moody algebra). Note that unlike
the case n “ 1, we now have many non-proportional classes, even for g
simple. Intuitively, they correspond to degree n ` 1 characteristic classes forprincipal G-bundles. As before, let φ be a representation of G. We prove(Theorem 5.5.9) that the determinantal line bundle detφ on RBunrig pX, xq
is acted upon by rg‚n,Pφ where Pφ pxq “ trpφpxqn`1 q{pn ` 1q! is the “pn ` 1q-th
component of the Chern character” of φ.
These results suggest that representations of the dg-Lie algebra g‚n should
produce geometric data on the derived moduli spaces of G-bundles on ndimensional manifolds.
4

(0.3) The stack BunG pXq can be seen as a version of the non-abelian firstcohomology H 1 pX, GpOX qq. When X is a curve, the above classical results
can be seen as forming a part of the “adelic approach” to the geometry of
curves. This approach consists in using the idealized “Čech covering” of X
p and X ˝ “ X ´ txu, with “intersection” xp˝ , to calculate the H 1 .
formed by x
If X is a curve and G is semi-simple, then G-bundles on X ˝ (and certainly
on xp) are trivial, and we can write BunG pXq “ Gpp
xqzGppx˝ q{GpX ˝ q (stacktheoretic quotient on the left). We then similarly represent Bunrig
G pX, xq as˝˝˝
the coset space Gpp
x q{GpX q, with Gpp
x q “ GpKx q being a group ind-schemewith Lie algebra gx .
A generalization of the adelic formalism to varieties X of dimension n ą 1
was proposed by Parshin and Beilinson [Be1] [Hu] [Os]. In this approach thecompleted local fields (analogs of Kx for curves) are parametrized not bypoints, but by flags txu “ X0 Ă X1 Ă ¨ ¨ ¨ Ă Xn´1 Ă X of irreducible subvarieties in X. If all the Xi ’s are smooth, then the completion is isomorphic to
kppz1 qq ¨ ¨ ¨ ppzn qq, the iterated Laurent series field. As before, it can be seen
as a version of the Čech formalism for an idealized open covering formedby certain formal neighborhoods. However, the manipulations with iterated
Laurent series fields are quite complicated: in order to capture all the “adictopologies”, they should be considered as n-fold iterated ind-pro-objects (nTate spaces) [BGW] and every step involves many levels of technical work.
In a sense, our approach can be seen as a “simplified version” of theflag adele formalism, in which we keep track only of points x P X (justlike for curves) and package all the combinatorial data involving subvarietiesof dimensions ‰ 0, n, into a “black box” using the cohomological formalism.
This allows us to avoid working with iterated ind-pro-objects and deal insteadwith classical Tate spaces (just like for curves) at the small price of having topass to the derived category of such spaces, i.e., to study Tate dg-spaces (orTate complexes), see §4.1 for details. For example, A‚n is a Tate complex foreach n. Our treatment is an adaptation and development of the approachesof [Dr] [He3].
(0.4) To relate our approach to the idea of MappΣ, gq, we can use a particular model A‚n of the “abstract” commutative dg-algebra A‚n “ RΓpDn˝ , Oq.
This model is formed by relative differential forms on the Jouanolou torsor,
see §1.2B. Such torsors have been used in [BD] as a general tool. In our case,A‚n provides a very precise algebraic analog of Ω0,‚ pCn ´ t0uq, the B-algebra5

of Dolbeault forms on Cn ´ t0u. In particular, such features of classical complex analysis as the Martinelli-Bochner form or its “multipole” derivatives,
have direct incarnations in A‚n , see Proposition 1.4.7. Our algebraic approachallows us to include these features in the formal setting of Tate (dg-)spaces.
It also lends itself to a representation-theoretic analysis providing the analog
of representing elements of kppzqq as infinite sums of monomials (Theorem
1.4.2).
Restricting from Cn ´ t0u to the unit sphere S 2n´1 , we can see A‚n as
2n´1
q, the tangential Cauchy-Riemann complex
an algebraic analog of Ω0,‚
b pS
2n´1
q
(the B b -complex [BeGr][DT]) of the sphere. The degree 0 part of Ω0,‚
b pS
8
2n´1
2n´1
is C pSq, the algebra of smooth complex functions on S
. This meansthat the degree 0 part of g‚n can be seen as an algebraic version of the current
Lie algebra MappS 2n´1 , gq, and the entire g‚n as a natural derived thickeningof this current algebra.
Usually, considering MappΣ, gq with dimpΣq ą 1, produces Lie algebras
which, instead of interesting central extensions (classes in H 2 ) have interesting higher cohomology classes. These classes are typically given by versionsof the formulaż
(0.5)
γpf0 , . . . , fn q “
trpf0 df1 ¨ ¨ ¨ dfn q.
In our case (Theorem 3.2.1), we still use a version of this formula (withintegration over S 2n´1 , done algebraically) but the classes we get have total
degree 2 and so give central extensions, regardless of n. This happens because
we take into account the grading on the dg-algebra, In this sense our derived
approach embeds MappS 2n´1 , gq into an object whose properties are closer
to those of MappS 1 , gq.
All this suggests that our higher Kac-Moody algebras should have aninteresting representation theory.
(0.6) As in the 1-dimensional case, a key intermediate step for us is a local
analog of the Riemann-Roch theorem (Corollary 4.3.10). It has the form ofcomparison of two central extensions of g‚n for g “ glprq: one given by aversion of (0.5), the other induced from the “Tate class” of the the endomorphism dg-algebra of the Tate complex pA‚n q‘r . For r “ 1 this statement canbe seen as an analog, in our simplified adelic formalism, of the main resultof Beilinson [Be1]. Since we deal with the current algebras only, we detect6

only the Chern character; the Todd genus will naturally appear, as in [FT2],
after we include the dg-algebra RΓpDn˝ , Tq, see (0.8) below.
(0.7) We use three main technical tools. The first one is the general theory of derived stacks [TV]. It is necessary for us to work freely with quite
general derived stacks and even prestacks in order to study, for instance, thegroup object corresponding to g‚n . This is an (infinite dimensional) derived
group GpDn˝ q, see Proposition 5.3.6. In particular, for dealing with various
infinitesimal constructions (even such seemingly simple ones as “passing froma group to its Lie algebra”) we need to use Lurie’s formalism of formal moduli
problems [Lu3].
The second technical tool is cyclic homology of dg-categories, a conceptof great flexibility and invariance. It includes, in particular cyclic (and deRham) homology for schemes and at the same time, is related to the Liealgebra homology of endomorphism dg-algebras of objects in a dg-category.
Another important tool is the GLn -invariance of our cohomology classes.
In exploiting this invariance, the Jouanolou model for A‚n is more convenientin that it allows an explicit GLn -action which can be analyzed in detail atthe level of complexes. This level of detail is not available for more abstract
models, e.g., for the flag-adelic one.
(0.8) This paper is related to the idea, mentioned already in [BD] and developed in [FG], of generalizing the theory of chiral and holomorphic factorization algebras to higher dimensions. In this approach we are dealing withfinite collections pxi q of points moving in an n-dimensional variety X, with“singularities” developing when xi “ xj for some i ‰ j. These singularities
are of cohomological nature, representing classes in H n´1 pX N ´ tdiagu, Oq.
Note that the standard quantum field theory approach deals with collections of points in the Minkowski space with singularities developing whensome xi ´ xj lies on the light cone, see, e.g., [Tam] for a discussion fromthe factorization algebra point of view. However, our cohomological formulas in the Jouanolou model are, up to “details” involving the cohomologicalgrading, algebraically similar to this. For example, the role of the standard“propagator” 1{}z}2 is played by the Martinelli-Bochner form Ωpz, z ˚ q.
The next natural step in this direction would be to study the dg-Lie algebra RΓpDn˝ , Tq, the analog of the Witt algebra of formal vector fields on the7circle (as well as its central extensions). It should act on the derived moduli
stack of n-dimensional rigidified complex manifolds pX, x, pz1 , . . . , zn qq where
x P X is a marked point and pz1 , . . . , zn q is a formal coordinate system nearx. There is a natural combined version involving the derived Atiyah algebra(RΓ of the semidirect product of matrix functions and vector fields). We plan
to address these issues in a future paper. The additional technical difficulty
here is the need to work with (quite general) derived stacks in the analytic
context, as not all deformations of an algebraic variety are algebraic.
(0.9) We would like to thank A. Beilinson, L. Hesselholt, P. Schapira andB. Vallette for useful correspondence. B.H. is thankful to P. Safronov andM.K. to A. Polishchuk for interesting discussions.
The work of M.K. was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. It was also partially
supported by the EPSRC Programme Grant EP/M024830 “Symmetries andCorrespondences”.

1

Derived analogs of functions and series

1.1

Derived adelic formalism

A. Local part. We fix a base field k of characteristic 0. For n ě 1 we havethe n-dimensional formal disk Dn “ Spec krrz1 , . . . , zn ss and the punctured
formal disk Dn˝ “ Dn ´ t0u. We consider them as the completion of theaffine space An “ Spec krz1 , . . . , zn s and of the punctured affine space Ån “
An ´ t0u.
Fundamental for us will be the commutative dg-algebraspA‚rns , Bq “ RΓpÅn , Oq.

pA‚n , Bq “ RΓpDn˝ , Oq,

defined uniquely up to quasi-isomorphism. The cohomology of these dgalgebras is well known and can be obtained using the covering of Dn˝ by naffine open subsets tzi ‰ 0u.
Proposition 1.1.1. For n “ 1 the scheme D1˝ is affine with ring of functions

8

kppzqq and Å1 is affine with ring of functions krz, z ´1 s. For n ą 1 we have$’i “ 0;&krrz1 , . . . , zn ss,
i
˝
´1
´1
´1
´1
H pDn , Oq “ z1 ¨ ¨ ¨ zn krz1 , . . . , zn s, i “ n ´ 1;’%
0,otherwise.
Here the notation z1´1 ¨ ¨ ¨ zn´1 krz1´1 , . . . , zn´1 s can be seen as encoding the action of the n-dimensional torus Gnm on H n´1 pDn˝ , Oq.
The cohomology H i pÅn , Oq differs from the above by replacing krrz1 , . . . , zn ss
by krz1 , . . . , zn s.
Thus, although for n ą 1, the scheme Dn˝ . resp. Ån , is not affine and its
global functions are the same as for Dn , resp. An , the missing “polar parts”
are recovered in the higher cohomology of the sheaf O. The dg-algebras Arns
and An are, therefore, correct n-dimensional generalizations of the the ringsof Laurent polynomials and Laurent series in one variable.
We will also use the doubly graded dg-algebraspA‚‚n , B, Bq “àRΓpDn˝ , Ωp q,

pA‚‚
rns , B, Bq “p

àRΓpÅn , Ωp q,

pwith B being induced by the de Rham differential on forms (and increasing
the first grading) and B being the differential on RΓ (and increasing thesecond grading).
B. Global part. Let X be a smooth n-dimensional variety over k and
pX,x which isx P X a k-point. We then have the completed local ring O
pX,x
p “ Spec O
isomorphic (non-canonically) to krrz1 , . . . , zn ss. We denote x
˝
the formal disk near x and by xp “xp ´ txu the punctured formal disk nearx. We then form the commutative dg-algebrasàx˝ , Oq, A‚‚“RΓppx˝ , Ωp q.
A‚x “ RΓppxp

In particular, the Grothendieck duality defines a canonical linear functional(1.1.2)δx˝ , Ωn q ÝÑ RΓtxu pp
x, Ωn qr1s ÝÑ kr1 ´ ns.
Resx : RΓpp

Let now x “ tx1 , . . . , xm u Ă X be a finite set of disjointŮk-points. We denote
Ů ˝
p “
p˝ “
pi and xpi .
X ˝ “ X ´ x the complement of x, and write xxx
9In particular, we have the commutative dg-algebra RΓpX ˝ , Oq. Elements ofthis dg-algebra can be seen as n-dimensional analogs of rational functions
on a curve with poles in x1 , . . . , xm . Similarly, if E is a vector bundle onx˝ , Oq andx˝ , Eq over A‚x “ RΓpp
X, we have the dg-modules A‚x pEq “ RΓpp
˝
˝RΓpX , Eq over RΓpX , Oq. We have the canonical morphism of complexes(commutative dg-algebras for E “ O) given by the restriction:(1.1.3)

δ : RΓpX ˝ , Eq ‘ Γpp
x, Eq ÝÑ Ax pEq “màA‚xi pEq.

i“0This morphism can be seen as the dg-version of the adelic complex of Beilinson [Be1] [Hu] , in which the dependence on schemes of dimensions 1, . . . , n´1
has been “integrated away” and hidden in the cohomological formalism. The
proof of the next proposition can be seen as an explicit comparison.
Proposition 1.1.4. The cone of δ is identified with RΓpX, Eq.
Proof: For any Noetherian scheme Y of dimension n over k (proper or not)and any coherent sheaf F on Y , the construction of [Be1][Hu] provides an
explicit model C ‚ pFq for RΓpY, Fq. By definition,àC p pFq “
Ci0 ,...,ip pFq,
0ďi0 ă¨¨¨ăip ďn

where
Ci0 ,...,ip pFq “ź’

CY0 ,...,Yp pFq

Y0 Ă¨¨¨ĂYp ĂY
dimpYν q“iν

is the appropriate restricted product, over all flags Y0 Ă ¨ ¨ ¨ Ă Yp Ă Yof irreducible subschemes, of the completions CY0 ,...,Yp pFq as described in[Be1][Hu]. In particular, for p “ 0 and i0 “ 0, the summand C0 pFq is theusual product of Cy pFq “ Γppy , Fq over all 0-dimensional points y P Y . We
now take Y “ X, F “ E and represent(dC ‚ pFq “ Cone C1‚ ‘ C2‚ ÝÑ C3‚ ,identifying the three summands with the corresponding summands in (1.1.3)
and d with δ. Explicitly, we take C2 to be the direct sum of Cy pFq overy P x, we take C1‚ to be the direct sum of restricted products of CY0 ,...,Yp pFq
over all flags Y0 Ă ¨ ¨ ¨ Ă Yp such that Y0 is any subvariety (of any dimension)
10other than some y P x. Then C2‚ is the adelic complex for the restriction of E
Y “ X ´ x. Similarly, we take C3 to be the direct sum of restricted productsof CY0 ,...,Yp pFq over all flags Y0 Ă ¨ ¨ ¨ Ă Yp such that Y0 equals some y P x.
p. This proves theThen C3‚ is the adelic complex for the restriction of E to x
statement.
In particular, for E “ ΩnX , we have morphisms
Resxi ,X : RΓpX ˝ , Ωn q Ñ kr1 ´ ns,

i “ 1, . . . , m,which satisfy the residue theorem:
řProposition 1.1.5. m
i“1 Resxi ,X “ 0.
Proof: This is a standard feature of Grothendieck duality, cf. [Con]. By
degree considerations it suffices to look at the behavior on the pn ´ 1q-st
cohomology only. That is, we consider, for each i, the map H n´1 pResi q :H n´1 ppx˝i , Ωn q Ñ k induced by Resxi on the pn ´ 1qst cohomology, and provethat the compositions of these maps with the H n´1 pX ˝ , Ωn q Ñ H n´1 pp
x˝i , Ωn q
sum to zero.
Indeed, H n´1 pResi q can be represented as the composition
ιx

tr

i
n
n
H n pX, Ωn q ÝÑ k,
H n´1 ppx˝i , Ωn q » Htx
pp
xi , Ωn q “ HtxpX, Ωn q ÝÑ
iu
iu

where tr is the global Serre duality isomorphism. Now the statement follows
from the fact that in the long exact sequence relating cohomology with andwithout support in x,
ιx “

δ

ř

ιx

¨ ¨ ¨ Ñ H n´1 pX ˝ , Ωn q ÝÑ Hxn pX, Ωn q ÝÑ i H n pX, Ωn q Ñ ¨ ¨ ¨
the composition of any two consecutive arrows is zero.

1.2

Explicit models

We start with the “polynomial” dg-algebra A‚rns . By considering the fibrationÅn Ñ Pn´1 , we can write(1.2.1)A‚rns „

àRΓpPn´1 , Opiqq.

iPZ

11

From here, passing to the completion is easy: it is similar to passing from
Laurent polynomialsÀ
in one variable to Laurent series. More precisely, for agraded vector space iPZ V i we denoteÿi"´8

V i “ lim
ÝÑ lim
ÐÝ
a

b

b
à

Vi

i“´a

ř
the vector space formed by Laurent series i"´8 vi with vi P V i . Thenÿ
(1.2.2)A‚n „RΓpPn´1 , Opiqqi"´8

Applying various way of calculating the cohomology of Pn´1 , we get various
explicit models for An and related dg-algebras and modules.
A. The Čech model. Covering Pn´1 with open sets tzi ‰ 0u, or, what is
the same, covering Dn˝ with similar open sets right away, we get a model for A‚n
as the Čech complex of this covering. The Alexander-Whitney multiplication
makes this complex into an associative but not commutative dg-algebra.
We can use Thom-Sullivan forms to replace this by a commutative dgalgebra model for A‚n .
B. The Jouanolou model. Introduce another set of variables z1˚ , . . . , zn˚which we think of as dual to the zν , i.e., as the coordinates in the dual affine
q n . We write
space Aÿq n.
zz ˚ “zν zν˚ , z P An , z ˚ P A
qn “ Proj krz ˚ , . . . , z ˚ s
We form the corresponding “dual” projective space P
1nand consider the incidence quadricˇ(
qn ˇ zz ˚ “ 0 Ă An ˆ P
qn .
Q “ pz, z ˚ q P An ˆ P
qn q ´ Q by J and note that the projection
We denote the complement pAn ˆ P
to the first factor gives a morphism
π : J ÝÑ Ån12

whose fibers are affine spaces of dimension n ´ 1. We refer to J as the
Jouanolou torsor for Ån . For further reference let us point out that(q n ´ t0uq | zz ˚ “ 1 ,(1.2.3)J » pz1 , . . . , zn , z1˚ , . . . , zn˚ q P An ˆ pA
q n ´ t0u Ñ P
qn on the second factor.
the isomorphism given by the projection An
For any quasi-coherent sheaf E on Å we then have the global relative deRham complex
˘`A‚rns pEq “ Γ J, Ω‚J{Ån b π ˚ E .
The differential in A‚rns pEq (given by the relative de Rham differential) will
be denoted B.
Let alsopπJp “ J ˆAn Dn “ J ˆÅn Dn˝ ÝÑ Dn˝

be the restriction of J to the punctured formal disk. As before, Jp is an affine
scheme and an An´1 -torsor over Dn˝ . For any quasi-coherent sheaf E on Dn˝
we denote`˘˚p Ω‚p ˝ b πpA‚n pEq “ Γ J,
E
.
J{D
n

Proposition 1.2.4. (a) A‚rns pEq is a model for RΓpÅn , Eq, and A‚n pEq is amodel for RΓpDn˝ , Eq.
(b) The functor E ÞÑ A‚rns pEq (resp. E ÞÑ A‚n pEq) is a lax symmetricmonoidal functor from the category of quasi-coherent sheaves on Ån (resp.
on Dn˝ ) to the category of complexes of k-vector spaces. In particular, if E isa quasi-coherent commutative OÅn -algebra (resp. ODn˝ -algebra), then A‚rns pEq(resp. A‚n pEq) is a commutative dg-algebra.
Proof: (a) This is a classical argument. We consider the only case of A‚rns pEq.
Because J is affine, we have quasi-isomorphisms
A‚rns pEq „ RΓpJ, Ω‚J{Pn´1 b π ˚ Eq „ RΓpÅn , Rπ˚ pΩ‚J{Ån b π ˚ Eqq.
Because π is a Zariski locally trivial fibration with fiber An´1 , the Poincarélemma for differential forms on An´1 implies that the embeddings
E ãÑ π˚ pΩ‚J{Ån b Eq ãÑ Rπ˚ pΩ‚J{Ån b Eq
are quasi-isomorphisms of complexes of sheaves on Ån , whence the statement.
13

(b) Obvious by using the multiplication of differential forms.
By the above, the dg-algebrasA‚rns “ A‚rns pOÅn q,(1.2.5)A‚n “ A‚n pODn˝ q

are commutative dg-models for A‚rns and A‚n respectively. Their grading issituated in degrees r0, n ´ 1s. Let us reformulate their definition closer to(1.2.1) and (1.2.2). For this, let
J “ˇ( π n´1
qn´1 ˇ zz ˚ ‰ 0 ÝÑ
pz, z ˚ q P Pn´1 ˆ P
Pbe the classical Jouanolou torsor for Pn´1 . For a quasi-coherent sheaf F on
Pn´1 we define`
˘‚
˚RΓpJq pPn´1 , F q “ Γ J, ΩJ{Pn´1 b π F .
As before, this is a model of RΓpPn´1 , F q, depending on F in a way compatible with the symmetric monoidal structures.
Proposition 1.2.6. We have isomorphism of commutative dg-algebrasÿàRΓpJq pPn´1 , Opiqq.
A‚rns “RΓpJq pPn´1 , Opiqq, A‚n “ii"8C. The Jouanolou model, explicitly. Let
krz, z ˚ s “ krz1 , . . . , zn , z1˚ , . . . , zn˚ s,krrzssrz ˚ ss “ krrz1 , . . . , zn ssrz1˚ , . . . , zn˚ s

q n and Dn ˆ A
q n respectively.
be the algebras of regular functions on An ˆ A
Proposition 1.2.7. Let m “ 0, . . . , n ´ 1. The m-th graded component Am
rns(resp. A‚n ) is identified with the vector space formed by differential formsÿω “
fi1 ,...,im pz, z ˚ qdzi˚1 ¨ ¨ ¨ dzi˚m1ďi1 ă¨¨¨ăim ďn

where each fi1 ,...,im is an element of the localized algebra krz, z ˚ srpzz ˚ q´1 s(resp. krrzssrz ˚ srpzz ˚ q´1 s) such that:14

(1) ω is homogeneous in the zν˚ , dzν˚ of total degree 0, that is, each fi1 ,...,im
is homogeneous of degree p´mq.
ř(2) The contraction ιξ pωq of ω with the Euler vector field ξ “ zν˚ B{Bzν˚vanishes.
The differential B is given by
n
ÿ

B “

dzν˚

ν“1

B
.
Bzν˚

m
Proof: We prove the statement about Am
rns ; the statement about An is provedin the same way.
q n and the incidence quadric Q
r inside it given
Consider the product An ˆ A
˚nnqr All forms ωby the same equation zz “ 0 as Q. Let U “ pA ˆ A q ´ Q.
as in the proposition (not necessarily satisfying the conditions (1) and (2))
r Ωm n q.
form the space ΓpJ,
U {ÅNow, we have a projection p : U Ñ J of Ån -schemes with the multiplicative group Gm acting simply transitively on the fibers (i.e., U is a Gm -torsor
over J). The infinitesimal generator of this action is the Euler vector field ξ.
Therefore relative forms from Ωmare identified with sections ω of Ωm
J{Ån
U {Ånwhich satisfyBξ pωq “ 0, ιξ pωq “ 0,where Bξ is the Lie derivative, see, e.g., [GKZ]. These conditions translate
precisely into the conditions (1) and (2) of the proposition.
Corollary 1.2.8. The dg-algebra A‚rns carries a natural filtration “by theorder of poles”
ˇ
(
Fr Am
ω ˇ pzz ˚ qr`m fi1 ,...,im P krz, z ˚ s, @i1 , . . . , im .
n :“compatible with differential and product:
BpFr A‚ q Ă Fr A‚ , pFr A‚ q ¨ pFr1 A‚ q Ă Fr`r1 A‚ .

1.3

Comparison with the Dolbeault and B b -complexes.

A. Comparison with the Dolbeault complex. In this section we assume k “ C. The notation B for the differential in A‚rns is chosen to suggest
the analogy with the Dolbeault differential in complex analysis. In fact, wehave the following.
15

Proposition 1.3.1. Let Ω0,‚ pCn ´ t0uq be the smooth Dolbeault complex of
the complex manifold Cn ´ t0u. We have a (unique, injective) morphism ofcommutative dg-algebras ε : A‚rns Ñ Ω0,‚ pCn ´ t0uq which sends zν˚ to z ν anddzν˚ to dz ν , i.e.,f pz, z ˚ qdzi˚1 ¨ ¨ ¨ dzi˚m ÞÑ f pz, zq|Cn ´t0u dz i1 ¨ ¨ ¨ dz im .
The proof is obvious once we notice that zz ˚ is being sent to zz “ |z|2
which does not vanish on Cn ´ t0u.
Remark 1.3.2. The morphism ε is not a quasi-isomorphism: it identifies
H ‚ pA‚n q with the “meromorphic part” of H ‚ pΩ0,‚ pCn ´ t0uqq “ H ‚ pCn ´t0u, Ohol q.
We also notice that A‚rns is concentrated in degrees r0, n´1s while Ω0,‚ pCn ´
t0uq is situated in degrees r0, ns. To exhibit a better analytic fit for A‚rns , werecall some constructions from complex analysis.
B. Reminder on the B b -complex. Let X be an n-dimensional complex
manifold and S Ă X a C 8 real hyper surface (of real dimension 2n ´ 1). The
embedding S Ă X induces on S a differential geometric structure known as
the CR-structure [BeGr][DT].
More precisely, let x P S. The 2n ´ 1-dimensional real subspace Tx S in
the n-dimensional complex space Tx X has the maximal complex subspace
Txcom S “ Tx S X ipTx Sq Ă Tx X
of complex dimension pn ´ 1q. We get a complex vector bundle TScom on S
embedded into the real vector bundle TS . Its complexification splits, in thestandard way, as
pTScom q b C “ TS1,0 ‘ TS0,1 Ă TS b C.
So TS1,0 and TS0,1 are complex vector sub-bundles in TS b C, of complex dimension pn ´ 1q. Integrability of the complex structure on X implies thatthese sub-bundles are integrable in the sense of Frobenius, i.e., their sections
are closed under the Lie bracket of the sections of TS b C.
0,1 ˚
8
q
Let b Ω0,q
S be the sheaf of C -sections of the complex vector bundle Λ pTS q .
Integrability of TS0,1 gives, by the standard Cartan formulas, the exterior differentials “along” TS0,1
b 0,q`1
B b : b Ω0,q
S ÝÑ ΩS

16

making b Ω0,‚
S into a sheaf of commutative dg-algebras known as the tangential
Cauchy-Riemann complex (or B b -complex) of S. It is concentrated in degrees
r0, n ´ 1s. The complex of global C 8 -sections of b Ω0,‚
S is traditionally denoted0,‚by Ωb pSq and is also called the B b -complex.
C. Generalities on real forms. Let Y “ Spec R be a smooth irreducible
affine variety over C of dimension m. A real structure on Y is a C-antilinear
involution f ÞÑ f on R. Such a datum defines an antiholomorphic involution
σ : Y pCq Ñ Y pCq on C-points. The fixed point locus of σ is denoted Y pRq.
If nonempty, Y pRq has a structure of a C 8 -manifold of dimension m. In
this case we have an embedding  : R Ñ C 8 pY pRqq. Moreover, any f P
C 8 pY pRqq can be approximated by functions from pRq on compact subsets(Weierstrass’ theorem). In particular, if Y pRq is compact, then pRq is dense
in C 8 pY pRqq in any of the standard metrics of the functional analysis (e.g.,in the L2 -metric).
If, further, E is a vector bundle on Y (not necessarily equipped witha real structure), it gives a C 8 -bundle E|Y pRq on Y pRq and an embedding
ΓpEq Ñ ΓC 8 pE|Y pRq q with approximation properties similar to the above. A
differential operator D : E Ñ F between vector bundles on Y gives rise to a
C 8 differential operator DY pRq : E|Y pRq Ñ F |Y pRq .
D. The Jouanolou model and the B b -complex for S 2n´1 . We specialize
part B to X “ Cn with the standard coordinates zν . We take S to be theunit sphere S 2n´1 with equation }z}2 “ 1. We have therefore the B b -complex
pb Ω0,‚
S 2n´1 , B b q.
q n (with coordinates zν , z ˚ ),At the same time, we introduce, on An ˆ Aνa real structure by putting z ν “ zν˚ and zν˚ “ zν . The Jouanolou torsor J
q n as the hypersurface zz ˚ “ 1, so it inherits the real
is realized in An ˆ Astructure.
Proposition 1.3.3. (a) We have JpRq “ S 2n´1 . In other words, A0rns “ CrJs
is identified with the algebra of (real analytic) polynomial C-valued functions
on S 2n´1 .
(b) The C 8 -vector bundle ΩqJ{Ån |JpRq on JpRq “ S 2n´1 is identified withb

Ω0,q
S 2n´1 , and the differential induced by dJ{Ån , is identified with B b . In other17

words, A‚rns is identified with the dg-algebra of polynomial (in the same sense
as in (a)) sections of b Ω0,‚S 2n´1 .
Proof: Part (a) is obvious, as zz ˚ “ 1 translates to zz “ 1. To prove (b), we
notice that the sub-bundle
TJ{Ån |S 2n´1 Ă TJ |S 2n´1 “ TS 2n´1 b C
is equal to TS0,1
2n´1 .
2n´1Corollary 1.3.4. Aqrns is dense in the L2 -completion of Ω0,q
q. In parb pS02n´1
ticular, Arns is dense in the L2 pS
q.

1.4

Representation-theoretic analysis

A. The GLn -spectrum of A‚n . The Jouanolou model gives commutative
dg-algebras with a natural action of the algebraic group GLn . We now study
this action.
Let V be a k-vector space of finite dimension n. Recall [FH] that irreducible representations of GLpV q are labelled by their highest weights which
are sequences of non-increasing integers α “ pα1 ě ¨ ¨ ¨ ě αn q, possibly
negative (dominant weights). We will denote the underlying space of theirreducible representation with highest weight α by Σα V and regard Σα as
a functor (known as the Schur functor) from the category of n-dimensional
k-vector spaces and their isomorphisms, to Vectk . Here are some tie-ins withmore familiar constructions.
Examples 1.4.1. (a) For d ě 0 we have Σd,0,...,0 pV q “ S d pV q is the dth
symmetric power of V . For 0 ď p ď n let 1p “ p1, . . . , 1, 0, . . . , 0q (with p
occurrences of 1). Then Σ1p pV q “ Λp pV q is the pth exterior power of V .
(b) We have canonical identifications
Σα1 ,...,αn pV q˚ » Σα1 ,...,αn pV ˚ q » Σ´αn ,...,´α1 pV q.
In particular, V ˚ “ Σ0,...,0,´1 pV q. Further,
Σ1,0,...,0,´1 pV q » slpV q “

18(
A P EndpV q : trpAq “ 0 .

We now denote by V “ kn the space of linear combinations of the coordinate functions zi on An , so An “ Spec S ‚ pV q, and let GLn “ GLpV q. The
Jouanolou torsor J Ñ Pn´1 is acted upon by GLn , and so the dg-algebra A‚rns
as well as its completion A‚n , inherit the GLn -action. The following fact canbe seen as a higher-dimensional generalization of the representation of elements of krz, z ´1 s as linear combinations of Laurent monomials (irreducible
representations of GL1 ).
Theorem 1.4.2. (a) As a GLn -module, each Aprns has simple spectrum, thatis, each Σα pV q enters into the irreducible decomposition of Aprns no more thanonce.
(b) More precisely, Σα pV q enters into Aprns if and only if
α1 ě 0 ě α2 ě 0 ě ¨ ¨ ¨ ě 0 ě αn´p ě ´1 ě αn´p`1 ě ´1 ě ¨ ¨ ¨ ě ´1 ě αn .
Examples 1.4.3. (a) For n “ 1 the only possible p is p “ 0. In this casethe condition on α “ pα1 q P Z is vacuous and the theorem says that A0r1s “Àα´1αPZ Σ pkq “ krz, z s.
(b) Let n “ 2. In this case the theorem says that we have an identificationof complexes of GL2 -modules*"à α ,αà α ,α
B
‚
1 2
1 2
Σ
pV q .
Ar2s “Σ
pV q ÝÑ
α1 ě´1α2 ď´1

α1 ě0
α2 ď0

From this we see the identifications
à α ,0KerpBq “Σ 1 pV q “ krz1 , z2 s,
α1 ě0CokerpBq “àΣ´1,α2 pV q “ z1´1 z2´1 krz1´1 , z2´1 s,

α2 ď´1the other irreducible representations, common to A0r2s and A1r2s , are cancelledby the action of B.
(c) For n “ 3 the theorem identifies A‚r3s , as a complex of GL3 -modules,withàà α ,´1,αà α ,0,α B
B3
Σ 1 3 ÝÑ
Σα1 ,α2 ,α3 ÝÑ
Σ 1
.
α1 ě0
α3 ď0

α1 ě0ěα2 ě´1ěα319α1 ě´1α3 ď´1One can see, for example, the reason why the complex is exact in the middle:
if α2 “ 0, then Σα1 ,α2 ,α3 lies in ImpBq, while if α2 “ ´1, then Σα1 ,α2 ,α3 ismapped by B isomorphically to its image.
The proof of Theorem 1.4.2 is based on the following observation. We
use the definition of Arns in terms of the torsor J.
Proposition 1.4.4. As a variety with GLn -action, J » GLn {GLn´1 . Further, ΩpJ{Ån is identified with the homogeneous vector bundle on GLn {GLn´1associated to the representation Λp ppkn´1 q˚ q of GLn´1 .
Proof: The identification (1.2.3) exhibits Jr as a homogeneous space under
GLn . The stabilizer of the point pz, z ˚ q where z “ p1, 0, . . . , 0q and z ˚ “
p1, 0, . . . , 0q, consists of matrices of the formˆ
˙
1 0
, A P GLn´1 ,0 A
whence the first statement of the proposition. To see the second statement,
look at the action of the stabilizer subgroup GLn´1 on the fiber of ΩpJ{Ån overthe chosen point pz, z ˚ q above. By definition, this fiber is the pth exterior
power of the relative cotangent space at this point. It remains to notice thatthe relative tangent space is, as a GLn´1 -module, nothing but kn´1 .
We now prove Theorem 1.4.2. By Proposition 1.4.4,
p
n´1 ˚
n
Aprns “ ΓpJ, ΩpJ{Ån q q » IndGLq
GLn´1 Λ pk

(induction in the sense of algebraic groups, i.e., via regular sections of thehomogeneous bundle). So we can apply Frobenius reciprocity and obtain:
`
˘
`
˘
p
n´1 ˚
n
mult Σα kn , IndGLq“ mult Λp pkn´1 q˚ , Σα kn |GLn´1 ,GLn´1 Λ pk
where mult means the multiplicity of an irreducible representation. Now,
Λp pkn´1 q˚ “ Σ0,...,0,´1,...,´1 pkn´1 q (with p occurrences of p´1q). It remains toapply the following fact [Wey].
Proposition 1.4.5. For any dominant weight α for GLn the restriction ofΣα pkn q to GLn´1 has simple spectrum, explicitly given by:àΣα pkn q|GLn´1 »Σβ1 ,...,βn´1 pkn´1 q.
α1 ěβ1 ěα2 ěβ2 ě¨¨¨ěβn´1 ěαnTheorem 1.4.2 is proved.
20

B. The Martinelli-Bochner form and its multipoles. We use theanalogy with the Dolbeault complex as a motivation for the following.
Example 1.4.6. The Martinelli-Bochner formřn˚

Ω “ Ωpz, z q “ν´1 ˚
y˚ ^ ¨ ¨ ¨ ^ dz ˚zν dz1˚ ^ ¨ ¨ ¨ ^ dz
νn
ν“1 p´1q
˚npzz q

is an element of An´1
rns . For n “ 1 it reduces to 1{z1 .
Proposition 1.4.7. Let n ą 1.
(a) The class of Ω in H n´1 pA‚rns q “ H n´1 pÅn , Oq is a generator of the 1dimensional subspace z1´1 ¨ ¨ ¨ zn´1 of weight p´1, . . . , ´1q under the coordinate
torus, see Proposition 1.1.1.
n´1(b) The top degree part Arns
contains precisely one 1-dimensional irreducible representation of GLn , which is Σ´1,...,´1 pV q “ Λn pV ˚ q. This subspace is spanned by the Martinelli-Bochner form Ωpz, z ˚ q.

(c) Further, every element of H n´1 pA‚rns q can be represented as the classof a “multipole”
P pBz1 , . . . , Bzn qΩpz, z ˚ q
for a unique polynomial P py1 , . . . , yn q.
Proof: (b) follows from Theorem 1.4.2. To deduce (a) from (b), note thatthe class rz1´1 ¨ ¨ ¨ zn´1 s P H n´1 pÅn , Oq spans a 1-dimensional representation ofGLn , and so does Ωpz, z ˚ q (direct calculation). At the same time Σ´1,...,´1 pV q
´1
˚
´1
is not present in An´2
rns , so rΩpz, z qs is a nonzero scalar multiple of rz1 ¨ ¨ ¨ zn s.
To prove (c), we notice the following fact which complements Proposition
1.1.1 and is proved using the same standard affine covering tzi ‰ 0u. Notethat the ring krBz1 , . . . , Bzn s of differential operators with constant coefficientsacts naturally on OÅn and therefore on H n´1 pÅn , Oq.
Proposition 1.4.8. As a krBz1 , . . . , Bzn s-module, the space H n´1 pÅn , Oq is
free of rank 1, with generator δ “ rz1´1 ¨ ¨ ¨ zn´1 s P H n´1 pÅn , Oq.
Proposition 1.4.7 is proved.
Remark 1.4.9. The fact that A0rns has simple spectrum, allows us to define
a canonical GLn -equivariant projection S : A0rns Ñ krz1 , . . . , zn s along all the21

irreducible representations which do not enter into krz1 , . . . , zn s. This is thealgebraic analog of the classical Szegö projection from complex Hilbert space
L2 pS 2n´1 q to the Hardy space formed by the boundary values of functions
holomorphic in the ball }z}2 ă 1. See, e.g., [BdMG].

1.5

Residues and duality

A. Jouanolou model for forms. We denoteqpAp,qrns “ Arns pΩÅn q,(1.5.1)

A‚‚
rns “

n n´1
à
à

Ap,q
rns .

p“0 q“0nqnElements of Ap,q
rns can be viewed as differential forms on A ˆ P with polesr “ tzz ˚ “ 0u be the hypersurface
on the quadric Q given by zz ˚ “ 0. Let Q
q n lifting Q. By pulling back from P
qn to Aq n ´ t0u, we can viewin An ˆ A
q nq ´ Qr of the form
elements of Ap,q as differential forms on pAn ˆ A
rns(1.5.2)ω “ÿi ,...,ifj11,...,jqp pz, z ˚ qdzi1 ^ ¨ ¨ ¨ ^ dzip ^ dzj˚1 ^ ¨ ¨ ¨ ^ dzj˚q ,which have total degreezν˚ , dzν˚ and are annihilated by contractionř ˚0 in the
˚with the vector field zν B{Bzn .
It follows that the bigraded vector space A‚‚
rns is a graded commutative
algebra, with respect to multiplication of forms, with grading situated indegrees
ř r0, nsˆr0, n´1s. It is equipped withřtwo anticommuting differentials:
B “ dz ˚ νB{Bzν˚ of degree p0, 1q and B “ dzν B{Bzν of degree p1, 0q which
qn . Onecorrespond to exterior differentiation along the two factors in An ˆ P
can say that B is induced by the relative de Rham differential in ΩJ{Ån bπ ˚ Ωpand B corresponds to the de Rham differential d : ΩpÅn Ñ ΩÅp`1. Part (a) ofn
Proposition 1.2.4 implies:
Proposition 1.5.3. The bigraded dg-algebra A‚‚
rns is a commutative dg-modelfor A‚‚.
rns
Example 1.5.4. Let k “ C. Then we have an embedding of commutative
bigraded bidifferential algebras‚‚nε : A‚‚n ãÑ Ω pC ´ t0uq,

22where on the right we have the algebra of C 8 Dolbeault forms on Cn ´ t0u
with its standard differentials B and B. The value of ε on a form ω given by(1.5.2), isÿ i ,...,iεpωq “
fj11,...,jqp pz, zq|Cn ´t0u dzi1 ^ ¨ ¨ ¨ ^ dzip ^ dz j1 ^ ¨ ¨ ¨ ^ dz jq
B. The residue map. Since ΩnÅn is identified, as a GLn -equivariant coherent sheaf on Ån , with OÅn bk Λn pkn q, we have a GLn -invariant identification
n
n
‚
An,‚
rns » Arns bk Λ pk q.

We define the residue map
Res : An,n´1
ÝÑ k
rns

(1.5.5)as the compositionAn,n´1“ A‚rns bk Λn pkn q ÝÑ Λn pkn q˚ b Λn pkn q “ k,
rns
n
n ˚
where An´1
rns Ñ Λ pk q is the unique GLn -invariant projection which takes
the Martinelli-Bochner form Ωpz, z ˚ q to dz1˚ ^ ¨ ¨ ¨ ^ dzn˚ , see Proposition
1.4.7(b). Thus, by definition,`
˘(1.5.6)
Res Ωpz, z ˚ qdz1 ^ ¨ ¨ ¨ ^ dzn “ 1.

Note that An,n´1
is the last graded component of A‚rns pΩnÅn q which is a dgrns
model for RΓpÅn , Ωn q.
Proposition 1.5.7. For any f pzq P krz1 , . . . , zn s we have`˘Res f pzqΩpz, z ˚ qdz1 ^ ¨ ¨ ¨ ^ dzn “ f p0q(the algebraic Martinelli-Bochner formula).
Proof: Note that both sides of theequality are GLn -invariant funcÀproposeddtionals of f P krz1 , . . . , zn s “SpVq.
For d ą 0 the space S d pV q
dě0does not admit any GLn -equivariant functionals, so the LHS factors throughthe projection to d “ 0 which is nothing but the evaluation at 0. So ourstatement reduces to (1.5.6).
.
23

n,n´1
Proposition 1.5.8. Let k “ C. Then for any ω P Arns
we have¿pn ´ 1q!
Respωq “
εpωqp2πiqnS 2n´1

where the integral is taken over any sphere }z} “ R in Cn ´ t0u.
Proof: Each εpωq, ω P An,n´1, is a closed pn, n ´ 1q-form on Cn ´ t0u, so
rns
its integrals over all spheres as above are equal. We see that the RHS of
the proposed equality is a GLn pCq-invariant functional on An,n´1, and anyrnssuch functional, by Theorem 1.4.2, should factor through the projection toΛn pkn q˚ b Λn pkn q, i.e., through the residue map. This means the statement
holds up to a universal constant depending only on n. To see that thisconstant is 1, we invoke the classical Martinelli-Bochner formula [GH], whichgives¿
p2πiqn.
Ωpz, zqdz1 ¨ ¨ ¨ dzn “
pn ´ 1q!
S 2n´1

Proposition 1.5.9. Each irreducible representation of GLn enters into Ap,qrns
with at most finite multiplicity.
qp
Proof: As a GLn -module, Ap,q
rns “ Λ pV q b Arns . By the Pieri formula [GH],the irreducible components Σβ pV q of Λp pV q b Σα pV q all satisfy β “ α `ei1 ` ¨ ¨ ¨ ` eip for some 1 ď i1 ă ¨ ¨ ¨ ă ip ď n. Here ei is the ith basis
vector. So the allowed β situated in some fixed radius neighborhood of α inZn Ă Rn . This means that tensoring a simple spectrum representation with
Λp pV q gives a representation with finite multiplicities. So our statement
follows from Theorem 1.4.2.
ÀLet E » α Cα b Σα pV q be a representation of GLpV q with finite multiplicities (so Cα are finite-dimensional vector spaces). We define the restricteddual of E to beà ˚
E ‹ :“Cα b Σα pV q˚ .
αProposition 1.5.10. The (GLn -invariant) residue pairingn´p,n´1´qpα, βq ÞÑ Respα ¨ βq : Ap,qÝÑ k
rns bk Arnsn´p,n´1´q‹gives an isomorphism Arns
Ñ pAp,q
rns q .

24

‹
Proof: We first prove a weaker statement: pAp,q
rns q is isomorphic, as a GLn module, to An´p,n´1´q. Because of the non-degenerate pairing Λp pV q brnsΛn´p pV q Ñ Λn pV q, the statement reduces to the isomorphismn´1´qpAqrns q‹ » Arnsb Λn pV q˚ .

This isomorphism follows at once from inspecting the irreducible componentsof Aqrns and An´1´qgiven by Theorem 1.4.2. They are in bijection Σα Ø Σβ ,rns
so that(1.5.11)βi “ ´1 ´ αn´i , i “ 1, . . . , n,which means that Σβ pV q » Λn pV q˚ b pΣα pV qq˚ .
We now prove that the residue pairing is actually an isomorphism as
claimed. As before, we reduce to considering the pairing
n´1´q
n´1
Aqrns b Arns
ÝÑ ArnsÝÑ Λn pV q˚ .

Only the summands Σα pV q Ă Aqrns and Σβ pV q Ă An´1´qwhere α, β correrns
spond to each other as in (1.5.11), can pair in a non-trivial way. It remains
to show that they indeed pair non-trivially. If they pair trivially, then then´1´qsubspace Σα pV q Ă Aqrns is orthogonal to the entire Arns
. The easiest way to
see why this is impossible, is to reduce (by the Lefschetz principle) to k “ C.
In this case we can use the fact that An´1´q
is L2 -dense in Ω0,n´1´q
pS 2n´1 q,b
rns
see Corollary 1.3.4, and the non-degeneracy of the L2 -pairing on Ωb0,‚ pS 2n´1 q.
Proposition 1.5.12 (Algebraic Stokes formula). The residue functional Res :n´1,n´1n,n´2An,n´1
Ñ k vanishes on BpArns
q ` BpArns
q.
rns
Proof: As before, the easiest proof is to reduce to k “ C and to embed‚‚nA‚‚
rns into Ω pC ´ t0uq. Then we can use the classical Stokes formula forn´1,n´1
are annihilated by B, whiled “ B ` B, noticing that elements of Arnsn,n´2elements of Arns are annihilated by B.
A purely algebraic proof, by inspection of the possible relevant irreducible
components, is left to the reader. In this inspection we find that An,n´2
does
rns
n´1,n´1
not contain the trivial representation, while Arns
does contain it but thecorresponding subspace is annihilated by B (it represents H n´1 pPn´1 , Ωn´1 q).

25

2

The residue class in cyclic cohomology

2.1

Cyclic homology of dg-algebras and categories

In this subsection we compare various definitions of cyclic homology of associative dg-algebras without any restriction on the grading. Care is needed,since existing treatments of some issues apply only to Zď0 -graded algebras
and rely on spectral sequences which may not converge in the general case.
For general background on cyclic homology (ungraded and Zď0 -gradedcases), see [Lo].
A. General definitions. Our basic approach is that of Keller [Ke1]. Thatis, let pA, Bq be any associative dg-algebra over k, possibly without unit. The
Hochschild and the cyclic complexes of A, denoted C‚Hoch pAq and CC‚ pAq,
are defined similarly to what is described in [Lo] 5.2.2 for the Zď0 -gradedcase. That is, we form the total complex of the double or triple complex
obtained when we take into account the grading and differential on A. For
example,(bbbC‚Hoch pAq “ Tot ¨ ¨ ¨ ÝÑ Ab3 ÝÑ A b A ÝÑ A .
The total complex here and elsewhere is always understood in the sense of
direct sums. The new phenomenon compared to the Zď0 -graded case is that
direct sums can be infinite.
The definition of CC‚ pAq is similar. More precisely, let V be a cochain
complex of k-vector spaces with a Z{pn ` 1q-action, the generator of theaction denoted by t. We denote(1´t1`¨¨¨`tn
1´t
VC “ Tot ¨ ¨ ¨ ÝÑ V ÝÑ V ÝÑ V ,(the horizontal grading situated in degrees ď 0). Then(bbbCC‚ pAq “ Tot ¨ ¨ ¨ ÝÑ Ab3
ÝÑ
pA
b
Aq
ÝÑ
A
.
C
C
C
where the Z{pn ` 1q-action on Abpn`1q is given as in [Lo] (2.1.0), understood
via the Koszul sign rule.
The homology of C‚Hoch pAq and CC‚ pAq will be denoted HH‚ pAq andHC‚ pAq. Each of these complexes has an exhaustive increasing filtration by
the number of tensor factors. This gives a convergent spectral sequence
E2 “ HC‚ pHB‚ pAqq ñ HC‚ pAq,
26

and similarly for HH‚ pAq. It follows that the functors HC‚ and HH‚ takequasi-isomorphisms of dg-algebras to isomorphisms and so descend to functors on the homotopy category of associative dg-algebras.
Lemma 2.1.1. Let V be a cochain complex of k-vector spaces with a Z{pn `1q-action. Then morphism VC Ñ VZ{pn`1q from the last term to the cokernelof p1 ´ tq, is a quasi-isomorphism.
Proof: This is well known if V in ungraded (Tate resolution). When Vis Zď0 -graded, it follows from a spectral sequence argument, as VC has anincreasing exhaustive filtration with quotients VCi . To prove the general case,it suffices to consider the case when k is algebraically closed. Assuming this,we consider the abelian category dgVectk Z{pn`1q formed by cochain complexes
with Z{pn ` 1q-action.
Lemma 2.1.2. Each object of dgVectk Z{pn`1q is isomorphic to a direct sumof (possibly infinitely many copies) of the following indecomposable objects:(1) A 1-term complex k (situated in some degree) on which Z{pn ` 1q acts
via some character.
Id(2) A 2-term complex k Ñ k (situated in a pair of adjacent degrees) on
which Z{pn ` 1q acts via the same character.
Lemma 2.1.2 implies Lemma 2.1.1 because the indecomposable objects
are bounded and so 2.1.1 holds for them.
Proof of Lemma 2.1.2: This is well known if n “ 0 (i.e., if we consider
just cochain complexes with no group action). Now, given V with action ofZ{pn ` 1q, we first split it (as a complex) into a direct sum of eigencomplexesVχ corresponding to the characters χ of Z{pn ` 1q. Then we decompose eachcomplex Vχ into indecomposables.
Corollary 2.1.3. The complex CC‚ pAq is quasi-isomorphic to the Connes’
complex(bbb
C‚λ pAq “ Tot ¨ ¨ ¨ ÝÑ Ab3
ÝÑ
pA
b
Aq
ÝÑ
A
.
Z{2
Z{3

27

B. Morita invariance. For unital dg-algebras, Keller [Ke1] proved thatcyclic homology is Morita-invariant. To formulate the results compactly, it isconvenient to extend the definition of Hochschild and cyclic complexes andcohomology to small dg-categories A (a unital dg-algebra is the same as a
dg-category with one object). That is, we define C‚Hoch pAq to be the total
complex of the double complex
b

¨¨¨ Ñ

à

n
â

b

Hom‚A pai , ai`1 q Ñ ¨ ¨ ¨ Ña0 ,...,an PObpAq i“0âHom‚A pa0 , a0 q,

a0 PObpAq(here we put an`1 :“ a0 ) and similarly for CC‚ pAq.
We recall [To4] that a functor f : A Ñ B is called a Morita equivalence,
if f˚ : Perf A Ñ Perf B is a quasi-equivalence. Recall further that Perf A
is essentially small, if A is small, and the canonical (Yoneda) embeddingυ : A Ñ Perf A is a Morita equivalence.
Proposition 2.1.4 (Keller, [Ke2]). (a) If f : A Ñ B is a Morita equivalence of small dg-categories, then HC‚ pf q : HC‚ pAq Ñ HC‚ pBq is an
isomorphism.
(b) In particular, if A is an essentially small dg-category, then HC‚ pA1 q
where A1 Ă A is an equivalent small dg-subcategory, are canonically
identified, and denoted HC‚ pAq.
(c) It follows that υ induces an isomorphism HC‚ pAq Ñ HC‚ pPerf A q.
(d) Therefore any dg-functor φ : Perf A Ñ Perf B induces a morphismφ˚ : HC‚ pAq Ñ HC‚ pBq which is an isomorphism, if φ is a quasiequivalence.
Let us note a more elementary instance of this proposition, cf. [Lo] 1.2.4
and 2.2.9.
Proposition 2.1.5. Let A be a unital dg-algebra and r ě 1. The collectionof the trace maps`
˘bpn`1q trA,n bpn`1qMatr pAqbpn`1q » Matr pkq bk A
ÝÑ A
,
`
˘
trA,n u0 a0 b ¨ ¨ ¨ b un an “ trpu0 ¨ ¨ ¨ un qa0 b ¨ ¨ ¨ b an ,defines quasi-isomorphisms of complexestrC‚Hoch pMatr pAqq ÝÑ C‚Hoch pAq,
28

tr

CC‚ pMatr pAqq ÝÑ CC‚ pAq.

The morphisms on C‚Hoch and CC‚ , induced by the embedding A ãÑ Matr pAq,
are quasi-inverse to tr.
C. Localization sequence.
Definition 2.1.6. A localization sequence of perfect dg-categories is a anhomotopy cofiber sequenceA ÝÑ B ÝÑ C
in the Morita model category of dg-categories (see Appendix), such that thefunctor A Ñ B is (quasi-)fully faithful.
Remark 2.1.7. Given such a localization sequence, the homotopy categoryrAs is a thick subcategory of the triangulated category rBs and rCs is equivalent to the Verdier quotient rBs{rAs.
Theorem 2.1.8 (Keller, [Ke2]). A localization sequence of perfect dg-categories
A Ñ B Ñ C induces a cofiber sequence
CCpAq ÝÑ CCpBq ÝÑ CCpCq
in the category of complexes. In particular we get a long exact sequence¨ ¨ ¨ ÝÑ HCp pAq ÝÑ HCp pBq ÝÑ HCp pCq ÝÑ HCp´1 pAq ÝÑ ¨ ¨ ¨

2.2

Cyclic homology of schemes

A. Sheaf-theoretic definition and Hodge decomposition. We recall
the basic constructions and results of Weibel.
HochDefinition 2.2.1 [W1]. Let X be a k-scheme. Denote by C‚,X
the complex of sheaves on the Zariski topology of X obtained by sheafifying (termby term) the complex of presheaves U ÞÑ C‚Hoch pOpU qq. We define the
Hochschild homology of X as the hypercohomology of this complex:

HHpXq “ Hk pX, C‚H pXqq.
We also define the bicomplex of sheaves CC‚,X as the sheafification of thecomplex of presheaves U ÞÑ CC‚ pOpU qq, and the cyclic homology of X as
HCpXq “ Hk pX, TotpCC‚,X qq.
29

So HH and HC fit into Mayer-Vietoris sequences by definition. Weibel
proved that for an affine scheme X “ SpecpAq, one has
HHpXq “ HHpAq,

HCpXq “ HCpAq.

Let us also recall here that the cyclic homology of a commutative algebra
has a Hodge decomposition (also called λ-decomposition, see [Lo]):àà
HH‚ pAq “
HH‚piq pAqandHC‚ pAq “HC‚piq pAqii

Weibel extends this decomposition to the case of schemes and describes it interm of HKR-isomorphism in the smooth case.
Theorem 2.2.2 ([W2]). Let X be a qcqs (quasi-compact and quasi-separated)k-scheme. There are Hodge decompositions
àà
HH‚ pXq “
HHpiq
andHC‚ pXq “HC‚piq pXq.
‚ pXq
i

i

Moreover, if X is smooth, then we have for any i and kpiq

HHk pXq “ H i´k pX, ΩiX q

and

piq

HCk pXq “ H2i´k pX, ΩďiX q.

B. Relation to HC of dg-algebras and categories. We first recall thefollowing result of Keller.
Theorem 2.2.3 ([Ke3]). For any qcqs scheme X we have
HH‚ pXq » HH‚ pPerfpXqq

andHC‚ pXq » HC‚ pPerfpXqq,

where Perf denotes the dg-category of perfect complexes.
Next, we use this result to relate cyclic homlogy of schemes with that ofdg-algebras of derived functions.
Definition 2.2.4. A scheme U is called quasi-affine if it is isomorphic to a
qcqs open subscheme of an affine scheme.
Theorem 2.2.5. For any quasi-affine scheme U , we haveHC‚ pU q » HC‚ pRΓpU, Oqq.
30

Proof: By Theorem 2.2.3 and Proposition 2.1.4, it is enough to show that
PerfpXq is equivalent to Perf RΓpU,Oq . Because perfect complexes (resp. dgmodules) are intrinsically characterized as compact objects in the derived
category of all quasicoherent sheaves (resp. all dg-modules), we reduce to:
Proposition 2.2.6. Let U be a quasi-affine scheme. The derived categoryof quasi-coherent sheaves on U is equivalent to the derived category of dgmodules on the cdga RΓpU, Oq.
Note that this statement is not true for more general schemes (e.g., for
U “ Pn ).
Proof of the proposition: Let us denote by A any model for the cdga RΓpU, Oq.
Let u : U Ñ X “ Spec B be an open immersion. In particular, we get a mapof cdgas p : B Ñ A. It induces an adjunction of 8-categoriesp˚ : dgModB Ô dgModA : p˚
We will prove the functor p˚ to be fully faithful and its essential image tocoincide with that of u˚ .
The element A is a compact generator of dgModA and both p˚ and p˚preserve small colimits. To show that p˚ is fully faithful, we consider the
adjunction map p˚ p˚ A Ñ A. This map is equivalent to the multiplication
map A bLB A Ñ A.
Lemma 2.2.7. The multiplication map AbLB A Ñ A is a quasi-isomorphism.
Proof of theŤ lemma: Let pfi qiPI denote a finite family of elements of B suchthat U “ Xfi Ă X, where Xfi “ SpecpBrfi´1 sq. For any non-empty subset
J Ă I, let BJ denote the (derived) tensor product
BJ “

iPJ
â
L

Brfi´1 s.

B

By definition, the cdga A is equivalent to the homotopy limit ÐholimÝÝÝ∅‰JĂI BJ .
The derived tensor product A bLB A Ñ A is then equivalent to
L
A bLB A » holimÐÝÝÝJ BJ bB holimÐÝÝÝJ 1 BJ 1 » holimÐÝÝÝJ holimÐÝÝÝJ 1 pBJ bA BJ 1 q »» holimÐÝÝÝJ holimÐÝÝÝJ 1 BJYJ 1 » holimÐÝÝÝJ holimÐÝÝÝJĂJ 2 BJ 2 » holimÐÝÝÝJ BJ » A.
31

Lemma 2.2.7 is proved.
To continue with Proposition 2.2.6, we can now identify dgModA with thefull stable and presentable 8-subcategory of dgModB generated by A. We
remark that the derived category of quasi-coherent sheaves on U identifies,
through the functor u˚ , with a full stable and presentable 8-subcategory ofdgModB containing u˚ OU » RΓpU, Oq » A. We hence get an adjunction
F :“ p˚ u˚ : Dqcoh pU q Õ dgModA : G “ u˚ p˚where the functor G is fully faithful. It therefore suffices to prove that F isconservative. Let then E P Dqcoh pU q such that F pEq “ 0. For any i, if wedenote by vi : SpecpBi q Ñ U the Zariski embedding, we getvi˚ pEq “ F pEq bA Bi “ 0.
As tSpecpBi quiPI is a cover of U , we deduce that E is acyclic and hence thatthe functor F is conservative. Proposition 2.2.6 and therefore Theorem 2.2.5,
is proved.

2.3

The residue class ρ P HC 1 pA‚n q

A. Definition via a cyclic cocycle. Let pA‚n , Bq » RΓpÅn , Oq be thecommutative dg-algebra from (1.2.5). We recall that A‚n is included inton‚the doubly graded differential algebra pA‚‚n , B, Bq » RΓpÅ , Ω q: we have0,‚‚
An “ An .
Consider the pn ` 1q-linear functional(2.3.1)r : pA‚n qbpn`1q ÝÑ k,

rpf0 , . . . , fn q “ Respf0 Bf1 ¨ ¨ ¨ Bfn q.

n´1,nHere Bfi P A1,‚
Ñ k is the residue map from (1.5.5). By
n , and Res : An
definition rpf0 , . . . , fn q is assumed to be equal to 0 unless f0 Bf1 ¨ ¨ ¨ Bfn lies inAn,n´1.
n

Proposition 2.3.2. The functional r is a degree 1 cocycle in the Connes
cochain complex Cλ‚ pA‚n q “ Homk pC‚λ pA‚n q, kq.
n,‚Proof: Degree 1: The expression
ř f0 Bf1 ¨ ¨ ¨ Bfn always lies in An . For it ton,n´1
lie in An, we must have degpfi q “ n ´ 1. The horizontal grading of‚ bpn`1q
pAn q
in the Hochschild chain complex of A, is p´nq. So r, as an elementof the dual complex, has degree `1.

32

Cyclic symmetry: With the Koszul sign rule taken into account, the condition
for r to lie in Cλ‚ pA‚n q is
rpf0 , . . . , fn q “ p´1qn`degpf0 qpdegpf1 q`¨¨¨`degpfn qq rpf1 , . . . , fn , f0 q.
This follows at once from the Leibniz rule for B and the fact that Res vanishes
on the image of B.
‚
pA‚n q, we need to show
r is a cocycle: As Cλ‚ pA‚n q is a subcomplex in CHoch‚. This differential is the sum
that r is closed under the differential in CHoch˚˚β ` B , where β is the Hochschild cochain differential, and B is induced by˚
the differential B in A‚n . We claim that both βprq and B prq vanish. Thesestatements follow from the Leibniz rule for B and B respectively and the factthat Res vanishes on the images of both B and B.

We denote by ρ P HC 1 pA‚n q the class of r and call it the residue class.
By construction, ρ is GLn -invariant.
B. Definition via the Hodge decomposition. By Theorems 2.2.5 and2.2.2, we have the Hodge decomposition(2.3.3)HC1 pA‚n q

n» HC1 pÅ q “àpiq
HC1 pÅn q

i

»

n
à

H2i´1 pÅn , Ωďi q.

i“1We consider the projection to the summand(2.3.4)pnq

HC1 pÅn q “ H2n´1 pÅn , Ωďn q “ H2n´1 pÅn , Ω‚ q “ k,which is the p2n ´ 1qst de Rham cohomology of Ån (e.g., for k “ C, thep2n ´ 1q-st topological cohomology of Cn zt0u „ S 2n´1 ).
Theorem 2.3.5. The class in HC 1 pA‚n q given by the projection of (2.3.3)
to (2.3.4), is the unique, up to scalar, GLn -invariant element in HC 1 pA‚n q.
In particular, it is proportional to the residue class ρ.
Proof: We analyze the GLn -module structure of the H2i´1 pÅn , Ωďi q. Applying the spectral sequence starting from H p pÅn , Ωq q and converging to
H2i´1 pÅn , Ωďi q, we reduce to the following statement.
Proposition 2.3.6. The vector space H p pÅn , Ωq q has no GLn -invariants
unless p “ q “ 0 or p “ n ´ 1, q “ n, in which case the space of invariants
is 1-dimensional.
33

3Higher current algebras and their central
extensions

3.1

Cyclic homology and Lie algebra homology

A. Reminder on the dg-Lie algebra (co)homology. Let l‚ be a dg-Lie
algebra with differential B and bracket c : Λ2 l Ñ l. The Chevalley-Eilenberg
cochain complex of l‚ is the symmetric algebra CE‚ plq “ pS ‚ pl‚ r1sqq˚ equippedwith the algebra differential dLie ` D, where:• D is the algebra differential on the symmetric algebra which extends
pBr1sq˚ by the Leibniz rule.
• dLie is the algebra differential given on the generators by the map ps ˝
cr2sq˚ : pl‚ r1sq˚ Ñ pS 2 pl‚ r1sqq˚ (i.e., extended from the generators by
the Leibniz rule). Here we suppressed the notation for dec2 from (A.2).
The cohomology of CE‚ plq (equipped with the total grading) will be denoted H‚Lie plq.
In fact, CE‚ plq is dual to the Chevalley-Eilenberg chain complex CE‚ plq “S ‚ pl‚ r1sq whose differential is defined by obvious “undualizing” of the formulas above. In other words, it can be described in terms of the coalgebra
structure on CE‚ plq. We have an increasing exhaustive filtration of CE‚ plq
by the number of tensor factors which gives a convergent spectral sequence‚, are quasiH‚Lie pHB‚ plqq ñ H‚Lie plq. This implies that H‚Lie , and therefore HLie
isomorphism invariant, in particular, descend to functors on the homotopy
category rdgLiek s. The following is then standard.
Proposition 3.1.1. We have a canonical identification
HomrdgLiek s pl, krnsq “ Hn`1
Lie plq.
In particular, H2Lie plq classifies central extensions of l.
B. Loday’s homomorphism θ. Let A be an associative dg-algebra (possibly without unit). The graded commutatorra, bs “ ab ´ p´1qdegpaq¨degpbq ba

34

makes A into a dg-Lie algebra which we denote ALie . If A is ungraded, then
H1Lie pALie q “ HC0 pAq “ A{rA, As,
Extending [Lo] (10.2.3) straightforwardly from the ungraded case, we include
this in the following.
Proposition 3.1.2. For any associative dg-algebra A there is a natural morphism of complexesθA : CE‚`1 pALie q ÝÑ C‚λ pAq,ÿ
a0 ^ ¨ ¨ ¨ ^ an ÞÑ
sgnpσqrpId, σq˚ pa0 b ¨ ¨ ¨ b an qs.
σPSn

Here, for σ P Sn , we denote by pId, σq P Sn`1 the permutation of t0, 1, . . . , nu
fixing 0 and acting on 1, 2, . . . , n as σ. The notation rxs means the class of x
in the coinvariant space of Z{pn ` 1q. In particular, this gives natural maps
θA “ θiA : HLiei`1 pALie q Ñ HCi pAq.

3.2

Higher current algebras and their central extensions

Let g be a finite-dimensional reductive Lie algebra over k. We consider thedg-Lie algebrag‚n “ g bk A‚n » RΓpÅn , g bk Oq.
We call g‚n the nth derived current algebra associated to g. For n “ 1 we getthe Laurent polynomial algebra g1 “ grz, z ´1 s.
Let P P S n`1 pg˚ qg be an invariant polynomial on g, homogeneous ofdegree pn ` 1q. We will also consider P as a symmetric pn ` 1q-linear form
px0 , . . . , xn q ÞÑ P px0 , . . . , xn q on g.
Theorem 3.2.1. (a) Consider the functional γP : pg‚n r1sqbpn`1q Ñ k given
by
`
˘
`
˘
γP px0 bf0 qbpx2 bf1 qb¨ ¨ ¨bpxn bfn q “ P px0 , . . . , xn q¨Res f0 ¨Bf1 ^¨ ¨ ¨^Bfn .
‚Here xν P g, fν P A‚n , B is the degree p1, 0q differential in A‚‚
n Ą An , andp,qwe consider Res as a functional on the entire A‚‚n vanishing on all An withpp, qq ‰ pn, n ´ 1q.

35

This functional is of total degree 2, is symmetric and is annihilated by bothdifferentials dLie and D. Therefore it is a cocycle in CE2 pg‚n q and defines aclass rγP s P H2Lie pg‚n q.
(b) Assume g semisimple. The correspondence P ÞÑ rγP s given an embedding S n`1 pg˚ qg Ă H2Lie pg‚n q.
Example 3.2.2. let g “ glr and Ptr pxq “ trpxn`1 q or, in the polarized form,
Ptr px0 , . . . , xn q “

ÿ1
trpxsp0q ¨ ¨ ¨ xspnq q.
pn ` 1q! sPSn`1

In this case γPtr is the image of the residue cocycle ρ P Cλ1 pA‚n q under thecomposite map
θ A‚

tr˚

n
Cλ1 pA‚n q ÝÑ Cλ1 pglr pA‚n qq ÝÑ
CE2 pglr pA‚n qq,

where tr˚ is dual to the trace map trA‚n ,n from Proposition 2.1.5, and θA‚n is‚dual to the map θAn from Proposition 3.1.2. In particular, we see that γPtr
is a Chevalley-Eilenberg cocycle satisfying all the conditions of part (a) of
Theorem 3.2.1.
Proof of Theorem 3.2.1: (a) Let φ : g Ñ glr be a representation of g. TakePφ pxq “ trpφpxqn`1 q. In this case γPφ is induced from γPtr P CE2 pglr pA‚n qq
by φ and therefore satisfies the conditions of (a). Further, notice that γP
depends on P in a linear way. Now, the statement follows from the next
lemma, to be proved further below.
Lemma 3.2.3. Fix m ě 0. Any P P S m pg˚ qg is a linear combination ofpolynomials of the form Pφ,m : x ÞÑ trpφpxqm q.
(b) Note that CE‚ pg‚n q is the total complex of a bicomplex CE‚‚ pg‚n q with
CEpq pg‚n q “ Hompk pΛq g‚n , kq,
and with differentials dLie and D. We consider the corresponding spectral
sequence˘` ‚‚ ‚ ˘
`
p`q ‚
p,q
pgn q.
CE pgn q “ Hompk Λq HB‚ pg‚n q, k ñ HLie
(3.2.4)
E1pq “ HD36

Since g‚n has B-cohomology only in degrees 0 and pn´1q, the spectral sequenceis supported in the fourth quadrant, on the horizontal linesq ě 0, p “ lp1 ´ nq, l ě 0.
Since γP is annihilated by both differentials, it gives a permanent cycle inE1´n`1,n`1 , denote it pγP q. For the class rγP s in H2Lie pg‚n q to be zero, pγP q must
be killed by some differential of the spectral sequence. From the shape of it,
the only possible such differential is dn : En0,1 Ñ En´n`1,n`1 . But, denoting
grzs “ grz1 , . . . , zn s, we have
`
L“
‰˘˚
1
1
E20,1 “ HLie
pHB0 pg‚n qq “ HLiepgrzsq “ grzs grzs, grzs
and this vanishes for a semi-simple g. Therefore En0,1 “ 0 and pγP q cannot
be killed. Theorem 3.2.1 is proved.
Proof of the lemma 3.2.3: Consider the completion Sp‚ pg˚ q, i.e., the ring offormal power series on g near 0, with its natural adic topology. To any
representation φ of g we associate the invariant series
Qφ pxq “ trpe

φpxq

8
ÿ
1
q “
Pφ,m pxq P Sp‚ pg˚ qg .
m!
m“0

By separating the series into homogeneous components, the lemma is equivalent to the following statement.
Lemma 3.2.5. The k-linear space spanned by series Qφ pxq for finite dimensional representations φ of g, is dense in Sp‚ pg˚ qg .
Proof of Lemma 3.2.5: Let G be a reductive algebraic group with Lie algebra
pG,1 , the comg. The exponential map exp : g Ñ G identifies Sp‚ pg˚ q with O
pleted local ring of G at 1, in a way compatible with the adjoint g-action on
pG,1 . Let
both spaces. Now, krGs, the coordinate algebra of G, is dense in O
Φ : G Ñ GLr be an algebraic representation of G. Then the character trpΦq
is a g-invariant element in krGsg , and such elements span krGsg . Therefore
pg . If now φ is the representathe space spanned by the trpΦq, is dense in O
G,1tion of g tangent to Φ, then the image of trpΦq under the above identificationpg Ñ Sp‚ pg˚ qg , is precisely the series Qφ pxq. So the space of suchexp˚ : OG,1series is dense in Sp‚ pg˚ q, as claimed.
37

Example 3.2.6 (Heisenberg dg-Lie algebra). Let g “ gl1 (abelian) andP pxq “ xn`1 . In this case the (2-)cocycle`
γpf0 , . . . , fn q “ Res f0 Bf1 ¨ ¨ ¨ Bfn q
defines a central extension Hn of the abelian dg-Lie algebra g‚n “ A‚n . The
dg-Lie algebra Hn is the n-dimensional analog of the Heisenberg Lie algebra
associated to the vector space kppzqq equipped with the skew-symmetric form
rf0 , f1 s “ Respf0 df1 q.

4

The Tate extension and local Riemann-Roch

4.1

Background on Tate complexes

A. The quasi-abelian category of linearly topological spaces. We
will use the concept of a quasi-abelian category [Sch], a weakening of that of anabelian category. In particular, in a quasi-abelian category A any morphisma : V Ñ W has categorical kernel Kerpuq, cokernel, image Impaq “ KertW Ñ
Cokerpaqu and the coimage Coimpaq “ CokertKerpaq Ñ V u but the canonicalmorphism Coimpaq Ñ Impaq need not be an isomorphism. If it is, a is called
strict.
As pointed out in [Sch], any quasi-abelian category A has an intrinsic
structure of exact category in the sense of Quillen.
Example 4.1.1. Let TopVectk be the category of Hausdorff linearly topological k-vector spaces with a countable base of neighborhoods of 0. Here k
is considered with discrete topology. See [Le], Ch. 2 for background; in thispaper we additionally impose the countable base assumption.
For a morphism a : V Ñ W , the kernel Kerpaq Ă V is the usual kernel
with induced topology, Impaq is the closure of the set-theoretic image, andCoimpaq is the set-theoretical quotient of V by Kerpaq with the quotient
topology.
The category TopVectk is analogous to several categories treated in theliterature [Pro1] [ProS] [Pro2]. In particular, arguments similar to [Pro2]
Cor. 3.1.5 and Prop. 3.1.8, give:
Proposition 4.1.2. (a) The category TopVectk is quasi-abelian.
(b) A morphism a : V Ñ W in TopVectk is strict, if its set-theoreticalimage is closed and a is quasi-open. That is, for any open subspace U Ă V
there is an open subspace U 1 Ă W such that apU q Ą apV q X U 1 .
38Every quasi-abelian category A gives rise to the (bounded) derived category Db pAq equipped with a canonical t-structure whose heart ♥ A (called
left heart in [Sch]), is a natural abelian envelope of A. It is equipped with afully faithful left exact embedding h : A Ñ ♥ A (so h preserves kernels but
not cokernels).
Objects of ♥ A can be thought of as formal “true cokernels” of monomorphisms a in A and in fact have the form Coker♥ A paq (actual cokernels in ♥ A).
See [Sch] Cor. 1.2.21 or, in the more general framework of exact categories,[La], Def. 1.5.7.
Example 4.1.3. The formal quotient krrtss{krts represents an object of theabelian category ♥ TopVectk . The short exact sequence0 Ñ krts ÝÑ krrtss ÝÑ krrtss{krts Ñ 0
represents a nontrivial extension in ♥ TopVectk .
One can identify Db pAq with the localization(4.1.4)

Db pAq » K b pAqrqis´1 sof K b pAq, the homotopy category of bounded complexes over A, with respect
to quasi-isomorphisms (understood in the sense of complexes over ♥ A), see[La] [Sch]. Further, the natural functor Db pAq Ñ Db p♥ Aq into the usual
bounded derived category of A, is an equivalence ([Sch] Prop. 1.2.32).
Remarks 4.1.5. (a) We note that K b pAq has a natural dg-enhancement: itis the H 0 -category of the dg-category of bounded complexes over A. Therefore (4.1.4) can be used to represent Db pAq as the H 0 -category of a dgcategory, by using the Gabriel-Zisman localization for dg-categories ([To5],§2.1) which is the dg-analog of the Dwyer-Kan simplicial localization for
categories [DK1][DK2].
(b) The concept of a quasi-abelian category is self-dual. Therefore there
exist another abelian envelope A♥ “ p♥ pAop qqop (the right heart) whose
objects can be thought as formal “true kernels” of epimorphisms in A, anda right exact embedding A Ñ ♥ A. In our example A “ TopVectk , thecategorical kernels coincide with set-theoretical ones so it is natural to use
the left heart to keep the kernels unchanged.

39

Proposition 4.1.6. Let A be quasi-abelian. A bounded complex V ‚ over A
has all H i pV ‚ q P A Ă ♥ A, if and only if all the differentials are strict. In
particular, for a monomorphism a : V Ñ W in A, the cokernel Coker♥ A paq
lies in A if and only if a is strict.
Proof: See [Pro1], Cor. 1.13.
B. Tate spaces and Tate complexes. For V P TopVectk we have thetopological dual V _ “ Homk pV, kq (continuous linear functionals, with weak
topology). The functor V ÞÑ V _ is not a perfect duality on TopVectk ; however, the canonical morphism V Ñ V __ is an isomorphism on the following
full subcategories in TopVectk :(1) The categoryÀVectk of discrete (at most countably dimensional) vector
spaces V » iPI k.
ś(2) The category LCk of linearly compact spaces V » iPI k.
(3) The category Tak of locally linearly compact spaces ([Le], Ch. 2, §6)
which we will call Tate spaces. Each Tate space V can be represented
as V » V d ‘ V c with V d P Vectk and V c P LCk .
Thus, the topological dual identifies
Vectopk » LCk ,LCopk » Vectk ,Taopk » Tak .

In particular, since Vectk is abelian, so is LCk , while Tak is a self-dual quasiabelian (in particular, exact) category, cf. [Be2] [BGW]. Let us add two
more examples to the above list:(4) The quasi-abelian category ILCk formed by inductive limits of linearlycompact spaces.
(5) The quasi-abelian category PVectk formed by projective limits of discrete vector spaces or, what is the same, objects in TopVect which,considered as topological vector spaces, are complete.
Example 4.1.7. The space of Laurent series kppzqq “ krrzssrz ´1 s is an objectof Tak . The localized ring krrz1 , z2 ssrpz1 z2 q´1 s is an object of ILCk but not of
Tak . Similarly, the ring krrzssrz ˚ srpzz ˚ q´1 s, see §1.2 C, is an object of ILCk .
40

Proposition 4.1.8. We have
ILCopk » PVectk ,PVectopk » ILCk ,Tak “ ILCk X PVectk ,the first two identifications given by forming the topological dual.
Proof: Let us prove the third statement. It is clear that Tak Ă ILCk XPVectk .
Let us prove the inverse inclusion. Suppose
(
q3
q2
q1
V “ lim
¨
¨
¨
ÝÑ
V
ÝÑ
V
ÝÑ
V
P PVectk
2
10
ÐÝ
is represented as the projective limit of a diagram pVi q of discrete vector
spaces and surjections qi . Then V is Tate, if and only if Kerpqi q is finitedimensional for all but finitely many i. Suppose this is not so. Then we can,without loss of generality, assume that all Kerpqi q are inifinite-dimensional,
by composing finite strings of the arrows in pVi q and getting a diagram withthe same projective limit.
With this assumption, suppose that V “ limÝÑj Lj where pLj qjě0 is anincreasing chain of linearly compact subspaces. Then, for each j,
pjq
pjq
pjq (
Lj “ lim
¨
¨
¨
ÝÑ
V
ÝÑ
V
ÝÑ
V
2
1
0
ÐÝ
pjq

where Vi is the image of Lj in Vi , a finite-dimensional subspace in Vi . We
now construct an element v P V , i.e., a compatible system pvi P Vi q, by aversion of the Cantor diagonal process. That is, we take v0 “ 0, then takep1q
v1 from Kerpq1 q (an infinite-dimensional space) not lying in V1 (a finitedimensional space). Then take v2 P V2 with q2 pv2 q “ v1 in such a way that
p2q
v2 R V2 (this is possible since Kerpq2 q is infinite-dimensional), and so on.
piq
We get an element pvi q of the projective limit with vi R Vi for all i. Suchan element cannot lie in the union of the Li .
Definition 4.1.9. (a) A Tate complex over k is a bounded complex V ‚ over
ILCk whose cohomology groups H i pV ‚ q P ♥ ILCk belong to ♥ Tak . We denote
by Tatek the full dg-subcategory in C b pILCk q formed by Tate complexes.
(b) A Tate complex V ‚ is called strict, if all the H i pV ‚ q belong to Tak Ă♥
Tak or, what is the same, if its differentials are strict (Prop. 4.1.6).
Note that Tatek is a perfect dg-subcategory, i.e., rTatek s, the corresponding cohomology category, is triangulated and closed under direct summands.
41

Example 4.1.10. The Jouanolou complex A‚n is a strict Tate complex. Moreprecisely, the topology on each Apn is given by the convergence of series. An
explicit representation of Apn as an inductive limit of linearly compact spaces
is given by the filtration of Corollary 1.2.8. Thus A‚n is a complex over ILCk .
As we have seen, its cohomology groups are Tate vector spaces. For n ą 0, we
have only two cohomology spaces: H 0 , linearly compact and H n´1 , discrete.
More generally, for any finite dimensional vector bundle E on Dn˝ , thecomplex A‚n pEq is naturally made into a strict Tate complex.
C. Tate complexes, algebraically. For a category C we denote by IndpCq
and PropCq the category of countable ind- and pro-objects in C, see [AM], [KS]for general background. In particular, we will use the notation “ limÝÑ ”iPI Ci
for an object of IndpCq represented by a filtering inductive system pCi qiPI
over C. Similarly for “ limÐÝ ”iPI Ci , an object of PropCq.
Assume C is abelian. Then so are IndpCq and PropCq. In this case wedenote by Inds pCq, Pros pCq the full subcategories formed by ind- and proobjects which are essentially strict i.e., isomorphic to objects “ limÝÑ ”iPI Ci ,resp. “ ÐlimÝ ”iPI Ci where pCi q is an filtering inductive (resp. projective) systemformed by monomorphisms (resp. epimorphisms). These are quasi-abelian
but not, in general, abelian categories.
Let Vectfk be the category of finite-dimensional k-vector spaces.
Proposition 4.1.11. (a) We haveInds pVectfk q “ IndpVectfk q » Vectk ,Pros pVectfk q “ PropVectfk q » LCk .

(b) Further, the taking of inductive or projective limits in TopVectk gives
identifications
Inds pLCk q » ILCk ,Pros pVectk q » PVectk .

(c) The abelian envelopes of the semi-abelian categories in (b) are identified as♥
ILCk » IndpLCk q, ♥ PVectk » PropVectk q(the categories of all, not necessarily strict, ind- and pro-objects).
Proof: (a) This is well known. For instance, because of the Noetherianand Artinian property of Vectfk , and ind- or pro-object in this category isessentially strict. Part (b) is also clear.
42

(c) Let us prove the first statement, the second one is dual. Consider theabelian category A of arbitrary chains of morphisms (inductive systems)((4.1.12)
V “ V0 ÝÑ V1 ÝÑ V2 ÝÑ ¨ ¨ ¨in LCk , i.e., of graded krts-modules in LCk . Let As Ă A be the semiabelian subcategory formed by chains of monomorphisms, i.e., torsion-freekrts-mod-ules. Every object V of A can be represented as the cokernel of amonomorphism i in As . More precisely, we have a short exact sequence(4.1.13)ic

0 Ñ K ÝÑ V bk krts ÝÑ V Ñ 0.

Here V bk krts is the free krts-module generated by V as a graded vector space,c is the canonical map given by the krts-module structure and K “ Kerpcq.
This implies that any object in IndpLCk q is the cokernel of a monomorphismin Inds pLCk q.
Proposition 4.1.14. (a) The objects of ILCk Ă ♥ ILCk “ IndpLCk q are
projective. In particular, Tate spaces are projective objects in the abeliancategory ♥ Tak .
(b) The abelian categories IndpLCk q, PropVectk q have homological dimension 1.
Proof: We start with three lemmas. As in the proof of Proposition 4.1.11(c),let A be the category of diagrams as in (4.1.12). We have a functor“Ýlim
Ñ ” : A ÝÑ IndpLCk q,

V ÞÑ “ ÝlimÑ ”Vi .

Lemma 4.1.15. “ limÝÑ ” is an exact, essentially surjective functor. Furthermore, any epimorphism in IndpLCk q is isomorphic (in the category of arrows)
to an image of an epimorphism in A.
Proof of the lemma: Exactness of “ limÝÑ ” and the statement about epimorphisms are general properties of ind-categories of abelian categories, see [KS],Lemmas 8.6.4 and 8.6.7. Next, any countable filtering category admits a cofinal map from the poset Z` “ t0, 1, 2, . . . u. This means that any countably
indexed ind-object is isomorphic to the object of the image of “ Ýlim
Ñ ” asabove.
Lemma 4.1.16. Objects of As are projective in A.
43

Proof of the lemma: A diagram V as in (4.1.12) formed by embeddings, is afree graded krts-module in LCk . Indeed, since LCk “ Vectop
k is a semisimple
abelian category, the embedding Vi´1 Ñ Vi admits a direct sum complement
Wi . Considering W “ pWi q as a Z` -graded object in LCk , we see that
V » W bk krts is free. This, and semisimplicity of LCk , implies projectivityof V .
Let us call an object V P A essentially strict, if “ ÝlimÑ ”pV q is an essentiallystrict object of IndpLCk q, i.e., is isomorphic to “ lim”pM
q where M P As .
ÝÑ
Lemma 4.1.17. If V P A is essentially strict, then there is M P As andan epimorphism q : V Ñ M in A such that “ limÝÑ ”pqq is an isomorphism in
IndpLCk q.
Proof of the lemma: For each i consider the diagram of epimorphisms
Vi ÝÑ ImtVi Ñ Vi`1 u ÝÑ ImtVi Ñ Vi`2 u ÝÑ ¨ ¨ ¨ .
If V is essentially strict, these epimorphisms eventually become isomorphisms, so the terms of the diagram stabilize to some Mi P LCk , and we
get the diagram of monomorphisms
M “ tM0 ÝÑ M1 ÝÑ ¨ ¨ ¨ u.
We see that M P As equipped with a natural epimorphism q : V Ñ M in A
which induces an isomorphism on the “ limÝÑ ”.
We now prove part (a) of Proposition 4.1.14. Let f : A Ñ B be an
epimorphism in IndpLCk q with B essentially strict. We prove that f splits.
By Lemma 4.1.15, f is isomorphic to f 1 “ “ Ý
limwhere g : N Ñ V
Ñ ”pgq1
is a surjection in A. It is enough to prove that f splits. Now, V P A isessentially strict, so by Lemma 4.1.17, there is a surjection q : V Ñ M in
A with M P As and such that “ Ýlim
Ñ ”pqq 2is an isomorphism. It is enoughtherefore to prove that the epimorphism f “ “ Ýlim
Ñ ”pqgq splits in IndpLCk q.
But qg : N Ñ M is a surjection in A with M projective. So qg splits in A
and therefore f 2 splits in IndpLCk q. This proves part (a).
Part (b), follows by considering the 2-term resolution (4.1.13). Proposition 4.1.14 is proved.

44

Corollary 4.1.18. (a) Every object of Db pILCk q » Db p♥ ILCk q is quasiisomorphic to its graded object of cohomology equipped with zero differential.
(b) The triangulated category Tatek is equivalent to the bounded derived
category of the abelian category ♥ Tak .
(c) Any strict Tate complex V ‚ can be split into a direct sum of complexes(over ILCk ) V ‚ “ H ‚ ‘ E ‚ , where E ‚ is exact and H ‚ has zero differential(and is thus a graded Tate space isomorphic to H ‚ pV ‚ q.)
Inside Tatek we have the full dg-subcategories: Perf k , complexes with finitedimensional cohomology; Dk , complexes with discrete cohomology; Ck , complexes with linearly compact cohomology. Thus Perf k “ Ck X Dk and theassociated cohomology categories are identified asrPerf k s » Db pVectfk q,

rDk s » Db pVectk q,

rCk s » Db pLCk q.

Proposition 4.1.19. rTatek sis the smallest strictly full triangulated subcategory in Db pILCk q which contains rDk s, rCk s and is closed under direct
summands.
Proof: Let T be the smallest subcategory in question. Then rTatek s Ă T ,because any object of the quasi-abelian category Tak is a direct sum of an
object from Ck and an object from Dk . Conversely, ♥ Tak Ă ♥ ILCk is closed
under extensions. Therefore forming cones and direct summands, startingwith ObpCk qYObpDk q will always give complexes whose cohomology objects
lie in ♥ Tak .

4.2

Tate objects in dg-categories

A. Ind- and pro-objects in dg-categories. For background on indand pro-objects in a (stable) 8-category C we refer to [Lu7], §5.3. In thispaper we consider only ind- and pro-objects represented by countable filtering
diagrams. Such objects form new 8-categories IndpCq, PropCq. Thus, objectsof IndpCq can be represented by symbols “holim
ÝÝÝÑ”iPI xi where I is a countablefiltering 8-category and pxi qiPI is an 8-functor I Ñ C. Similarly, objects of
PropCq can be represented by symbol “holim
ÐÝÝÝ”iPI yi , where I is as before andpyi qiPI is an 8-functor I op Ñ C.
We apply these concepts to dg-categories by converting them to k-linear
8-categories by using the dg-nerve construction [C][Fao]. The resulting(countable) ind- and pro-categories associated to a dg-category A will be45denoted IndpAq and PropAq. They are still k-linear 8-categories that we
can and will see as dg-categories. We note that, with this understanding,
IndpAq and PropAq are perfect, whenever A is perfect. Let us point out thefollowing more explicit description.
Proposition 4.2.1. Let A be a dg-category. Then:(I0) IndpAq is quasi-equivalent to its full dg-subcategory whose objects are“holimÝÝÝÑ”xi , wherepxi q “

f

f

f

01
12
23
x0 ÝÑ
x1 ÝÑ
x2 ÝÑ
¨¨¨

(

is a diagram consisting of objects xi P A, i ě 0 and closed degree 0
morphisms fi,i`1 : xi Ñ xi`1 .
(I1) We have quasi-isomorphisms (“-δ formula”):`
˘‚
Hom‚IndpAq “holim”x,“holim”y» holim
i
j
ÝÝÝÑ
ÝÝÝÑ
ÐÝÝÝi holimÝÝÝÑj HomA pxi , yj q,
where the homotopy limits on the right are taken in the model category
dgVectk of complexes.
Remarks 4.2.2. (a) The holimÝÝÝÑj in (I1) is the same as the naive inductive
limit in dgVectk , see Proposition A.3(a).
(b) By duality we get a description of PropAq “ IndpAop qop in terms ofsymbols “holim
ÐÝÝÝ”xi where pxi q is a diagram of objects and closed degree 0
morphisms in the order opposite to that of (I0).
Proof of Proposition 4.2.1: By definition, objects of IndpAq are representedby 8-functors I Ñ A where I is a filtering 8-category. In our setting weassume that ObpIq is at most countable. As in the classical case, any filtering
I admits a cofinal 8-functor from a filtering poset ([Lu7], Prop. 5.3.1.16),
and in the countable case we can take this poset to be Z` . Now, the category
corresponding to the poset Z` , is freely generated by the elementary arrowsi Ñ i ` 1. Therefore any 8-functor x : Z` Ñ A can be replaced by anhonest functor obtained by extending x from these elementary arrows. Suchfunctors are precisely the data in (I0). Finally, the appearance of the “-δ
formula” in (I1) from conceptual properties of IndpAq is explained in [Lu7],
p.378.
46

B. Tate A-modules. Let A be a Zď0 -graded commutative dg-algebra. For
any m P Z we then have the full dg-category Perf ďm
A Ă Perf A formed by those
perfect dg-modules over A which, as complexes, are situated in degrees ď m.
Recall that the duality functor M ÞÑ M _ “ RHomA pM, Aq identified Perf A
with its dual. We define the full subcategories in Perf A :ď´l _Perf ěl
A “ pPerf A q ,rl,msPerf Aďm“ Perf ělA X Perf A ,l ď m.

We define then the 8 (or dg-) categories
ď
ď
rl.ms
rl,ms
DA “
IndpPerf A q Ă IndpPerf A q, CA “
PropPerf A q Ă PropPerf A q
l,m

l,m
rl,ms

formed by ind- or pro-diagrams of which all terms belong to Perf A for somel, m (depending on the diagram). These categories are dual to each other.
The category DA is tensored over the category Vectk of all (possibly
infinite-dimensional) Àk-vector spaces. In particular, with each object M it
contains krzs b M “ 8i“0 M , the direct sum of infinitely many copies of M .
The category CA is tensored over the category of linearly compact topologicalp “
k-vector
spaces. In particular, with each object M it contains krrzssbM
ś8
i“0 M , the direct product of infinitely many copies of M .
We define the dg-category TateA of Tate A-modules as the perfect envelope of the full dg-subcategory in IndpPropPerf A qq whose class of objects is
ObpCA q Y ObpDA q. Since IndpPropPerf A qq is a perfect dg-category, we canview TateA as the minimal dg-subcategory in IndpPropPerf A qq containing
CA , DA and closed under forming shifts, cones and homotopy direct summands.
Remark 4.2.3. This is a slight modification of the definition in [He3] in that,besides restricting to countable ind- or pro-diagrams, we force all objects tobe “bounded complexes”. In particular, an infinite resolution of a non-perfect
A-module M (and thus such an M itself) would be an object in IndpPerf A q
but not in DA . In particular, it is, a priori, not an object of TateA .
Proposition 4.2.4. Let A “ k. Then, in comparison with the constructions
from §4.1:(a) Dk is quasi-equivalent to Dk » C b pVectk q and Ck to Ck » C b pLCk q;(b) TateA (for A “ k) is quasi-equivalent to the category of Tate complexes from §4.1.
47

Proof: (a) Let us show the first identification, the second one follows by
duality. It is known that IndpPerf k q “ dgVectk . (Both homotopy limits in
the RHS of (I1) can be replaced by the ordinary projective resp. inductive
limits because of Proposition A.3(b1).) Then Dk and C b pVectk q are fullsubcategories of dgVectk and have the same objects.
We now prove part (b). Let us denote temporarily by TateA“k the dgcategory obtained by specializing the above definition of TateA to A “ k, as
distinguished from the dg-category Tatek defined in §4.1. Let
ď
rl,ms
IPk “
IndpPropPerf k qq Ă IndpCk q.
l,mThen TateA“k is the perfect envelope of CDk , the full dg-subcategory in IPk
on the class of objects ObpCk q Y ObpDk q. Similarly, since C b pILCk q is aperfect dg-category, Proposition 4.1.19 implies that Tatek is identified withthe perfect envelope of CDk , the full subcategory in C b pILCk q on the classof objects ObpCk q Y ObpDk q. Let us denote both identifications Dk Ñ Dk
and Ck Ñ Ck from part (a) by the same letter λ (taking the limit). Then it
is enough to prove the following,‚‚
object of Dk and W “ “holim
Lemma 4.2.5. Let V “ “holim
ÐÝÝÝ” WiÝÝ‚ÝÑ” Vi be an‚
be an object of Ck , with Vi , resp. Wi being an inductive resp. projectiverl,mssystem over Perf k . Then the natural morphisms of complexes

Hom‚IPk pV, W q ÝÑ Hom‚C b pILCk q pλpV q, λpW qq,
Hom‚IPk pW, V q ÝÑ Hom‚C b pILCk q pλpW q, λpV qq
are quasi-isomorphisms.
Proof of the lemma: We can assume that pVi‚ q consists of injective morphisms
and pWi‚ q consists of surjective morphisms of perfect complexes. Then, applying the formula (I1) from Proposition 4.2.1 twice, we realize Hom‚IPk pV, W q
as double ÐholimÝÝÝ of a diagram of perfect complexes and surjective maps. Applying Proposition A.3(b) once, we replace the first Ð
holim
lim
ÝÝÝ by ÐÝ and get a
holim
of
a
diagramofcomplexes
andsurjective
maps.
Applying
PropositionÐÝÝÝ
A.3(b) once again, we replace the second holimÐÝÝÝ by lim
ÐÝ. After this the result
reduces to the set of continuous morphisms of complexes λpV q Ñ λpW q.
Similarly, we realize Hom‚IPk pW, V q as double holimÝÝÝÑ of a diagram of perfect complexes and injective maps. Applying Proposition A.3(a), we reduce48

it to double ÝlimÑ which again gives the space of continuous morphisms ofcomplexes λpW q Ñ λpV q. This proves the lemma and Proposition 4.2.4.
p Ω‚
p˚ Eq
Proposition 4.2.6. The construction E ÞÑ A‚n pEq “ ΓpJ,
p ˝ b πJ{Dndefines an exact functor
RΓ : Perf Dn˝ ÝÑ Tatek .
This statement, as well as the stronger Proposition 5.5.1 later, can beseen as analogs, in our setting, of Theorem 7.2 of [Dr].
p Ω‚ ˝ b ´q is naturally made into a (strict) functor
Proof. The functor ΓpJ,
J{DnModB ÝÑ Ind Pro Perf kp Deriving this functor,
where B is the ring of function of the affine scheme J.
we get an exact dg-functorp » dgModB ÝÑ Ind Pro Perf kDqcoh pJq
p˚ to the announced functor
which restricts along π
RΓ : Perf Dn˝ ÝÑ Tatek .

4.3

The Tate class, the residue class and the local
Riemann–Roch

A. The Tate class in cyclic cohomology. We start with a delooping
result.
Theorem 4.3.1. There is a canonical isomorphismHC˚ pTatek q » HC˚´1 pPerf k q » HC˚´1 pkq.
Proof. Let us consider the morphism of localization sequences (of perfect
dg-categories):/ Ck/ Ck { Perf k
Perf k

Dk/ Tatek49α/ Tatek {DkIt follows from [He3, Proposition 4.2] that the functor α is an equivalence.
We will deduce the result from the following lemma.
Lemma 4.3.2. If A is a perfect dg-category with infinite direct sums, thenthe cyclic complex CCpAq is acyclic. In particular HC˚ pAq » 0.
Let us postpone the proof of the lemma for now. The categories Ck andDk both admit infinite sums, and therefore have vanishing cyclic homology.
Using the localization invariance of HC (see Theorem 2.1.8), we get quasiisomorphisms„„„

CCpTatek q ÝÑ CCpTatek {Dk q ÐÝ CCpCk { Perf k q ÝÑ CCpPerf k qr1s.
This concludes the proof of Theorem 4.3.1.
Proof (of Lemma 4.3.2): We use the following fact (see [Ke2]): if f, g are
two functors from A to B and B is perfect (in particular, has direct sums),
then the action of f ‘ g onis the sum of the action of f and of g.
ÀHC
8Specifying to the case f “ i“0 Id and g “ Id, then f ‘ g » f and we get
HCpgq “ HCpIdq “ 0.
Definition 4.3.3. Let us denote by τ the class τ P HC 1 pTatek q given by theimage of the trace class through the isomorphism HC 0 pkq » HC 1 pTatek q.
For any object V P Tatek we have a class τV P HC 1 pEndpV qq induced by τ .
Remark 4.3.4. The class τV vanishes (by construction) as soon as V belongs
to either Ck or Dk .
B. Comparison with the residue class and local Riemann-Roch.
The Tate complex A‚n is, as the same time, a commutative dg-algebra . Wenote that (left) action of A‚n on itself gives rise to a morphism of associative
dg-algebras which we call the regular representationl : A‚n ÝÑ EndTatek pA‚n q.
In other words, for each p and each a P Apn , the multiplication operator
lpaq : A‚n Ñ A‚n rps, is a continuous morphism of graded objects of ILCk . This
follows from the fact that the filtration on A‚n from Corollary 1.2.8 which, by
Example 4.1.10, gives the inductive limit representation of A‚n , is compatible
with multiplication.
50

Theorem 4.3.5. There exists a non-zero constant λ P k with the following
property. The pullback l˚ τEndpA‚n q of the Tate class τEndpA‚n q with respect to l
is a class in HC 1 pA‚n q equal to λ ¨ ρ, where ρ is the residue class, see §2.3.
Proof: For any associative dg-algebra R‚ let us denote by BR‚ the corresponding dg-category with one object. The global section functor
RΓ : Perf A‚n » Perf Dn˝ ÝÑ Tatek
from Proposition 4.2.6 is compatible with the map l. In other words, we havea commutative diagram of dg-categories/ Perf A‚

BA‚n


n

l



RΓ

/ Tatek

BpEndpA‚n qq

where the horizontal functors map the one object onto A‚n . In particular wehavel˚ τEndpA‚n q “ RΓ˚ τ P HC 1 pPerfpA‚n qq » HC 1 pA‚n q.
Recall that the dg-category Perf A‚n » Perf Dn˝ is the dg-quotient of Perf Dn
by the full subcategory Perf Dn ,t0u spanned by perfect complexes supported
at 0. The global section functor ΓDn : Perf Dn Ñ Ck hence induces a functor
G and a commutative diagram
Perf t0u pDq/ PerfpDq

Γt0u



ΓDn



Perf k

/ Ck





Dk

P



G

/ Ck { Perf k» αi/ Tatek/ PerfpD ˝ qQ/ Tatek {Dk .

Lemma 4.3.6. The composite functor Q ˝ RΓ is equivalent to the composite
α ˝ G.
Proof: From the universal property of PerfpDn˝ q as a quotient of PerfpDn q,
it suffices to compare Q ˝ RΓ ˝ P and α ˝ G ˝ P » Q ˝ i ˝ g. The inclusion
H 0 pA‚n q Ñ A‚n induces a canonical natural transformation α ˝ g Ñ RΓ ˝ P .
51

In turn, it induces the required equivalence (as the pointwise kernel of thenatural transformation is killed by the projection Q).
It follows from this lemma that RΓ˚ τ equals the composite
Γt0u

δ

HC1 pPerfpDn˝ qq ÝÑ HC0 pPerf Dn ,t0u q ÝÑ HC0 pPerf k q.
Let us write zÀ‚ for the coordinate system z1 , ¨ ¨ ¨ , zn on Dn and denote, as in§1.4, by V “
kzi the space spanned by the zi . In what follows we will pay
attention to the GLn -action on various spaces arising.
The dg-category Perf Dn ,t0u is compactly generated by k (considered as atrivial krrz‚ ss-module), and we have a GLn -equivariant identification
R Endkrrz‚ ss pkq » Ext‚krrz‚ ss pk, kq » S ‚ pV ˚ r´1sq(4.3.7)

with the exterior algebra of V graded by its degree. Using [SS], we get a
Morita equivalence between Perf Dn ,t0u and R Endkrrz‚ ss pkq (a version of theclassical S ´Λ duality [GM]). Under this Morita equivalence, the functor Γt0u
amounts to the augmentation morphism S ‚ pV ˚ r´1sq Ñ k. The induced map
HC0 pPerf Dn ,t0u q Ñ HC0 pPerf k q is thus non-trivial. It is also GLn -invariant.
Lemma 4.3.8. HC0 pPerf Dn ,t0u q admits a unique GLn -coinvariant class.
Proof: Consider the Hodge decomposition of HC‚ pS ‚ V ˚ q. The part of weightp is computed by the complex S ‚ pV ˚ r´1sq b Λďp pV ˚ r´1sqr2ps with the de
Rham differential. The de Rham differential is GLn -equivariant. Therefore,
to compute the GLn -coinvariant elements in the cyclic homology at hand, it
suffices to understand the action on each degree of the graded vector space
‚

˚

ďp

˚

S pV r´1sq b Λ pV r´1sqr2ps »

p
n à
à

Λj V ˚ b S i V ˚ r2p ´ p2i ´ jqs

j“0 i“0The GLn -representation Λj V ˚ b S i V ˚ has simple spectrum and admits coinvariants if and only if i “ j “ 0. We get#
k if m “ 2pppq
HCmpS ‚ pV ˚ r´1sqqGLn “
0 else.
In particular HC0 pS ‚ pV ˚ r´1sqq has exactly one invariant 1-dimensional subspace.
52

We now finish the proof of Theorem 4.3.5. The residue HC1 pAn q Ñ
HC0 pkq is GLn -invariant and vanishes when restricted to HC1 pkrrz‚ ssq. Thelong exact sequenceδ¨ ¨ ¨ ÝÑ HC1 pkrrz‚ ssq ÝÑ HC1 pAn q ÝÑ HC0 pS ‚ pV ˚ r´1sqq ÝÑ ¨ ¨ ¨implies that δ maps isomorphically the unique invariant line of HC1 pAn q onto
the unique invariant line of HC1 pS ‚ pV ˚ r´1sqq. This concludes the proof of
Theorem 4.3.5.
Remark 4.3.9. Since k is allowed to be an arbitrary field of characteristic 0,
we have λ P Q˚ . We expect λ “ ˘1. This can be possibly proved either bydirect calculation or by upgrading some of the considerations of this paper
to fields of arbitrary characteristic.
Recall (Proposition 3.1.2) that the class τV P HC 1 pEndpV qq, V P Tatek
gives a Lie algebra cohomology class θ˚ τV in H2Lie pEndpV qq. We will also callθ˚ τV the Tate class. Let us consider the following particular case.
Let r ě 1 and V “ pA‚n q‘r . As before, we have then the regular representation (morphism of dg-algebras) lr : Matr pA‚n q Ñ EndTatek ppA‚n q‘r q which
we can also see as a morphism of dg-Lie algebras.
Corollary 4.3.10 (Local Riemann–Roch). The pullback lr˚ pθ˚ τpA‚n q‘r q is equal
to the class of the cocycle λ¨γPr , where λ is the same as in Theorem 4.3.5 andγPr is the special case, for P pxq “ Pr pxq “ trpxn`1 q{pn ` 1q!, of the cocycle
defined in Theorem 3.2.1.
We note that Pr pxq is the degree pn ` 1q component of trpex q, the “Chern
character” of x. Corollary 4.3.10 can be therefore seen as a simplified version
of a local Riemann-Roch-type theorem where we take into account the infinitesimal symmetries of a vector bundle but not of the underlying manifold.

5

Action on derived moduli spaces

In the rest of the paper we will relate the dg-Lie algebra g‚n and its central
extensions, with derived moduli spaces of principal bundles on n-dimensional
manifolds. We start with recalling the general setup.
53

5.1

Background on derived geometry

A. General conventions. We will work in the framework of derived algebraic geometry. For general results on the subject, we refer to [TV]. For acomprehensive survey, the reader may look at [To2].
Derived algebraic geometry can be though as algebraic geometry, whererings are being replaced by “homological rings”. Namely, the category ofk-algebras will be replaced the category Cdgaď0
k formed by Zď0 -graded commutative dg-algebras over k (up to quasi-isomorphisms). It is naturally madeinto a model category. Moreover, the usual notions of Zariski open or closed
immersions, flat, smooth or étale morphisms extend to morphisms in Cdgaď0k .
In particular, one can form an étale Grothendieck topology.
Given any commutative algebra, one can consider it as an object inCdgaď0k concentrated in degree 0. On the other hand, for any object A P
0Cdgaď0k , the cohomology space H A is a commutative algebra. We also get
0a canonical morphism A Ñ H A.
Let also sSet be the category of simplicial sets. Given two objects A, B PCdgaď0k , we get a simplicial set of morphisms MappA, Bq. In particular, we geta Yoneda functor mapping A to a covariant functor Spec A : Cdgaď0k Ñ sSet.
ď0
A derived prestack is a covariant functor Cdgak Ñ sSet. A derived stack
is a derived prestack satisfying the natural homotopy étale descent condition.
A derived stack representable by a cdga is called a derived affine scheme. We
will denote by dAffk the category of derived affine schemes. It is equivalent
to the opposite category of Cdgaď0k .
B. Derived stacks and derived categories. The category of derived
stacks will be denoted by St. It will be considered either as a model category,or as an 8-category. Given A P Cdgaď0k , the category of A-dg-modules isendowed with a standard model structure. We denote by dgModA the (klinear) dg-category of fibrant-cofibrant A-dg-modules. Given a derived stackY , we define, following [To3]
Dqcoh pY q “ holimÐÝÝÝSpec AÑY dgModA(homotopy limit in the model category of dg-categories). An object inDqcoh pY q can be informally described as the data of a A-dg-module MA,φfor any A P Cdgaď0k and any map φ : Spec A Ñ Y , together with natural homotopy glueing data. We also define the Zď0 -graded derived category
Dď0qcoh pY q Ă Dqcoh pY q formed by pMA,φ q consisting of Zď0 -graded dg-modules.
54

Example 5.1.1. Any M P Dď0qcoh pY q gives rise to the dual number stack Y rM s
defined by gluing SpecpA ‘ MA,φ q (the trivial square zero extension).
Note that for any map f : Y Ñ Z, one gets an adjunction
Lf ˚ : Dqcoh pZq Õ Dqcoh pY q : Rf˚For example, for φ : SpecpAq Ñ Y and M P Dqcoh pY q, the object Lφ˚ M isjust the structure datum MA,φ (we identify Dqcoh pSpec Aq with dgModA ).
Another example: if Z “ Spec k and f is the canonical projection, thenRf˚ : Dqcoh pY q Ñ dgModk computes the cohomology of Y with values in agiven object in Dqcoh pY q.
We will also need not necessarily quasi-coherent sheaves. The functor
A ÞÑş dgModA lands in dg-categories, hence in (k-linear) 8-categories. Letξ : dgMod Ñ dAffk denote the associated Cartesian Grothendieck construction (see [Lu7, Chap. 3]).
Definition 5.1.2. Let Y be a derived stack. We define its derived
ş categoryof OY -complexes DpY q as the 8-category of sections dAffk {Y Ñ dgMod ofξ over Y .
Note that by [Lu7, 3.3.3.2], the 8-category Dqcoh pY q is the full subcategory of DpY q spanned by Cartesian sections. Informally, an object in
M P DpY q is the data of A-dg-modules MA,φ for any map φ : Spec A Ñ
Y , together with coherence maps ζf : MA,φ bLA B Ñ MB,φ˝f for any map
f : Spec B Ñ Spec A and higher coherence data. The module M is thenquasi-coherent if and only if all the maps ζf are quasi-isomorphisms.
The category DpY q admits internal homs that we will denote by R HomOY .
C. Geometric objects and tangent complexes. For the definition ofgeometric derived stacks (or, what is the same, derived Artin stacks) we refer
to [TV].
This class includes, first all derived schemes, that is, derived stacks that
are Zariski locally equivalent to derived affine schemes. Following [Lu1],
one can represent derived schemes in terms of ”homotopically” ring spaces.
Namely, a derived scheme X is a topological space together with a sheaf (upto homotopy) of Zď0 -graded cdga’s OX such that pX, H 0 pOX qq is a scheme.
In fact, a derived Artin stack is a derived stack that can be obtained fromderived affine schemes by a finite number of smooth quotients.
55

The cotangent complex LY of a derived stack Y is an object of Dqcoh pY q
defined (when it exists) by the universal property
MapDqcoh pY q pLY , M q » MapY { St pY rM s, Y q,

M P Dď0qcoh pY q.

Here Y { St is the comma category of derived stacks under Y . The object LY
is known to exist [TV] when Y is geometric (no smoothness assumption).
The tangent complex TY is defined as the dual
TY “ R HomOY pLY , OY q P DpY q.
If Y is locally of finite presentation [TV], then LY is a perfect complex andhence so is TY . In particular TY is an object of Dqcoh pY q.
For a k-point iy : y ãÑ Y we will write
TY,y “ Li˚y pTY q “ R HomOY pLY , Oy q
for the tangent complex of Y at y. This is a complex of k-vector spaces.
D. Derived intersection: Given a diagram X Ñ Z Ð Y , we have thederived (or homotopy) fiber product X ˆhZ Y . If X, Y, Z are affine, so ourdiagram is represented by a diagram A Ð C Ñ B in Cdgaď0k , then`˘X ˆhZ Y “ Spec A bLC B .
We will be particularly interested in the following situation. Let f : X Ñ Y
be a morphism of derived stacks, and y P Y be a k-point. Then we have thederived stack (a derived (affine) scheme, if both X and Y are derived (affine)schemes)Rf ´1 pyq “ X ˆhY tyu.
It will be called the derived preimage of y. It is the analog of the homotopy
fiber of a map between spaces in topology.

5.2

The Kodaira–Spencer homomorphism

A. Group objects and actions. By a group stack we will mean a stack G
together with simplicial stack G‚ such that G0 » Spec k, G1 » G and which
satisfies the Kan condition: the morphisms corresponding to the inclusions56

of horns are equivalences. Intuitively, G‚ is the nerve of the group structure
on G, see [Lu6, §4.2.2] for more details.
Similarly, an action of a group stack G (given by G‚ ) on a stack Y is a simplicial stack Y‚ together with a morphism q : Y‚ Ñ G‚ with an identification
Y0 » Y such that, for any m, the morphism
pqm , BtmuãÑt0,1,...,mu q : Ym ÝÑ Gm ˆ Y0
is an equivalence. In this case Y‚ satisfies the Kan condition. Intuitively,Y‚ is the nerve of the “action groupoid”. The “realization” of Y‚ , i.e., thederived stack associated to the prestack A ÞÑ |Y‚ pAq|, is the quotient derived
stack rY {Gs. In particular, we have the stack BG “ r˚{Gs, the classifying
stack of G.
Examples 5.2.1. (a) Let Y be a derived stack and y P Y be a k-point. The
pointed loop stackΩy Y “ tyu ˆhY tyu : A ÞÑ ΩpY pAq, yq
is a group stack. The corresponding simplicial stack pΩy Y q‚ is the (homotopy) nerve of the morphism tyu Ñ Y , i.e.,pΩy Y qm “ tyu ˆhY tyu ˆhY ¨ ¨ ¨ ˆhY tyu » pΩy Y qm
(pm ` 1q-fold product).
(b) Let Y be any derived stack. Its automorphism stack is the group stack
RAutpY q : A ÞÑ Mapeq
St{ Spec A pY ˆ Spec A, Y ˆ Spec Aq
Here the superscript “eq” means the union of connected components of the
mapping space formed by vertices which are equivalences. Alternatively, we
can describe it as the functor
A ÞÑ ΩpSt{ Spec A, Y ˆ Spec Aq,
the based loop space of the nerve of the category of derived stacks over Spec A
with the base point being the object Y ˆ Spec A. By construction, we havean action of RAutpY q on Y ; an action of a group stack G on Y gives amorphism of group stacks G Ñ RAutpY q.
57

Proposition 5.2.2. Let f : X Ñ Y be a map of derived stacks and y P Y
be a k-point. Then the group stack Ωy Y has a natural action on the derived
preimage Rf ´1 pyq.
Proof: We define the simplicial stack Rf ´1 pyq‚ as the nerve of the morphism
Rf ´1 pyq Ñ Y , i.e.,Rf ´1 pyqm “ Rf ´1 pyq ˆhX Rf ´1 pyq ˆhX ¨ ¨ ¨ ˆhX Rf ´1 pyq »» tyu ˆhY tyu ˆhY ¨ ¨ ¨ ˆhY X » pΩy Y qm ˆ Rf ´1 pyq.
All the required data and properties come from contemplating the commutative diagram/X
Rf ´1 pyq



/ Y.

tyu

Example 5.2.3 (Eilenberg-MacLane stacks). Let Π be a commutative
algebraic group (in our applications Π “ Gm ). For each r ě 0 we then havegroup stack EMpΠ, nq, known as the Eilenberg-MacLane stack. It is definedin the standard way using the Eilenberg-MacLane spaces for abelian groups
ΠpAq for commutative k-algebras A.
Thus EMpΠ, 0q “ Π as a group stack, i.e., the corresponding simplicial
stack EMpΠ, 0q‚ “ Π‚ is the simplicial classifying space of Π. Similarly (theunderlying stack of the group stack) EMpΠ, 1q is identified with BΠ. In
general, if we denote EMpΠ, nq‚ the simplicial stack describing the groupstructure on EMpΠ, nq, then |EMpΠ, nq| “ EMpΠ, n ` 1q.
Definition 5.2.4. Let G be a group stack and Π a commutative algebraic
group. A central extension of G by Π is a morphism of group stacks φ : G ÑBΠ or, what is the same, a morphism of stacks BG Ñ EMpΠ, 2q.
A central extension φ gives, in a standard way, a fiber and cofiber sequenceof group stacks
r ÝÑ G Ñ 1,
1 Ñ Π ÝÑ G
r is the fiber of φ.
where G58

B. Formal moduli problems.
We recall Lurie’s work [Lu3] on formal
moduli problems which serve as infinitesimal analogs of derived stacks.
Definition 5.2.5. A cdga A P Cdgaď0k is called Artinian, if:• The cohomology of A is finite dimensional (over k);• The ring H 0 A is local and the unit induces an isomorphism between k
and the residue field of H 0 A.
In particular, any Artinian cdga admits a canonical augmentation (theunique point of Spec A). Artinian cdga’s form an 8-category which we will
denote by dgArtk .
Definition 5.2.6. A formal moduli problem is a functor (of 8-categories)
F : dgArtk Ñ sSet
such that:(1) F pkq » ˚ is contractible.
(2) (Schlessinger condition): For any diagram A Ñ B Ð C in dgArtk
with both maps surjective on H 0 , the canonical map F pA ˆhB Cq Ñ
F pAq ˆhF pBq F pCq is an equivalence.
We denote by Fun˚ pdgArtk , sSetq the (8-)category of functors from dgArtk
to simplicial sets satisfying the condition (1) of Definition 5.2.6, and by FMP
the full subcategory of formal moduli problems. General criteria for representability of functors imply (see [Lu3], 1.1.17) that we have a left adjoint(the “formal moduli envelope”)
L : Fun˚ pdgArtk , sSetq ÝÑ FMP(5.2.7)to the embedding functor.
For a formal moduli problem F we define its tangent complex TF (at the
only point ˚ of F ). This is a complex of k-vector spaces defined as follows.
First, we define
ppq

TF“ F pkrp s{2p q,

degpp q “ ´p,p ě 0.

These simplicial sets are actually simplicial vector spaces, forming a spectrum
in the sense of homotopy theory, that is, connected by morphisms (in fact,
ppq
pp`1q
by equivalences) TF Ñ ΩTF. We define the complex TF to correspondppq
to the spectrum pTF q by the Dold-Kan equivalence.
59

Theorem 5.2.8 (Lurie). For any formal moduli problem F , the complex
TF r´1s has a homotopy Lie structure. Moreover, the data of F is equivalent(up to homotopy) to the data of a complex T “ TF r´1s and of a dg-Lie
structure on this complex.
For future reference we recall from [Lu3] §2, the characterization of the
Lie algebra structure on TF r´1s. For a graded k-vector space M ‚ let FLpM ‚ q
be the free graded Lie algebra (with zero differential) generated by M ‚ . Denote by FLieě1
k be the full (8-) subcategory in dgLiek whose objects are
FLpM ‚ q where M ‚ is finite-dimensional and situated in degrees ě 1. For‚
any L P FLieě1
k the Chevalley-Eilenberg cochain algebra CE pLq is an object‚of dgArtk . In fact, CE pLq is quasi-isomorphic to the dual number algebrak ‘ pM ‚ q˚ r´1s, see [Lu3] (2.2.15). We then have
(5.2.9)MapdgLiek pL, TF r´1sq “ F pCE‚ pLqq,

L P FLieě1k(identification of functors on FLieě1k ). Note that this defines the dg-Lie algebra structure uniquely.
We will also need the following global analog of the Schlessinger condition,see [TV, Def. 1.4.2.1].
Definition 5.2.10. A derived stack Y is infinitesimally cartesian if the folď´1lowing condition holds. Let A P Cdgaď0andk , M P dgModAδM ÝÑ B ÝÑ A ÝÑ M r1s
be a square zero extension of A by M . Then the square
F pBq/ F pAq





F pAq

F p1q

F p1`δq/ F pA ‘ M r1sq

is cartesian.
It is known that any derived Artin stack is infinitesimally cartesian [TV,Prop. 1.4.2.5].

60

Example 5.2.11. Any infinitesimally cartesian stack Y and any k-point y P
Y defines a formal moduli problem
Ypy : A ÞÑ Y pAq ˆhY pkq tyu.
Its tangent complex TYpy is identified with TY,y “ HompLY , ky q, if LY exists(e.g., if Y is geometric). In other cases it can be considered as the definition ofTY,y . By Theorem 5.2.8, the shifted complex TY,y r´1s carries a homotopy Lie
structure. It is an analog of the fundamental group π1 pY, yq of a topologicalspace Y at a point y.
For any derived stacks Y and X we define the mapping stack
RMappY, Xq : A ÞÑ MapSt{ Spec A pY ˆ Spec A, X ˆ Spec Aq.
Notice that RAutpY q is an open substack in RMappY, Y q, that is, the pullback of it under any morphism U Ñ RMappY, Y q from an affine derived
scheme U , is an open subscheme in U .
Proposition 5.2.12. (a) If Y is an infinitesimally cartesian derived stack,then so is RMappX, Y q for any X. Moreover, suppose Y is geometric andf : X Ñ Y is any morphism, representing a k-point rf s of RMappX, Y q.
Then the tangent complex TRMappX,Y q,rf s (defined as the tangent complex of
the associated formal moduli problem) is identified with RΓpX, f ˚ TY q.
(b) Further, if Y is infinitesimally cartesian, then so is RAutpY q. In thiscase TRAutpXq,Id is RΓpX, TX q.
Proof. (a) Consider the full subcategory C of St spanned by those X P St
such that RMappX, Y q is infinitesimally cartesian. Since Y is infinitesimallycartesian, C contains all derived affine schemes. The category C is moreover
stable by (homotopy) colimits. We get C “ St. The identification of thetangent complex is the standard argument using the universal property of
LY and the appropriate adjunctions.
(b) Follows from (a) because RAutpXq is a open in RMappX, Xq and sotheir completions are identified.
C. The Lie dg-algebra of a group stack. Let G be a group stack (represented by a simplicial stack G‚ , as in part A). Assume that the completion
p1 is a formal moduli problem (this is true, for example, if G is infinitesiGmally cartesian). In this case we have the tangent complex LiepGq “ TG,1 ,61

naturally made into a dg-Lie algebra. Explicitly, we form the “classifying
p1 q as the formal moduli envelope (5.2.7) of theformal moduli problem” BpG
functorx
A ÞÑ holimÝÝÝÑ pGn q1 pAq, dgArtk ÝÑ sSet.
Then LiepGq is, as a dg-Lie algebra, identified with TBpGp 1 q r´1s.
Examples 5.2.13. (a) This construction extends the classical correspondence between group schemes and Lie algebras. Further, if Π is a commutative algebraic group, then LiepEMpΠ, nqq » LiepΠqrns, an abelian Liealgebra LiepΠq put in degree n.
(b) For the group stack ΩY,y of loop spaces, we have LiepΩy Y q “ TY,y r´1s,
because BppΩY,y q1p q is identified with Ypy .
p1 is a formal moduli problem,(c) Let G be a group stack such that G
and g “ LiepGq. A central extension of G by Gm , i.e., a morphism of group
stacks φ : G Ñ BpGm q gives, after passing to Lie dg-algebras, a morphism
Liepφq : g Ñ kr1s, since LiepGm q “ k. By Proposition 3.1.1, Liepφq gives rise
to a cohomology class γφ P H2Lie pg, πq.
D. The Kodaira-Spencer morphism. Let f : X Ñ Y be a morphismof derived stacks and y P Y be a k-point. Assume that Ypy is a formal moduli
problem and the homotopy fiber Rf ´1 pyq is geometric. In particular, wehave the cotangent complex LRf ´1 pyq .
Definition 5.2.14. The Kodaira-Spencer morphism of f is the morphism ofhomotopy Lie algebras
κ : TY,y r´1s ÝÑ RΓpRf ´1 pyq, TRf ´1 pyq q
obtained as the differential, at the identity element, of the action(5.2.15)a : Ωy Y ÝÑ RAutpRf ´1 pyqq

from Proposition 5.2.2. Here we use Proposition 5.2.12(c) to identify thetangent to RAut.
Proposition 5.2.16. Assume X is geometric, so we have the normal fiber
sequence in DpRf ´1 pyqq
TRf ´1 pyq ÝÑ TX bLOX ORf ´1 pyq ÝÑ TY,y bk ORf ´1 pyq .
62Let δ be the coboundary map of this triangle. Then κ is identified, as amorphism of complexes, with the composite of δr´1s with the adjunction map
TY,y Ñ RΓpRf ´1 pyq, TY,y b ORf ´1 pyq q.
Proof: We deduce the claim from a more general local statement. Let i :F Ñ X be any morphism of geometric stacks. We then have the relative
tangent complex TF {X on X fitting into the exact triangle(5.2.17)c

TF {X ÝÑ TF ÝÑ i˚ TX ÝÑ TF {X r1s.

Let us form the “groupoid stack” Z‚ with Zn “ F ˆhX ¨ ¨ ¨ ˆhX F (pn ` 1q
factors). Thus Z0 “ F (the objects), Z1 “ Z :“ F ˆhX F (the morphisms)
and so on. The second projection p2 : Z Ñ F defines an action of thegroupoid Z‚ on F which we write asa : Z ˆF F ÝÑ F.
The tangent, at the identities of the groupoid (which are represented by the
diagonal embedding e : F Ñ Z) of a is a map
da : e˚ pTZ{F q ÝÑ TF
of which the source is identified, via the projection p1 : Z Ñ F , with TF {X .
The following is then obvious.
Lemma 5.2.18. In the above situation, da is identified with the canonicalmorphism c : TF {X Ñ TF .
i

We now deduce Proposition 5.2.16 from the lemma. Let F “ Rf ´1 pyq ÑX be the inclusion of the homotopy fiber. By definition, the action a from
(5.2.15) comes from the action a, while δr´1s class from the normal sequence,
as in the proposition. We now note that the normal sequence is canonically
identified with the shift of the triangle (5.2.17), that is, TY,y b OF (the“normal bundle” to F ) is the same as TF {X r1s. This finishes the proof.

5.3

Derived current groups and moduli of G-bundles

A. Derived moduli spaces. Let G be a reductive algebraic group overk with Lie algebra g. Let X be a smooth irreducible algebraic variety over63

k and x P X be a k-point. Recall the notations x
p “ Spec OX,x » Dn and˝˝p “xp ´ txu » Dn for the (punctured) formal neighborhood of x in X.
x
Recall that principal G-bundles are classified by the 1-stack BG “ r˚{Gs:for any scheme Y , the (nerve of the) groupoid of principal G-bundles on
Y is equivalent to the simplicial set of maps Y Ñ BG. This allows us todefine the notion of principal G-bundles over a derived scheme Y : first, wedenote by RBunG pY q the simplicial set of morphisms Y Ñ BG. A vertex inthat simplicial set is called a principal G-bundle on Y . We then define thederived moduli stack of principal G-bundles over Y as the following functorfrom Cdgaď0k to simplicial sets:
RBunG pY q “ RMappY, BGq : A ÞÑ RBunG pY ˆ Spec Aq.
Fundamental for us will be the derived stack RBunG pXq.
Proposition 5.3.1. Let P be a principal G-bundle on X and rP s P RBunG pXq
be the corresponding k-point. Then
TrP s RBunG pXq » RΓpX, AdpP qqr1s.
Proof: Follows from Proposition 5.2.12 with Y “ BG.
xq as the functor
We then define RBunG pp
pX,x qq,
pk O
A ÞÑ RBunG pSpecpA b

pX,x “ holimpA bk OX {I n q. ă
pk O
Ab
ÐÝÝÝ

Here I is the ideal of x.
xq is not the result ofRemark 5.3.2. Note that the definition of RBunG pp
p. More precisely,
applying the construction RBunG pY q to the scheme Y “ xpnqnthfor any integer n, denote by x “ SpecpOX {I q, the n infinitesimal neighborhood of x in X. Thenpnq
p
RBunG ppxq » holimÐÝÝÝn RBunG px q “ RBunG pSpf OX,x q

is a particular case of RBun construction but for a formal scheme.
We have the restriction map
xq.
λ : RBunG pXq Ñ RBunG pp
64

Definition 5.3.3. The derived stack of G-bundles on X rigidified at x isdefined to be´1hRBunrig
G pX, xq “ RBunG pXq ˆRBunG px̂q tTrivu “ Rλ pTrivq,

where Triv “ X ˆG denotes the trivial bundle. For any cdga A, the simplicial
set of A-points of RBunrig
G pX, xq is
RBunG pX ˆ Spec Aq ˆhRBun pSpecpAbp Op
G

X,x qqtTrivu.

In other words, this is the groupoid formed by G-bundles on X ˆ Spec A
pX,x q.
pO
endowed with a trivialization on SpecpA b
Proposition 5.3.4. (a) The derived stack RBunrig
G pX, xq is represented bya derived Artin stack.
(b) If X is projective, then RBunrig
G pX, xq is represented by a derived
scheme of amplitude r0, n ´ 1s.
Proof: We first deduce from [He1, Lemma 4.2.4] that the inclusion of thetrivial bundle tTrivu Ñ RBunG pp
xq is a smooth atlas. In particular, we getthat the derived stack RBunrig
pX,
xq is representable by a derived ArtinG
stack.
Using Lurie’s representability criterion for derived schemes (see [Lu5,
Theorem 3.1.1]), we can hence reduce to the non-derived moduli problem. It
is known that BunG pXq is a geometric stack of amplitude r´1, n ´ 1s, which
locally is represented as the quotient of a scheme of finite type by an algebraic
group. When we introduce the rigidification, we kill the stack structure.
x˝ q as the functor
We next define the derived stack RBunG pp
´
`
˘ `
˘¯
pX,x ´ SpecpAq ˆ txu .
pk O
A ÞÑ RBunG Spec A b
Let us fix the notations X ˝ “ X ´ txu.
Proposition 5.3.5. The natural morphisms˝hRBunrigx˝ q tTrivu
G pX, xq Ñ RBunG pX q ˆRBunG pp

is an equivalenceThis is proved in [HPV, Theorem 6.20].
65

B. Derived current groups and their action. We define Gpp
x˝ q as thefunctorˆ˙pp OX,x q ´ pSpecpAq ˆ txuq, G .
A ÞÑ Map SpecpAb
This is group object in derived stacks.
Proposition 5.3.6. (a) Gpp
x˝ q is a derived affine ind-scheme, identified withthe group of automorphisms of the point Triv in the derived stack RBunG ppx˝ q.
(b) The Lie algebra of Gpp
x˝ q is identified with
LiepGppx˝ qq “ Te Gpp
x˝ q “ g b RΓpp
x˝ , Oq » g‚x » g‚nstudied earlier.
Proof:(a) The first claim follows from [He1]. The second is obvious.
x˝ q with the mapping stack(b) We use (a) and compare RBunG pp
x˝ , BGq : A ÞÑ Map pp
x˝ ˆ Spec A, BGq .
RMap ppx˝ , BGq has Lie algebra idenThe group of automorphisms of Triv in RMap pp
tified with g b RΓpp
x˝ , Oq by Proposition 5.2.12(a).
We now consider the morphism of derived stacks
x˝ , BGq ÝÑ RBunG pp
x˝ q
f : RMap pp
induced by the canonical maps
(5.3.7)

pX,x Ñ A b
pX,x .
pk O
A bk O

We note that (5.3.7) is a quasi-isomorphism whenever A has finite dimensional cohomology. This implies that f is formally étale at the point Triv
and therefore the induced morphism of tangent Lie algebras at Triv is aquasi-isomorphism.
Consider the projection of derived stacks
g : RBunG pX ˝ q ÝÑ RBunG ppx˝ q.
´1
Proposition 5.3.5 identifies RBunrig
G pX, xq with the homotopy fiber Rg pTrivq.
Therefore Propositions 5.3.6 and 5.2.2 imply:

66

Theorem 5.3.8. (a) The derived group stack Gpp
x˝ q acts on RBunrig
G pX, xq
by changing the trivialization.
(b) The Kodaira-Spencer map of g gives rise to a morphism of dg-Lie
algebras
β : g‚x ÝÑ RΓpRBunrig
G pX, xq, Tq.
Remark 5.3.9. In particular, β induces a morphism on the pn ´ 1q-st cohomology of the dg-Lie algebra g‚x :
`
˘
pX,
xq,
T
.
HBn´1 pg‚x q ÝÑ Hn´1 RBunrig
G
Consider the first non-classical case n “ 2, when X is a surface. In this case
we are dealing with H1 of the tangent complex which has the meaning of thespace of deformation of RBunrig
G pX, xq. Theorem 5.3.8 produces therefore aclass of deformations of the rigidified derived moduli space labelled by the2
pg b OX q.
space of “polar parts” HB1 pg‚x q “ Htxu
A natural way of deforming the moduli space would be to “twist” the´1cocycle condition gik
gjk gij “ 1 defining G-bundles by replacing it with
´1
gik gjk gij “ λijk with some “curvature data” λ “ pλijk q. Considering such
twisted bundles is standard when G “ GLr and λ consists of scalar functions(we then get modules over an Azumaya algebra). In our case, elements of2
Htxu
pg b OX q can be seen as providing infinitesimal germs of more general(non-abelian) twistings and thus deformations of the rigidified moduli space.

5.4

Central extensions associated to Tate complexes

A. The group stack GLpV q and its Lie algebra. The categories of Tate
modules assemble into two prestacks
Tate : A ÞÑ TateA ,Tate : A ÞÑ TateequivA

where “equiv” means the maximal 8-groupoid in the 8-categorical envelope(nerve) of the dg-category.
In particular, for any object V of Tatek we have a group prestack ofautomorphisms GLpV q “ ΩV Tate.
Example 5.4.1. If V is the Tate space kppzqq in degree 0, then GLpV q is thenon-derived group ind-scheme GLp8q studied in [Ka]. For a general V , we
get a derived analog of GLp8q.
67

Proposition 5.4.2. (a) Each prestack GLpV q is an infinitesimally cartesian
stack.
(b) The Lie algebra LiepGLpV qq is identified with EndpV q, the algebra ofendomorphism of V in Tatek made into a Lie algebra.
Proof: (a) let A Ñ B be a homotopy étale cover in Cdgaď0k . Denote Bn “BAbn , so that B‚ “ pBn q is a cosimplicial object in Cdgaď0k . The naturalfunctor F : TateA Ñ holimÐÝÝÝ TateBn is fully faithful by embedding into thefunctor
IndpPropPerf A qq ÝÑ holimÐÝÝÝn IndpPropPerf Bn qq.
Fully faithfulness of this last functor follows from étale descent for the categories of dg-modules. Now, fully faithfulness of F implies that each GLpV q
is a stack.
Next, we show that GLpV q is infinitesimally cartesian. This goes in thesimilar way, using that the stack of right bounded dg-modules is infinitesimally cartesian [Lu2, Theorem 7.2].
(b) We construct a morphism of dg-Lie algebras(5.4.3)

φ : LiepGLpV qq ÝÑ EndpV q.

For this we construct a natural transformation Φ of the functors represented
by both Lie algebras on L P FLieě1
k . For such L we have
MappL, LiepGLpV qq “ LpTateVp qpCE‚ pLqq,
where L is the formal moduli envelope functor (5.2.7). We will construct,
first, a natural morphism of simplicial sets
ΨL : TateVp pCE‚ pLqq Ñ MapdgLiek pL, EndpV qq.
By definition, an object of TateVp pCE‚ pLqq is a Tate complex W over CE‚ pLq
together with an identification k bLCE‚ pLq W Ñ V . Note that we have acanonical identification (Koszul duality)(5.4.4)

U pLq » RHomCE‚ pLq pk, kq.

Therefore, for any such W , the associative dg-algebra U pLq, and therefore,
the dg-Lie algebra L, acts on V , giving a morphism L Ñ EndpV q. Thisdefines ΨL on vertices, and the standard arguments of Morita theory extend
68it to a morphism of simplicial sets, in fact to a natural transformation Ψ offunctors FLieě1 Ñ sSet.
Let eEndpV q be the formal moduli problem corresponding to the dg-Lie
algebra EndpV q by Theorem 5.2.8. We then can write each ΨL as a morphismΨL : TateVp pCE‚ pLqq ÝÑ eEndpV q pCE‚ pLqq.
Next, we notice that each ΨL descends to a natural morphism of simplicial
sets
ΦL : LpTateVp qpCE‚ pLqq ÝÑ eEndpV q pCE‚ pLqq.
This is just because eEndpV q is a formal moduli problem. Now, the collectionof the ΦL extends, in a canonical way, to the natural transformation Φ andtherefore to a morphism φ of dg-Lie algebras as claimed in (5.4.3).
We now prove that φ is an equivalence or, what this the same, that ΨL is
an equivalence whenever L is free on one generator of degree d ě 1. In thiscase CE‚ pLq “ krs is the algebra of dual numbers with generator  “ d´1of degree 1 ´ d.
Applying the Morita theory associated with (5.4.4), we include ΨL into
an adjunction
ψL : IndpPropPerf krs qq Ø tRepresentations of L in IndpPropPerf k qqu : ηL
ψL pW q “ k bLkrs W,

ηL pV, f : V Ñ V rdsq “ CE‚ pL, V q » Conepf qr´1s.

Here f is the action of (the generator of) L on V . We note that Conepf qr´1s
can be written as V rs “ V bk krs (free krs-module) with an additional
differential given by f .
Note that ψL is fully faithful. This is a formal consequence of the factthat
ψLPerf : Perf krs Ø Representations of L in Perf kis fully faithful. In fact, ψLPerf is an equivalence (Koszul duality).
Therefore ΨL is fully faithful. Let us prove that it is essentially surjective.
Lemma 5.4.5. Let pV, f q be a representation of L in Tatek . Then ηL pV, f q P
Tatekrs .
Proof: We can assume that V is a graded Tate space with a zero differential,
and so f is a morphism of graded Tate spaces. So V has two lattices V1c Ă V2c69

so that f induces a morphism of pro-finite-dimensional spaces f c : V1c Ñ
V2c rds. We make Conepf c qr´1s into a (pro-perfect) krs module by making 
acts by the embedding V1c Ñ V2c .
Similarly, f induces a morphism of ind-finite-dimensional spaces f d :dV1 Ñ V2d where Vid “ V {Vic . Note that we have the quotient map V1d Ñ V2d
and so Conepf d qr´1s is made into an (ind-perfect) krs-module. Now wehave a short exact sequence
0 Ñ Conepf c qr´1s ÝÑ Conepf qr´1s ÝÑ Conepf d qr´1s Ñ 0which implies that ηL pV, f q “ Conepf qr´1s P Tatekrs .
We now prove that the canonical map c : ψL pηL pV, f qq Ñ pV, f q is aquasi-isomorphism in the category of representations of L. i.e., that C “
pConepcq, gq is contractible. By the above, ηL pcq is a quasi-isomorphism, i.e.,ηL pC, gq “ Conepgq is contractible. So g is a degree d quasi-isomorphism of
C to itself. But C is bounded by our assumption. So C is contractible. Thisfinishes the proof of Proposition 5.4.2.
B. K-theoretic extensions. Recall the stack of categories Perf definedbyPerf pAq “ Perf A
for any A P Cdgaď0k .
For a perfect dg-category A we denote by KpAq the space of K-theoryof A, so that πi KpAq “ Ki pAq. Explicitly, we can define KpAq “ Ω|S‚ pAq|
as the loop space of the Waldhausen S-construction (in which all Sn pAq are
understood as 8-groupoids).
We now make K-theory into a prestack
K “ K ˝ Perf : A ÞÑ KpPerfpAqq.
By composing K with Tate we get the prestack
KTate : A ÞÑ KpTateA q.
We have the morphism of prestacks
Tate ÝÑ KTate70

induced by the identification
TategrpA ÝÑ S1 pTateA q.
The following is proven in [He3].
Theorem 5.4.6. KTate is identified with BpKq “ |S‚ pPerf q| (after stackification on Nisnevich topology).
Remark 5.4.7. Theorem 5.4.6 is a geometric analog of Theorem 4.3.1. In
particular, as Theorem 4.3.1 allows us to build central Lie algebra extensions,
Theorem 5.4.6 gives us group central extensions. The construction goes asfollows.
We have the determinantal Gm -torsor Det Ñ K or, equivalently, a groupmorphism K Ñ BGm . Applying the classifying stack on both ends, we get
the determinantal gerbe(5.4.8)Detp2q : KTate ÝÑ EMpGm , 2q.

This gerbe gives, for any V P Tatek , a central extension of group prestacks
Ą q ÝÑ AutpV q Ñ 1.
1 Ñ Gm ÝÑ AutpV
Recall that using Theorem 4.3.1, we built in Definition 4.3.3 a cyclic classτV P HC 1 pEndpV qq for each V P Tatek . With Loday’s map from Proposition2
pEndpV qq. Those
3.1.2, we get Lie algebra cohomology classes θpτV q P HLie
classes give central extensions of Lie algebrasĄ q Ñ EndpV q Ñ 0.
0 Ñ k ÝÑ EndpVĄ q
Theorem 5.4.9. Let V be a strict Tate complex. The Lie algebra of AutpV
Ą q.
is identified with EndpV
Remark 5.4.10. It is very natural to expect that the cyclic homology of a
dg-category A can be recovered from its K-theory by some functorial procedure (“taking the tangent space”) so that, in particular, the trace class
tr P HC 0 pkq corresponds to the determinantal character (the identification
det : K1 pkq Ñ k˚ ). Then one could argue that the Tate class (the deloopingof tr) is similarly “tangent” to the determinantal gerbe (the delooping of71det), thus obtaining a very natural proof of Theorem 5.4.9. This would also
justify the name “additive K-theory” for cyclic homology.
However, such a direct construction seems to be unknown. The closest statement in this direction is the recovery (due to L. Hesselholt) of the
Hochschild homology of a (dg-)algebra R in terms of the rational K-theoryof the ring of dual numbers Rrs{2 , see [DGM].
We therefore dedicate the rest of this section to a proof of Theorem 5.4.9
by a series of reductions.
C. Primitivity of the Lie cohomology classes. For V P Tatek wedenote by γV P H2Lie pEndpV qq the class corresponding to the Lie algebra ofĄ q. We need to prove the equality
AutpV(5.4.11)γV “ θ˚ pτV q

where τV P HC 1 pEndpV qq is induced by the Tate class τ P HC 1 pTatek q andθ is the Loday homomorphism, see §3.1B.
Note that the statement is known (and classical) in the case when V “kppzqq is the most standard example of a Tate space. We will now reduceto this case by showing that the system of classes γV satisfies compatibilities
that hold for the system of θ˚ pτV q.
Definition 5.4.12. Let ηV P H2Lie pEndpV qq, V P Tatek be a system of Liealgebra cohomology classes. We say that pηV q is a primitive system if, forany direct sum decomposition V » V1 ‘ V2 in the abelian category of strict
Tate complexes we have
ηV |EndpV1 q‘EndpV2 q “ p˚1 ηV1 ` p˚2 ηV2 .
Here pν : EndpV1 q ‘ EndpV2 q Ñ EndpVν q is the projection.
We note that direct sum decompositions with V2 “ 0 are given by isomorphisms φ : V Ñ V1 , so a primitive system satisfies, in particular, thecompatibility condition: Ad˚φ pηV1 q “ ηV . HereAdφ : EndpV q Ñ EndpV1 q,

u ÞÑ φ ˝ u ˝ φ´1 .

Lemma 5.4.13. The classes γV form a primitive system.

72

Proof: This is because the determinantal gerbe, being a K-theory datum, is“group-like”, i.e., gives a local system on the Waldhausen space of TateA for
any A. That is, for any triangle (simplest cell on the Waldhausen space)
V1 Ñ V Ñ V2
in TateA we have an isomorphism of Gm -gerbes over Specpkq
Detp2q pV1 q b Detp2q pV2 q Ñ Detp2q pV q
satisfying coherent compatibilties. In particular, for V » V1 ‘ V2 , a direct
sum decomposition in Tatek , we have
Ą 1 q ‹ AutpV
Ą 2q
Ą 1 ‘ V2 qAutpV qˆAutpV 1 q » AutpV
AutpV
1
2
(Baer sum). This, by differentiation (passing to the Lie algebras of group.
stacks), implies that the system pγV q is primitive.
Lemma 5.4.14. The classes θ˚ τV form a primitive system as well.
Proof: This is a general property of cyclic homology. Let A “ EndpV q and
Ai “ EndpVi q. We then have the enbedding of dg-algebras A1 ‘ A2 Ñ A.
It is enough to prove that the restriction of the Loday homomorphism θA
2
to HLie
2 pA1 ‘ A2 q is equal to the sum of the restrictions on HLie pA1 q and2
HLie pA2 q. This restriction is the left path in the commutative diagram
HLie
2 pA1 ‘ A2 q


HLie
2 pAq

θA1 ‘A/ 2

HCr´1 pA1 ‘ A2 qθA
/ HCr´1 pAq.

Looking at the right path we see that HC1 pA1 ‘ A2 q being identified with
HC1 pA1 q ‘ HC1 pA2 q, the composition splits into the direct sum of the two
restrictions, as claimed.
D. Comparison of Lie cohomology classes. It remains now to prove
the following statement.
Proposition 5.4.15. Let η and η 1 be two primitive system of classes in
1
H2Lie pEndpV qq, V P Tatek . Suppose that ηkppzqq “ λ ¨ ηkppzqq
for some λ P k.
1
Then ηV “ λ ¨ ηV for any strict Tate complex V .
73

We notice first:
Proposition 5.4.16. Let pηV q be a primitive system. If V » V1 ‘ V2 asbefore, then the pullback of ηV to EndpV1 q Ă EndpV q, is equal to ηV1 .
Let now V ‚ be a strict Tate complex. Decomposing it as V ‚ “ H ‚ ‘ E ‚as in Corollary 4.1.18(c), we have an isomorphism of associative dg-algebras(and hence of dg-Lie algebras) EndpH ‚ q Ñ EndpV ‚ q. It implies an isomorphism H‚Lie pEndpV ‚ qq » H‚Lie pEndpH ‚ qq. Since H ‚ has no differential,
EndpH ‚ q is a graded Lie algebra without differential.
Proposition 5.4.17. Let H be a graded Tate space (situated in finitely
many degrees) which is neither discrete nor linearly compact. Assume thegraded components of H are of dimension either 0 or 8. Then we have
H2Lie pEndpHqq » k.
Proof: This is a modification of the result of [FT1] which can be considered
as corresponding to V being kppzqq in degree 0. We first relate HC 1 pEndpHqq
with HC 1 pTatek q » k. More precisely, we note:
Lemma 5.4.18. Let H be as above.
(a) The functor
ρ “ R HompH, ´q : Tatek ÝÑ dgModEndpHq .
takes values in perfect dg-modules over EndpHq.
(b) This functor gives a quasi-equivalence between Tatek and Perf EndpHq .
In particular, HC‚ pEndpHqq » HC‚ pTatek q is spanned by generators in degrees 1, 3, 5, . . . .
Proof: (a) Since H is neither linearly compact nor discrete, it decomposes asC‘D, where C P Ck and D P Dk have at leastÀ one graded component infinitedimensional. That is,śD admits a shift of Z` k as a direct summand, while
C admits a shift of Z` k as a direct summand. It follows that any Tatecomplex W can be obtained, up to a quasi-isomorphism, by a finite numberof extensions and retracts from H. This means that R HompH, W q can beobtained by a finite number of extensions and retracts from R HompH, Hq “
EndpHq, so it is perfect.
(b) Denote A “ EndpHq. We first prove that ρ is fully faithful in the dgsense, i.e., induces quasi-isomorphisms on Hom-complexes. This is certainly
74true for the complex HompH, Hq which is sent by ρ to HomA pA, Aq “ A.
Further, ρ is exact and takes direct summands (retracts) to direct summands.
So ρ induces a quasi-isomorphism on HompW1 , W2 q, where W1 and W2 are
any Tate complexes obtained from H by a finite number of extensions andretracts. But by the above, all Tate complex are obtained in such a way.
Next, we show that ρ is essentially surjective. This is immediate since
Perf A is generated, under extensions and retracts by A “ ρpHq itself.
We now prove Proposition 5.4.17 by the same arguments as in [FT1]. We
keep the notation A “ EndpHq and apply the dg-algebra analog of the LodayQuillen-Tsygan theorem [Bur] which gives that HLie
‚ pgl8 pAqq is the symmetricalgebra on the graded space HC‚´1 pAq. Next, because each component of
H is infinite-dimensional, H » pHq‘r and therefore A » glr pAq for eachr ě 1. This allows us to pass from gl8 pAq to A itself and conclude that
H2Lie pAq » HC 1 pAq “ k. Proposition 5.4.17 is proved.
1, we
We now finish the proof of Proposition 5.4.15. If ηkppzqq “ λ ¨ ηkppzqq
1
have ηV “ λ ¨ ηV for any V “ kppzqq bk F , where F is a finite-dimensional
graded k-vector space. In other words, V is a direct sum of shifts of kppzqq.
Indeed, up to a shift kppzqq is a direct summand of V and so the statement
follows from Proposition 5.4.16. Further, if V is any strict Tate complex, thenthere exists an F as above such that V is a direct summand of kppzqq bk F ,and so the statement again follows from Proposition 5.4.16. Proposition
5.4.15 and Theorem 5.4.9 are proved.

5.5

Action on determinantal torsors

A. Global sections on Dn˝ . Let us denote by Perf Dn˝ the derived prestackin categories
Perf Dn˝ : A ÞÑ Perf SpecpArrz1 ,...,zn ssq´t0u
It is actually a stack [HPV, Theorem 6.10].
Proposition 5.5.1. The global section functor RΓ from Proposition 4.2.6
naturally extends to a morphism of prestacks
RΓ : Perf Dn˝ ÝÑ Tate.
Proof. For any A P Cdgaď0k and p ě 0, denote by Ap the Kozsul resolutionpp
associated to pz1 , . . . , zn q in Arz1 , . . . , zn s. The cdga Arrzss is the homotopy
75

limit of the diagram q ÞÑ Arzs{Iq , where Iq is the ideal generated by degree q
monomials. For any p, we have pz1p , . . . , znp q Ă Ip and Inp Ă pz1p , . . . , znp q. We
getArrzss » holimÐÝÝÝAp .
Let Perf Dn denote the functor A ÞÑ Perf Arrzss and Perf Dnppq be the functor
A ÞÑ Perf Ap . It follows from [Lu4, §5.1] that the canonical morphism ofstacksPerf Dn ÝÑ holimÐÝÝÝp Perf Dnppq
is an equivalence. For any A P Cdgaď0k and any p P N, the category Perf Ap
is canonically equivalent to the category of Ap -modules in Perf A (as Ap isperfect on A). In particular, it embeds fully faithfully into the category of
Ap -modules in Ind Pro Perf A . We get
IPP
Perf Dn Ă holimÐÝÝÝp ModOp ,where ModIPP
is the functor A ÞÑ holimOpÐÝÝÝp ModAp pInd Pro Perf A q. Denote

top
by ModIPP
Orrzss the functor A ÞÑ ModArrzsstop pInd Pro Perf A q where Arrzss
is ”holim
ÐÝÝÝ”Ap , considered as a commutative algebra in Pro Perf A . The basechange natural transformation
IPP
ModIPP
Orrzss ÝÑ holimÐÝÝÝp ModOp

admits a pointwise right adjoint ψ. However ψ is not a natural transformation
as it does not commute with base change. It does once restricted to Perf Dnthough and we get a natural transformation
Perf Dn ÝÑ ModIPP
Orrzss .
We now consider A‚n “ RΓpDn˝ , Oq as a krrzsstop -module in Ind Pro Perf k .
Tensoring with A‚n defines an endotransformation of ModIPP
Orrzss . We thendeduce the proposition from the following lemma.
Lemma 5.5.2. The composite natural transformation´bA‚

Forgetn
IPP
η : Perf Dn ÝÑ ModIPP
Orrzss ÝÑ ModOrrzss ÝÑ Ind Pro Perf

has values in Tate and is null-homotopic once restricted to perfect complexes
supported at 0 P Dn .
76

Proof. Let A P Cdgaď0k . The functor ηA has by construction values in TateA .
It now suffices to prove that the image ηA pAq vanishes (where A is seen as a
Arrzss-modules with the trivial action). Using base-change, we can assume
A “ k.
The krrzsstop -complex A‚n is, by Čech descent, the homotopy limit ofmodules of the form krrzsstop rzI´1 s for a none empty I Ă t1, . . . , nu. Resolving
k as a krrzss-module using the natural Kozsul complex, we get
k bLkrrzsstop krrzsstop rzI´1 s » 0
for any I ‰ H. The functor ηk therefore maps k to an acyclic complex.
We now finish the proof of Proposition 5.5.1. Notice that for any A P
˝ pAq is a quotient of the category Perf D pAq byCdgaď0n
k , the category Perf Dn
the stable full subcategory of perfect complexes supported at 0. It follows
from the lemma that η factors through the morphism of prestacks
RΓ : Perf Dn˝ ÝÑ Tate
which coincides with the functor from Proposition 4.2.6 over k-points. Thisconcludes the proof of Proposition 5.5.1.
Remark 5.5.3. Note that the above construction can be mimicked to define
a global section morphismRΓ : Perf xp˝ Ñ Tate
for any k-point x in a variety of dimension n.
B. Determinantal torsors and determinantal gerbes. Let φ : G Ñ
GLr be a representation of G. Each G-bundle E on any derived stack Z
induces a vector bundle φ˚ E P Perf Z .
As before, X is a smooth projective variety, dimpXq “ n. We constructa Gm -torsor detφ P PicpRBunG pXqq as the morphism RBunG pXq Ñ BGm
defined as the composition
φ˚

RΓ

det

RBunG pXq ÝÑ Perf X ÝÑ Perf k ÝÑ KPerf k ÝÑ BGm .
We also denote by detφ the pullback of this torsor to RBunrig
G pX, xq.
77

The determinantal gerbe Detφ : RBunG pp
x˝ q Ñ KpGm , 2q is the composition
φ˚

Detp2qRΓ

RBunG ppx˝ q ÝÑ Perf xp˝ ÝÑ Tate ÝÑ KTate ÝÑ KpGm , 2q
where Detp2q defined in (5.4.8).
Proposition 5.5.4. (a) The determinantal gerbe Detφ comes with canonicaltrivializations τp and τ ˝ over RBunG pp
xq and RBunG pX ˝ q.
(b) The determinantal torsor detφ is equivalent to HomDetφ pp
τ , τ ˝ q.
Proof. (a) Let us first deal with τp. We have a canonical natural transformationαp : RΓxp “ RΓppx, ´q ÝÑ RΓxp˝ “ RΓpp
x˝ , ´qof maps Perf xp Ñ Tate. For any A P Cdgaď0k , the K-theory of TateA is equivalent to that of the quotient TateA {DA . In particular, the natural transforp induces an equivalence of morphisms between the two composites
mation αRΓPerf xp ÝÑxp Tate ÝÑ KTate

and

RΓ ˝

p
x
Tate ÝÑ KTate.
Perf xp ÝÑ

The LHS composite factors through KC : A ÞÑ KpCA q which vanishes, as thecategories of compact complexes admit infinite sums. The RHS composite
appears in the restriction of Detφ to RBunG pp
xq and this identification induces
the trivialization τp.
The case of τ ˝ is done similarly, using the natural transformationx˝ , ´q.
α˝ : RΓpX ˝ , ´q ÝÑ RΓpp
This concludes the proof of (a).
(b) By construction, the torsor HomDetφ pp
τ , τ ˝ q is induced by the map
Perf X Ñ Tate mapping a family of perfect complexes E on X to the homotopy equalizer of the mapsp
αRΓpX ˝ , Eq ‘ RΓpp
x, Eq ÝÑx˝ , Eq.
ÝÑ RΓppα˝This homotopy equalizer is equivalent to RΓpX, Eq, by a family version of
Proposition 1.1.4 (which follows from [HPV, Corollary 6.13]). The result
follows.
78

C. The action of central extensions. Let X be a projective variety ofdimension n, and x P Xpkq a k-point. Let φ : G Ñ GLr be a representation. Pulling back the Tate class from Definition 4.3.3 along the functor
RΓ : Perf xp˝ Ñ Tatek , we get a classx˝ q » HC 1 pA‚x q.
τx P HC 1 pp
x˝ , Oq.
Recall the definition A‚x “ RΓpp
Definition 5.5.5. The class τx induces a central extension of g‚x “ g b A‚xthat we will denote by rg‚x,φk ÝÑ rg‚x,φ ÝÑ g‚x .
This extension has a geometric counterpart. Recall the determinantal
gerbe Detφ : RBunG pp
x˝ q Ñ KpGm , 2q. We denote by rDetφ s its total spacerDetφ s “ RBunG pp
x˝ q ˆhKpGm ,2q t˚u.
The diagonal map and the trivial bundle define a k-point d P rDetφ spkq.
r x˝ qφ denote the group stack Ωd rDetφ s. It comes
Definition 5.5.6. Let Gpp
r x˝ qφ Ñ Gpp
x˝ q “ ΩTriv RBunG pp
x˝ q. The
with a natural projection π : Gpp
homotopy fiber of π at the unit is the group scheme Gm , so that we have acentral group extensionr x˝ qφ ÝÑ Gpp
Gm ÝÑ Gppx˝ q.
r x˝ qφ is classified by the group
Remark 5.5.7. Note that the extension Gpp
morphism Gpp
x˝ q Ñ BGm obtained by taking the pointed loops of the map
φx˝ q Ñ KpGm , 2q.
Det : RBunG pp
The following is a direct consequence of the above definitions and of
Proposition 5.4.2.
r x˝ qφ q and r
Proposition 5.5.8. The Lie algebra extensions LiepGppg‚x,φ of
LiepGppx˝ qq » g‚x are equivalent.
We denote by rdetφ s the total space of the determinantal torsor detφ on
RBunrig pX, xq.
79

r x˝ qφ acts on rdetφ s in a way compatible
Theorem 5.5.9. (a) The group Gpp
r x˝ qφ Ñ Gpp
with the projections Gpp
x˝ q and rdetφ s Ñ RBunrig
G pX, xq, and withthe action from Proposition 5.3.8.
(b) The dg-Lie algebra rg‚x,φ acts infinitesimally on rdetφ s in a way compatible with the infinitesimal action of g‚x on RBunrig pX, xq.
r x˝ qφ is the pointed loop group of theProof. (a) By construction the group Gpp
φpoint d in rDet s. Moreover, the trivialization τ ˝ and Proposition 5.5.4 givea homotopy cartesian squarerdetφ s/ RBunG pX ˝ q





τ˝

/ rDetφ s.

tdu

Furthermore, the inclusion tdu Ñ rDetφ s factors as
τp

tdu “ Spec k “ tTrivu ÝÑ RBunG pp
xq ÝÑ rDetφ s.
r x˝ qφ on
In particular, Proposition 5.2.2 defines the announced action of Gpp
φrdet s. The above diagram being compatible with the various projections,
we see that the action is indeed compatible with the one from Proposition
5.3.8.
(b) It follows from (a) and the Kodaira–Spencer morphism.

A

Model categories of dg-algebras and dgcategories(A.A) Conventions on complexes. We recall that k is a field of characteristic 0. We follow the usual sign conventions on (differential) gradedk-vector spaces, their tensor products, Koszul sign rule for symmetry and soon. The degree of the differential is always assumed to be `1. The degreeof a homogeneous element v will be denoted |v|. Note, in particular, theconvention(A.1)

pf b gqpv b wq “ p´1q|g|¨|v| f pvq b gpwq
80

for the action of the tensor product of two operators f : V Ñ V 1 and g :W Ñ W 1.
The shift of grading of a graded vector space V ‚ is defined by V ‚ rns “
krns b V ‚ , where krns is the field k put in degree p´nq. So the basis of krns
is formed by the vector 1rns. For v P V ‚ we denote vrns “ p1rnsq b v P V ‚ rns.
This gives the suspension morphism (an “isomorphism” of degree n)sn : V ÝÑ V rns,v ÞÑ vrns.

With respect to tensor products, we have the decalage isomorphism (of degree0)dec : V1‚ r1s b V2‚ r1s b ¨ ¨ ¨ b Vn‚ r1s Ñ pV1‚ b V2‚ b ¨ ¨ ¨ b Vn‚ qrns
given by:řn

decpspv1 q b ¨ ¨ ¨ b spvn qq “ p´1q i“1 pn´iq|vi | sn pv1 b ¨ ¨ ¨ b vn q
This isomorphism induces an isomorphism of graded vector spaces
`
˘
`
˘
(A.2)
decn : S n V ‚ r1sq ÝÑ Λn V ‚ rns.
(A.B) Model structures and categories of dg-algebras.
We willfreely use the concept of model categories, see, e.g., [Lu7] for background.
For a model category M we denote by rMs “ MrW ´1 s the corresponding
homotopy category obtained by inverting weak equivalences.
We denote by dgVectk the category of differential graded vector spaces(i.e., cochain complexes) over k with no assumptions on grading. This is asymmetric monoidal category.
Let P be a k-linear operad. By a dg-algebra of type P we will mean a Palgebra in dgVectk . They form a category denoted dgAlgP . Thus, for P being
one of the three operads As, Com, Lie, describing associative, commutativeand Lie algebras, we will speak about associative dg-algebras, resp. commutative dg-algebras, resp. dg-Lie algebras (over k). The categories formedby such algebras will be denoted by dgAlgk , Cdgak and dgLiek respectively.
They have products, given by direct sums over k, and coproducts given by
free products of algebras, denoted A ˚ B. For commutative dg-algebras,A ˚ B “ A bk B.
The category dgAlgP carries a natural model structure [Hi], in which:81

• Weak equivalences are quasi-isomorphisms.
• Fibrations are surjective morphisms of dg-algebras.
Cofibrations are uniquely determined by the axioms of model categories. In
particular, any dg-algebra A of type P and any graded vector space V we canform FP pV q, the free algebra of type P generated by V , and the embedding
A Ñ A ˚ FP pV q is a cofibration.
As usual, the model structure allows us to form homotopy limits andcolimits in the categories dgAlgP . They will be denoted by Ð
holim
holim
ÝÝÝ and ÝÝÝÑ.
In particular, if F is a sheaf of algebras of type P on a topological space S,
then
RΓpS, Fq “ holimÐÝÝÝ U ĂS FpU q
open

is canonically defined as an object of the homotopy category rdgAlgP s. An
explicit way of calculating the homotopy limit of a diagram of algebras represented by a cosimplicial algebra (this includes RΓpS, Fq) is provided by theThom-Sullivan construction, see [HiS].
The above includes (for P being the trivial operad) the category dgVectk
itself. In particular, let us note the following fact about homotopy limits in
dgVectk indexed by Z` “ t0, 1, 2, ¨ ¨ ¨ u.
Proposition A.3. (a) Let pEi‚ q be an inductive system over dgVectk , in‚‚
dexed by Z` . Then the natural morphism holimÝÝÝÑ Ei Ñ lim
ÝÑ Ei is a quasiisomorphism.
(b) Let pEi‚ q be a projective system over dgVectk , indexed by Z` . Then‚‚the natural morphism limÐÝÝÝ Ei is a quasi-isomorphism in each ofÐÝ Ei Ñ holim
the following two cases:(b1) Each Ei‚ is a perfect complex.
(b2) The morphisms in the projective system pEi‚ q are termwise surjective.
Proof: (a) holimlimÝÝÝÑ is the left derived functor of ÝÑ. Therefore part (a) follows
Vectkfrom the fact that the functor lim
is exact on the category of inductiveÝÑsystems of vector spaces indexed by Z` .
(b) holimlimÐÝÝÝ is the right derived functor of ÐÝ. Therefofe we have thespectral sequenceqp`q‚
pRq limÐÝÝÝi Ei q.
ÐÝi qpEi q ñ H pholim82

As well known, the functors Rq lim
ÐÝi for1 countable filtering diagrams can benonzero only for q “ 0, 1. Further, R lim
ÐÝi vanishes for diagrams of finitedimensional spaces as well as for any diagrams formed by surjective maps.
(A.C) Model structure on category of dg-categories. We denote by
dgCatk the category of k-linear dg-categories. For a dg-category A we denote
by rAs the corresponding H 0 -category: it has the same objects as A, while
HomrAs px, yq “ H 0 Hom‚A px, yq. This notation, identical with the notation
for the homotopy category of a model category, does not cause confusion:
when both meanings are possible, the result is the same.
We equip dgCatk with the Morita model structure of Tabuada [Tab].
Weak equivalences in this structure are Morita equivalences. Fibrant objects
are perfect dg-categories, i.e., dg-categories quasi-equivalent to Perf B where
B is some small dg-category. We recall two additional characterizations ofperfect dg-categories.
First, A is perfect, if and only if the Yoneda embedding A Ñ Perf A is aquasi-equivalence.
Second, A is perfect, if and only if A is pre-triangulated and rAs (whichis then triangulated) is closed under direct summands.

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G. F., B. H.: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111
Bonn, Germany.
Email: G. F.: gf77@mpim-bonn.mpg.de, B. H.: hennion@mpim-bonn.mpg.de
M.K.: Kavli IPMU, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583 Japan.
Email: mikhail.kapranov@ipmu.jp88