LetAbe a connective unital dg-algebra overk. Recall from Section 1.1, Section (B), that the generalized trace map is a (functorial) morphism
Tr : CEk•(gl∞(A))−→HCk•(A)[1].
A morphism CEk•(gl∞(A))→ HCk•(A)[1]such as Tr amounts to an L∞-morphism gl∞(A) → HCk•(A), where the RHS is considered with its abelian L∞-structure.
It corresponds to a mapTr :gl∞(A)→HCk•(A)in the∞-categorydgLiek, or equiv-alently to a map
gl∞(A)[1]−→θ(HCk•(A)[1])
in the ∞-category dgLieΩk of shifted dg-Lie algebras. Recall that θ: dgModk → dgLieΩk maps ak-dg-module to the abelian shifted dg-Lie algebra built on that mod-ule.
In this section, we will prove that the generalized traceTris tangent (in the sense of formal moduli problems) to the canonical morphism of functors BGL→Kmapping a vector bundle to its class inK-theory. We will see thatBGL→Kinduces a tangent morphismT:gl∞(A)→θ(HCk•(A))of dg-Lie algebras overkand thatT is homotopic to the generalized traceTr. See Theorem 4.2.1 below for a precise statement.
4.1. The tangent Lie algebra ofBGL. — LetAA:dgArtk→dgAlg60
Q denote the functorB7→A⊗kB. Denote byAAits augmentation idealAA(B)'A⊗kAug(B).
Definition4.1.1. — Letn∈N∪ {∞}and letA: dgArtk →dgAlg60
Q be any func-tor. We denote byBGLn(A)the functordgArtk →sSetsmappingB ∈dgArtk to
BGLn(A)(B) := hofib(BGLn(A(B))−→BGLn(A(k))).
Remark4.1.2. — The canonical morphism colimn∈NBGLn(A) →BGL∞(A)is an equivalence.
Example4.1.3. — We assumeA =AA. Denote byexp(gln(AA))(B)the (nilpotent) subgroup of GLn(AA(B))of matrices of the form1 +M, whereM is a matrix with coefficients in AA(B) ' A⊗k Aug(B). We can then identify the homotopy fiber BGLn(AA)with
BGLn(AA)'GLn(AA(k))
GLn(AA)'B exp(gln(AA)).
For later use, we will work in a slightly bigger generality. Let R be a (possibly non-unital) discretek-algebra. We denote bygR the functor
gR:dgAlg6k0,nu−→dgLie6k0
mapping a connective dg-algebra C to the Lie algebra underlying the associative algebraC⊗kR. Examples include the case whereRis the algebra ofn×n-matrices with coefficients in k, where we findgR =gln. Obviously, the construction R 7→gR is functorial.
For an algebraR, we also denote byBGR(AA)∈PFMPk the functor BGR(AA) :B7−→B exp gR AA(B)
= B exp gR(A⊗kAug(B)) ,
whereexp(gR(A⊗kAug(B)))is the subgroup of(A⊗kB⊗kR)×consisting of elements of the form1 +M withM ∈A⊗kAug(B)⊗kR.
Lemma4.1.4. — There is a functorial (in R) equivalence
`(BGR(AA))'gR(A)[1]∈dgLieΩk.
Proof. — Denote byR+ =knR the unitalk-algebra obtained by formally adding a unit toR. Consider the functorF:dgArtk →sSets mapping B to the maximal
∞-groupoid in
PerfR+⊗kA⊗kB.
Deformations of the perfect moduleR⊗kAare then controlled by the functor Def(R⊗kA) :B7−→F(B)⊗F(k){R⊗A}.
It follows from [Lur11, Cor. 5.2.15 and Th. 3.3.1] that the deformations ofR⊗kAare controlled by the dg-Lie algebragR(A) = End(R⊗kA). Moreover, we have a natural transformationBGR(AA)→Def(R⊗kA)which induces an equivalence on the loop groups. It is therefore an`-equivalence and the result follows.
Corollary 4.1.5. — For any n ∈ N∪ {∞}, the (shifted) tangent Lie algebra of BGLn(AA) isgln(A)[1].
4.2. The generalized trace. — Let A: dgArtk → dgAlg60
Q be any functor.
It comes with a canonical natural transformationBGL∞(A)→Ω∞K(A)mapping a vector bundle to its class. We can now state the main result of this section:
Theorem 4.2.1. — Let A be a connective unital dg-algebra over k. We denote by AA: dgArtk → dgAlg6Q0 the functor B 7→ B⊗k A. The natural transformation BGL∞(AA)→Ω∞K(AA)induces, by taking the tangent Lie algebras, a morphism T:gl∞(A)[1]'`(BGL∞(AA))−→`(Ω∞K(AA))−→θ `Ab(K(AA))'θ(HCk•(A)[1]) indgLieΩk. The morphismT is homotopic to the generalized trace mapTr.
Proof. — Recall that the equivalence`Ab(K(AA))'HCk•(A)[1]is built through the relative Chern characterK(AA)→jQHCQ•(AA)[1]. In particular, the morphismT is induced by the natural transformation
chGL: BGL∞(AA)−→Ω∞K(AA)−→ν(HCQ•(AA)[1])
where ν ' Ω∞◦jQ:PFMPQk → PFMPk is the forgetful functor. Denote byQ[−]
the left adjoint toν(so that it computes pointwise the rational homology of the given simplicial set). The natural transformationchGL then factors as
chGL: BGL∞(AA)−→νQ[BGL∞(AA)]−→νQ[X(AA,AA)] νchQ
−−−−−→νHCQ•(AA)[1].
Recalling the construction ofchQand using Malcev’s theory forgl∞(AA), we get the commutative diagram
BGL∞(AA) νQ[BGL∞(AA)] νQ[X(AA,AA)] νHCQ•(AA)[1]
νCEQ•(gl∞(AA)) νcolimn,σCEQ•(tσn(AA,AA)) νHCQ•(AA)[1].
'
chQ
'
i Tr
By functoriality of the generalized trace map, we find that the composite map Tr◦i is homotopic to
νCEQ•(gl∞(AA)) Tr
−−−→HCQ•(AA)[1] ηHC
−−−−→HCQ•(AA)[1].
All in all, we get that the natural transformationchGLis homotopic to the composite BGL∞(AA)−→νCEQ•(gl∞(A))−→HCQ•(AA)[1] ηHC
−−−−→HCQ•(AA)[1].
Passing to the tangent morphism and using the Beck-Chevalley natural transforma-tion, we find
T:gl∞(A)[1]−→`νCEQ•(gl∞(AA))−→θ `QCEQ•(gl∞(AA))−→θ `QHCQ•(AA)[1]
−→∼ θ `QHCQ•(AA)[1]'θHCk•(A)[1].
Lemma4.2.2. — LetRbe a (possibly non-unital)k-algebra andAbe a (possibly non-unital) connective dg-algebra over k. The natural morphism
`QCEQ•(gR(AA))−→CEk•(gR(A)) induced by the lax monoidal structure oneQ is an equivalence.
Remark4.2.3. — We have gR(A⊗k−) '(R⊗kA)⊗k−. Up to replacing A with R⊗kA, we can assume that R =k. Note that gk =gl1 computes the underlying dg-Lie algebra of a given dg-algebra.
Proof. — By the above remark, we restrict ourselves to the caseR=k. The functor AA is pointed. In particular, the morphism `Q(A⊗AQp)→A⊗kp is an equivalence for anyp(see Proposition 2.3.9). Since`Q preserves colimits, we find
`Q(Symp
Q(A⊗pA [1]))'Sympk(A[1]).
The Chevalley-Eilenberg complex comes with a natural filtration whose graded parts are symmetric powers, and the result follows (since`Qpreserve colimits).
As a consequence, we get a commutative diagram, for anyA∈dgAlg6k0,nu
`Q(CEQ•(gl∞(AA))) `Q(HCQ•(AA)[1]) CEk•(gl∞(A)) HCk•(A)[1].
`Q(Tr)
' `Q(αHC)◦βHC
Tr
'
In particular, the proof of Theorem 4.2.1 now reduces to the following
Lemma4.2.4. — For any (possibly non-unital)k-algebraRand anyA∈dgAlg6k0,nu, the tangent morphism of the composite
BGR(AA)−→Q[BGR(AA)]'CEQ•(gR(AA))
is homotopic (as a morphism gR(A)[1] → θ `Q(CEQ•(gR(AA))) ' θCEk•(gR(A)) in dgLieΩk) to the unit of the adjunction CEΩ•(−) = CEk•(−[−1])aθ.
Proof. — By Remark 4.2.3, we may assume R =k. We are to study a morphism A[1]→θCEk•(A)in dgLieΩk. By adjunction, it suffices to identify the corresponding morphismCEk•(A)→CEk•(A)with the identity. Note that this is the linear tangent of the Malcev quasi-isomorphism Q[BGR(AA)]'CEQ•(gR(AA)). It is actually enough to study the functorial morphism
ξA: CEk•(A)−→CEk•(A) obtained by adjoining a unit on both sides.
This construction is functorial in A∈dgAlg6k0,nu and moreover the functorA7→
CEk•(A) preserves geometric realizations. Since every connective dg-algebra can be obtained as the geometric realization of a diagram of discrete algebras, the natural transformationξis determined by its values on discrete algebras. We thus restrict to discrete algebras.
Malcev’s quasi-isomorphism is compatible with the standard filtrations. It follows that so is ξ. The associated graded part of weightp of both the source and target ofξA vanishes forp60and is canonically isomorphic to Symp(A[1]) = (ΛpL)[p] for positivep’s. In particular, both source and target ofξAlive in the heart of Beilinson’s t-structure on filtered complexes (see the appendix of [Bei87]). We can thus work in the1-category of complexes.
With Malcev’s quasi-isomorphism being compatible with the coalgebras struc-tures, the natural transformation ξ also consists of morphisms of coalgebras. Since CEk•: Liek →coAlgfiltk from discrete Lie algebras to filtered coalgebras is fully faith-ful, it follows that ξ lifts to a self natural transformationζ:F ⇒F of the functor F: Algnuk →Liek mapping an algebra to its underlying Lie algebra.
By forgetting down to the category of small sets, we obtain a natural self-trans-formation (still denoted ζ) of the forgetful functor G: Algk → Sets. The functorG
is representable by the non-unital algebrak[t]>1 of polynomials P overk such that P(0) = 0. By the Yoneda lemma, the natural transformation ζ is determined by P ∈k[t]>1. The fact thatζisk-linear implies thatP is of degree1, and the fact that it preserves the Lie brackets implies thatP =atwitha2=a. In particular, the natural transformationζ(and therefore alsoξ) is either the identity or the0transformation.
Sinceξis the image by`Q of the Malcev equivalenceQ[BGk(A−)]'CEQ•(gk(A−)),
it certainly cannot vanish.
This concludes the proof of Theorem 4.2.1.
References
[Bei87] A. A. Beilinson– “On the derived category of perverse sheaves”, inK-theory, arithmetic and geometry (Moscow, 1984–1986), Lect. Notes in Math., vol. 1289, Springer, Berlin, 1987, p. 27–41.
[Bei14] , “Relative continuousK-theory and cyclic homology”,Münster J. Math.7(2014), no. 1, p. 51–81.
[BKP18] A. Blanc, L. Katzarkov&P. Pandit – “Generators in formal deformations of categories”, Compositio Math.154(2018), no. 10, p. 2055–2089.
[Blo73] S. Bloch– “On the tangent space to QuillenK-theory”, inAlgebraicK-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lect. Notes in Math., vol. 341, Springer, 1973, p. 205–210.
[Bur86] D. Burghelea– “Cyclic homology and the algebraicK-theory of spaces. I”, inApplications of algebraicK-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, American Mathematical Society, Providence, RI, 1986, p. 89–115.
[Cat91] J.-L. Cathelineau– “λ-structures in algebraicK-theory and cyclic homology”,K-Theory 4(1990/91), no. 6, p. 591–606.
[CHW09] G. Cortiñas, C. Haesemeyer&C. A. Weibel– “Infinitesimal cohomology and the Chern character to negative cyclic homology”,Math. Ann.344(2009), no. 4, p. 891–922.
[CW09] G. Cortiñas&C. A. Weibel– “Relative Chern characters for nilpotent ideals”, inAlgebraic topology, Abel Symp., vol. 4, Springer, Berlin, 2009, p. 61–82.
[DL14] G. Donadze&M. Ladra– “The excision theorems in Hochschild and cyclic homologies”, Proc. Roy. Soc. Edinburgh Sect. A144(2014), no. 2, p. 305–317.
[FHK19] G. Faonte, B. Hennion&M. Kapranov– “Higher Kac-Moody algebras and moduli spaces of G-bundles”,Adv. Math.346(2019), p. 389–466.
[FT87] B. L. Feigin&B. L. Tsygan– “AdditiveK-theory”, inK-theory, arithmetic and geometry (Moscow, 1984–1986), Lect. Notes in Math., vol. 1289, Springer, Berlin, 1987, p. 67–209.
[Goo86] T. G. Goodwillie– “Relative algebraicK-theory and cyclic homology”,Ann. of Math. (2) 124(1986), no. 2, p. 347–402.
[GG96] J. A. Guccione&J. J. Guccione – “The theorem of excision for Hochschild and cyclic homology”,J. Pure Appl. Algebra106(1996), no. 1, p. 57–60.
[Lod92] J.-L. Loday–Cyclic homology, Grundlehren Math. Wiss., vol. 301, Springer-Verlag, Berlin, 1992.
[LQ83] J.-L. Loday&D. Quillen – “Homologie cyclique et homologie de l’algèbre de Lie des matrices”,C. R. Acad. Sci. Paris Sér. I Math.296(1983), no. 6, p. 295–297.
[Lur09] J. Lurie–Higher topos theory, Annals of Math. Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009.
[Lur11] , “Derived algebraic geometry X: Formal moduli problems”, available at https:
//www.math.ias.edu/~lurie/papers/DAG-X.pdf, 2011.
[Lur16] , “Higher algebra”, available at https://www.math.ias.edu/~lurie/papers/HA.
pdf, 2016.
[Mal49] A. I. Mal’cev– “Nilpotent torsion-free groups”,Izv. Akad. Nauk SSSR Ser. Mat.13(1949), p. 201–212.
[Pri10] J. P. Pridham– “Unifying derived deformation theories”,Adv. Math.224 (2010), no. 3, p. 772–826, Corrigendum:Ibid.228(2011), no. 4, p. 2554–2556.
[Pri16] , “Smooth functions on algebraic K-theory”, 2016,arXiv:1602.06222.
[SS03] S. Schwede&B. Shipley– “Equivalences of monoidal model categories”,Algebraic Geom.
Topol.3(2003), p. 287–334.
[Sus81] A. A. Suslin – “On the equivalence of K-theories”, Comm. Algebra 9 (1981), no. 15, p. 1559–1566.
[SW92] A. A. Suslin&M. Wodzicki– “Excision in algebraicK-theory”,Ann. of Math. (2)136 (1992), no. 1, p. 51–122.
[Tam18] G. Tamme– “Excision in algebraic K-theory revisited”, Compositio Math. 154 (2018), no. 9, p. 1801–1814.
[TV08] B. Toën&G. Vezzosi – Homotopical algebraic geometry. II. Geometric stacks and ap-plications, Mem. Amer. Math. Soc., vol. 193, no. 902, American Mathematical Society, Providence, RI, 2008.
[Tsy83] B. L. Tsygan– “Homology of matrix Lie algebras over rings and the Hochschild homology”, Uspehi Mat. Nauk38(1983), no. 2(230), p. 217–218.
[Vol71] I. A. Volodin– “AlgebraicK-theory as an extraordinary homology theory on the category of associative rings with a unit”,Izv. Akad. Nauk SSSR Ser. Mat.35(1971), p. 844–873.
[Wal85] F. Waldhausen – “Algebraic K-theory of spaces”, in Algebraic and geometric topology (New Brunswick, N.J., 1983), Lect. Notes in Math., vol. 1126, Springer, Berlin, 1985, p. 318–419.
[Wei97] C. A. Weibel– “The Hodge filtration and cyclic homology”,K-Theory12(1997), no. 2, p. 145–164.
[Wod89] M. Wodzicki– “Excision in cyclic homology and in rational algebraicK-theory”,Ann. of Math. (2)129(1989), no. 3, p. 591–639.
Manuscript received 8th May 2020 accepted 16th March 2021
Benjamin Hennion, IMO - Université Paris-Saclay F-91405 Orsay Cedex, France
E-mail :benjamin.hennion@universite-paris-saclay.fr Url :https://www.imo.universite-paris-saclay.fr/~hennion/