MapdgArt
k(B1,−)×MapdgArt
k(B2,−),Q
'SQ(B1)⊗QSQ(B2)⊕SQ(B1)⊕SQ(B2).
Applying`Q on that equivalence, we find using Lemma 2.3.7 Aug B1⊗kB2∨
'`Q SQ(B1)⊗QSQ(B2)
⊕Aug(B1)∨⊕Aug(B2)∨. SinceB1 andB2are perfect ask-dg-modules, the LHS identifies with
Aug(B1)∨⊗kAug(B2)∨⊕Aug(B1)∨⊕Aug(B2)∨. The mapγSQ
(B1),SQ(B2) is thus a retract of the equivalence
`Q(SQ(B1⊗kB2))'Aug(B1⊗kB2)∨
and is therefore itself an equivalence.
Lemma 2.3.10. — Let C ∈ cdga6Q0 and V ∈ dgMod6C0. Let F: dgArtk → dgMod6C0 be a pre-FMP. Tensoring pointwise byV defines a new pre-FMPF⊗CV. The canonical morphism
`Q(F⊗CV)−→(`QF)⊗CV is an equivalence indgModk.
Proof. — Since the involved functors preserves all colimits, we can reduce to the
generating caseV =C, which is trivial.
Corollary2.3.11. — Let F, G ∈PFMPQk. IfF is pointed (i.e., F(k)'0)) and if
`Q(F)'0, then`Q(F⊗QG)'0.
Proof. — We split G into the direct sum G⊕G(k) where G is pointed and G(k) is a constant functor. We can therefore assume that G is either pointed or con-stant. The first case follows from Proposition 2.3.9, while the second case follows
from Lemma 2.3.10 (forC=Q).
3. Excision
In this section, we fixA an algebra object in PFMPQk. We also denote byA the associated pointed functor
A:B 7−→hofib(A(B)−→A(k)).
Note that A inherits a non-unital algebra structure. Finally, we denote by A the (non-unital) algebra`Q(A)'`Q(A)in dgModk.
3.1. Main theorem
Theorem3.1.1. — Consider the canonical morphisms HHQ•(A) αHH
−−−−→HHQ•(A) `Q(HHQ•(A)) βHH
−−−−→HHk•(A),
HCQ•(A) αHC
−−−−→HCQ•(A) `Q(HCQ•(A)) βHC
−−−−→HCk•(A).
The following holds:
(a) The morphismsβHH◦`Q(αHH)andβHC◦`Q(αHC)are equivalences.
(b) If A is H-unital, then the morphisms `Q(αHH) and `Q(αHC) are equivalences (and therefore so are βHH andβHC).
Remark3.1.2. — A direct consequence of Assertion (b) is that the tangent complex of Hochschild or cyclic homology of a given functor A does not depend on A(k).
This is an excision statement similar to Theorem 1.2.2.
Corollary3.1.3. — Let C∈dgAlgnuk be H-unital. Denote byAC the functorB 7→
(C⊗kB)60. There is a functorial equivalence
`Ab(K(AC))'HCk•(C)[1].
Proof. — We haveAC = eQ(C)and therefore`Q(AC)'`Q(AC)'C. We find
`Ab(K(AC))'`Ab(K(AC)) by Remark 2.3.4 '`Q(K(AC)∧Q) by Proposition 2.3.2 '`Q(HCQ•(AC)[1]) by Corollary 1.3.13 '`Q(HCQ•(AC))[1] by Remark 2.3.4
'HCk•(C)[1] by Theorem 3.1.1.
Remark3.1.4. — Using the notations of Definition 2.1.11 forC =Spthe category of spectra and the proof of Proposition 2.2.2, we get thatFMPSpk ('C3) is equivalent todgModk. We get an adjunction
`Sp:PFMPSpk FMPSpk 'dgModk : eSp.
It follows from Remark 1.3.2 and our main theorem that for any algebra objectA in PFMPQk such thatA:=`Q(A)is H-unital, we have
`Sp(Knc(A))'`Ab(K(A))'HCk•(A)[1].
Remark3.1.5. — The equivalence of Corollary 3.1.3 is defined through the relative Chern character. We know from [Cat91] and [CHW09] that the Chern character is compatible with the λ-operations on both sides. It follows that the equivalence of Corollary 3.1.3 is also compatible with theλ-operations.
Remark3.1.6(Goodwillie derivative). — Denote byfthe functorPerf6k0→dgArtk mapping a connective perfectk-complexM to the split square zero extensionk⊕M. Restricting along f defines a functor from PFMPSpk >0 to the category of functors
F: Perf6k0→Sp>0. Such functorsF satisfying some Schlessinger-like condition form a category equivalent to that ofk-complexes. We find a commutative diagram
Fct(Perf6k0,Sp>0) PFMPAbk
dgModk
− ◦f
eAb
φ
where φ (as well as eAb) is fully faithful. Now, the left adjoints `Ab and (say) ψ ofeAb andφrespectively, do not commute with− ◦f. The functorψcan actually be interpreted in terms of Goodwillie derivative. For instance, it can be proved to map K(Ak)◦f tok∈dgModk. Note thatk[1], seen as ak⊗k-module, corepresents the Goodwillie derivative of theK-theory ofk:
∂K'HH•(k,−)[1] :dgModk⊗k−→Sp.
More generally, one can prove ψ(K(AC)◦ f) ' HHk•(C)[1]. Moreover, the Beck-Chevalley transformation ψ◦(− ◦f) → `Ab is identified with the usual morphism HHk•(C)[1]→HCk•(C)[1].
We will not use the above remark in what follows, so we will not provide a proof.
Note however that understanding the relationship between ψ and `Ab (and a puta-tive circle action) may give us a less computational (and more conceptual) proof of Corollary 3.1.3, based directly on Goodwillie calculus.
Lemma3.1.7. — LetM ∈PFMPQk be anA-bimodule. Assume thatM(k)'0. We setM :=`Q(M)as an A-bimodule. The canonical morphisms
βαB: `Q BQ•(A,M)
−→B•k(A, M) βαH: `Q H•Q(A,M)
−→H•k(A, M) are equivalences.
Proof. — The augmented Bar complexBQ•(A,M)identifies as the homotopy cofiber of the augmentation BQ(A,M) → M, where BQ(−,−) denotes the reduced Bar complex. The latter is obtained as a homotopy colimit of the semi-simplicial Bar construction. Since`Q preserves colimits, we find using Proposition 2.3.9
`Q BQ(A,M)
'`Q colim[n]∈∆M ⊗QA⊗Qn
'colim[n]∈∆M⊗kA⊗kn 'Bk(A, M).
Taking the homotopy cofiber of the augmentation on both sides, we find the first claimed equivalence. Similarly, the functorH•Q(A,M)is again the homotopy colimit
of a standard semi-simplicial diagram.
of Theorem3.1.1(a). — The functor HHQ• is the homotopy cofiber of the natural transformation 1−t:B•Q → H•Q. In particular, the morphism βHH◦`Q(αHH) is
an equivalence because of Lemma 3.1.7 (forM =A) and the fact that`Q preserves cofiber sequences. In the case ofHC, we use Remark 1.1.8 and Lemma 2.3.10:
`Q HCQ•(A)
'`Q HHQ•(A)⊗Q[ε]Q
'`Q HHQ•(A)
⊗Q[ε]Q
'HHk•(A)⊗Q[ε]Q'HCk•(A),
whereQ[ε] := H•(S1,Q)∈cdga6Q0.
3.2. Proof of Assertion(b). — The proof of Assertion (b) in Theorem 3.1.1 is more evolved and relies on the ideas behind Wodzicki’s Theorem 1.2.2. We first reduce the study of Hochschild and cyclic homology to that of the complexes H and B from Section 1.1. We will use the following terminology:
Definition3.2.1. — A morphismf: F→G∈PFMPQk is an`Q-equivalence if`Q(f) is an equivalence.
Denote byαH andαBthe canonical morphisms H•Q(A) αH
−−−−→H•Q(A), B•Q(A) αB
−−−−→BQ•(A).
As in the previous section, we can easily reduce the proof of Theorem 3.1.1 (b) to proving that bothαH andαB are`Q-equivalences.
Lemma3.2.2. — If both αH andαB are`Q-equivalences, then so areαHH andαHC. We will now focus onαH andαBusing techniques from Section 1.2.
Proposition3.2.3. — The canonical morphisms
αB:BQ•(A)−→B•Q(A) and αH :H•Q(A)−→H•Q(A) are`Q-equivalences.
Lemma3.2.4. — Let M ∈PFMPQk be anA-bimodule. Assume thatM(k)'0 and that M :=`Q(M)is H-unital as an A=`Q(A)-bimodule, then
γH: H•Q(A,M)−→H•Q(A,M) and γB:B•Q(A,M)−→BQ•(A,M) are`Q-equivalences.
Proof. — We focus onγH, the case ofγBbeing identical. Denote byf the canonical natural transformation A → A(k) (where on the RHS is the constant functor).
We getA 'hofib(f). Recall from Section 1.2 (A) the filtration H•Q(A,M)'FH0 Q
•(f,M)−→ · · · −→FHnQ
•(f,M)−→ · · · with colimnFHn Q
•(f,M)'H•Q(A,M).
Applying`Q, we find
`Q H•Q(A,M)
'`Q FH0 Q
•(f,M)
−→ · · · −→`Q FHn Q
•(f,M)
−→ · · ·
Since`Q is a left adjoint and thus preserves colimits, we have colimn`Q FHn•Q(f,M)
'`Q colimnFHn•Q(f,M)
'`Q H•Q(A,M) .
It therefore suffices to prove that for any n > 0, the morphism FHnQ
• dgModk, the codomain of`Q, is a stable∞-category, it is enough to check that
Fn+1
is contractible. Using Lemma 1.2.3, we get FHn+1Q
Lemma3.2.5. — If A is H-unital, the four canonical morphisms H•Q(A,A(k)) δH images by`Q thus vanish, by Remark 2.3.4. It follows that their projections to0 are
`Q-equivalences. We now focus on the maps δH andδB. Letf:A →A(k)be the augmentation. Recall from Section 1.2 Section (B) the filtrations by quotients:
BQ•(A,A(k))'QB0Q
Since`Q preserves colimits, it suffices to prove that the transition morphismsQn → Qn+1 are`Q-equivalences. Denote byM(n)theA-bimoduleA(k)⊗Qn+1⊗QA and byM(n)its image by`Q. We get from Lemma 1.2.6 two fiber and cofiber sequences
B•Q(A,M(n))[n+ 1]−→QnBQ
Their image by`Qare still cofiber sequences, and it is now enough to prove that both BQ•(A,M(n))and H•Q(A,M(n))are canceled by`Q. We first observe thatM(n) is pointed:M(n)(k)'0. Moreover, we have by Lemma 2.3.10
M(n) =`Q(M(n))'A(k)⊗Qn+1⊗Q`Q(A) =A(k)⊗Qn+1⊗QA.
Remark 1.1.4 implies that M(n) is H-unitary over A. Using Lemma 3.2.4, we are reduced to the study ofB•Q(A,M(n))andH•Q(A,M(n)). By Lemma 3.1.7, we get
`Q BQ•(A,M(n))
'B•k(A, M(n))'0 and, since the left action ofAonM(n)is trivial:
`Q H•Q(A,M(n))
'H•k(A, M(n))'B•k(A, M(n))'0.
Proof of Proposition3.2.3. — We first observe thatαB andαH factor as B•Q(A) γB
−−−→BQ•(A,A) ηB
−−−→B•Q(A) H•Q(A) γH
−−−−→H•Q(A,A) ηH
−−−−→H•Q(A).
Since A is H-unital and A is pointed, Lemma 3.2.4 implies that γB and γH are
`Q-equivalences. The homotopy cofibers ofηBandηH are respectivelyB•Q(A,A(k)) and H•Q(A,A(k)). They are canceled by `Q because of Lemma 3.2.5. It follows thatηBand ηH are`Q equivalences, and so are αB andαH.
This concludes the proof of Proposition 3.2.3 and therefore the proof of
Theo-rem 3.1.1.
3.3. Non-connective K-theory and the case of schemes. — We now extend our main result (Corollary 3.1.3) to the case of quasi-compact quasi-separated (and pos-sibly derived) schemes.
Proposition3.3.1. — LetXbe a quasi-compact quasi-separated (and possibly derived) scheme over k. Denote by KncX: dgArtk → Sp the functor mapping an Artinian dg-algebraB to the non-connectiveK-theory spectrum of the derived schemeX⊗kB= X×Spec(B). Using the notations of Remark3.1.4: the Chern character induces an equivalence
`Sp(KncX)'HCk•(X)[1].
Proof. — We write X ' colimiSpec(Ai) as a finite colimit of Zariski open affine subschemes. By Zariski descent forK-theory, we haveKncX 'limiKnc(AAi). Since`Sp is an exact functor between stable ∞-categories, it preserves finite limits. We find, using descent for cyclic homology and Corollary 3.1.3:
`Sp(KX)'`Sp(lim Knc(AAi))'lim`SpKnc(AAi)'lim HCk•(Ai)[1]'HCk•(X)[1].
Example 3.3.2(Relation to the Picard stack). — Let us fix a compact quasi-separated (derived) scheme X and consider the (abelian) pre-FMPPicZX:
PicZX:B7−→ {0} ×
PicZ(X)
PicZ(X⊗B),
where PicZ is the graded Picard functor. It follows from Example 2.1.3 that PicZX satisfies the condition (S) and is therefore a formal moduli problem.
Recall that the determinant defines a functorial morphism of spectra fromK-theory to the graded Picard group det : Knc → PicZ. In particular, we get a morphism of
abelian pre-FMP’s detX: KncX → PicZX. Taking the tangent complex yields a mor-phism
trX: HCk•(X)[1]'`Sp(KncX)−→`Ab(PicZX)'T(PicZX)'RΓ(X,OX)[1].
The last equivalence is provided by Example 2.1.9 and the classical computation of the tangent of the Picard stack ofX at the trivial bundle.
The canonical morphism BGm → K similarly induces a section sX of trX. This identifiesRΓ(X,OX) as a summand ofHCk•(X), which corresponds to the weight 0 part in the Hodge decomposition of cyclic homology (see [Wei97]).