THE TANGENT COMPLEX OF K-THEORY

. — We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 ﬁeld k , is the cyclic homology (over k ). This equivalence is compatible with λ -operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any ﬁeld k of characteristic 0 . We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map BGL ∞ → K . The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham. Résumé (Le complexe tangent de la K-théorie) . — Dans cet article, nous prouvons que le complexe tangent de la K-théorie, en termes de problèmes de déformations formels et sur un corps k de caractéristique nulle, n’est autre que l’homologie cyclique sur k . Cette équivalence est de plus compatible aux λ -opérations. Nous démontrons également que le morphisme tangent du morphisme canonique BGL ∞ → K est homotope au morphisme de trace généralisée de Loday-Quillen et Tsygan. La démonstration s’appuie sur des résultats de Goodwillie, à l’aide du théorème d’excision pour l’homologie cyclique de Wodzicki et de la théorie des déformations formelles à la Lurie-Pridham.


Introduction
Computing the tangent space of algebraic K-theory has been the subject of many articles.The first attempt known to the author is due to Spencer Bloch [Blo73] in 1973.It was then followed by a celebrated article of Goodwillie [Goo86] in 1986.
Considering the following example, we can easily forge an intuition on the matter.Let A be a smooth commutative Q-algebra and G(A) be the group of elementary matrices.It admits a universal central extension by the second K-theory group of A. The group G(A) + is the Steinberg group of A. This above exact sequence can also be extended as an exact sequence (1) This exact sequence can be thought as the universal extension of GL ∞ by K-theory (both K 2 and K 1 here).This idea leads to Quillen's definition of K-theory through the +-construction.
Consider now the tangent Lie algebra gl ∞ of GL ∞ : B → GL ∞ (B).Its current Lie algebra gl ∞ (A) := gl ∞ ⊗ Q A also admits a universal central extension, this time by the first cyclic homology group The obvious parallel between these two central extensions leads to the idea that the (suitably considered) tangent space (or rather complex) of K-theory should be cyclic homology.We will give a meaning to this folkloric statement, prove it and provide a comparison between these extensions (see below).
In both of the aforementioned articles of Bloch and Goodwillie, the tangent space is considered in a rather naive sense: as if K-theory were an algebraic group.Bloch defines the tangent space of K-theory at 0 in K(A) (for A a smooth commutative algebra over Q) as the fiber of the augmentation where ε squares to 0. Goodwillie then extends and completes the computation by showing that relative (rational) K-theory is isomorphic to relative cyclic homology, in the more general setting where A is a simplicial associative Q-algebra.He shows that for any nilpotent extension A of A, the homotopy fibers of K(A )⊗Q → K(A)⊗Q and of HC Q • (A ) → HC Q • (A) are quasi-isomorphic.In this article, we give another definition of the tangent space of K-theory using deformation theory, over any field k containing Q.We then show that this tangent space is equivalent to the absolute and k-linear cyclic homology.Before explaining exactly how this tangent space is defined, let us state the main result.In this introduction, we will restrict for simplicity to the connective case (1) (or equivalently to the (1) For the unbounded case, we essentially replace in what follows A ⊗ k B with its connective cover (A ⊗ k B) 0 .J.É.P. -M., 2021, tome 8 case of simplicial algebras).With our definition of tangent complex, for any unital simplicial k-algebra A, we have , where the right-hand-side denotes the (shifted) k-linear (absolute) cyclic homology of A. Of course, for this to hold for any field k, the left-hand side has to depend on k.This dependence occurs by only considering relative K-theory of nilpotent extensions A of A of the form where B is a (dg-)Artinian commutative k-algebra with residue field k.This defines a functor from the category of dg-Artinian commutative k-algebras with residue field k to the category of connective spectra.The category of such functors dgArt k → Sp 0 admits a full subcategory of formal deformation problems -i.e., of functors satisfying a Schlessinger condition (see Definition 2.1.1).The datum of such a functor is now equivalent to the datum of a complex of k-vector spaces.The induced fully faithful functor dgMod k → Fct(dgArt k , Sp 0 ) admits a left adjoint -denoted by Abthat forces the Schlessinger conditions.We define (2) the tangent complex of K-theory (of A) as T K(A),0 := Ab (K(A A )) and our main theorem now reads Theorem 1 (see Corollary 3.1.3).-Let A be any (H-)unital dg-algebra over k, with char(k) = 0.There is a natural equivalence . This equivalence is furthermore compatible with the λ-operations on each side.
We later extend this result to the case where A is replaced by a quasi-compact quasiseparated scheme.
As a consequence, the K-theory functor fully determines the cyclic homology functors over all fields of characteristic 0.Moreover, it gives its full meaning to the use of "Additive K-theory" as a name of cyclic homology (see [FT87]).
Using our theorem, we then prove (see Theorem 4.2.1) that the canonical natural transformation BGL ∞ → K (encoding the aforementioned universal central extension of GL ∞ ) induces a morphism gl ∞ (A) → HC k • (A) of homotopical Lie algebras (i.e., a L ∞ -morphism).Such a morphism corresponds to a morphism of complexes

CE k
• (gl ∞ (A)) −→ HC k •−1 (A) from the Chevalley-Eilenberg homological complex to cyclic homology.We will show that this morphism identifies with the Loday-Quillen-Tsygan generalized trace.In particular, it exhibits HC k •−1 (A) as the kernel of the universal central extension of gl ∞ (A).As a consequence, the extensions (1) and (2) above are indeed tangent to one another.
Structure of the proof.-The proof of Theorem 1 goes as follows.First, we show in Section 2.3 that the tangent complex T K(A),0 only depends on the (relative) rational K-theory functor K ∧ Q.Using the work of Goodwillie [Goo86], we know the relative rational K-theory functor is equivalent (through the Chern character), with the relative rational cyclic homology functor: , and where Q is the left adjoint of the fully faithful functor Since cyclic homology if well defined for non-unital algebras, we also have a functor HC Q •−1 (A A ) mapping an Artinian B to the (shifted) cyclic homology of the augmentation ideal A A (B) := A ⊗ k Aug(B).
We then argue that the functor Q is (non-unitally) symmetric monoidal (once restricted to a full subcategory, see Proposition 2.3.9).Since We will then prove (see Theorem 3.1.1)that the induced morphism We use here the assumption that A is unital (or at least H-unital).This excision statement is close to Wodzicki's excision theorem for cyclic homology [Wod89].The structure of our proof relies on a paper by Guccione and Guccione [GG96], where the authors give an alternative proof of Wodzicki's theorem.Nonetheless, our Theorem 3.1.1 is strictly speaking not a consequence of Wodzicki's theorem, and the proof is somewhat more subtle.
Composing these quasi-isomorphisms, we find the announced theorem Possible generalizations.-In this article, we work (mostly for simplicity) over a field k of characteristic 0. The results will also hold over any commutative Q-algebra, or, more generally, over any eventually coconnective simplicial Q-algebra.
(3) This can (3) so that the theory of formal moduli problems would still work flawlessly.surely also be done over more geometric bases like schemes or stacks (and bounded enough derived versions of such).
A more interesting generalization would be to (try to) work over the sphere spectrum.Since we only need in what follows 'abelian' formal moduli problems, it is not so clear that the characteristic 0 assumption is necessary.There would, however, be significant difficulties to be overcome, starting with a Wodzicki's excision theorem for topological cyclic homology.
Notations.-From now on, we fix the following notations -Let k be a field of characteristic 0.
-Let dgMod k denote the category of (cohomologically graded) complexes of k-vector spaces.Let dgMod 0 k be its full subcategory of connective objects (i.e., V • such that V n = 0 for n > 0).Let dgMod k and dgMod 0 k denote the ∞-categories obtained from the above by inverting the quasi-isomorphisms.
-Let dgAlg nu k be the category of (possibly non-unital) associative algebras in dgMod k (with its usual graded tensor product).We denote by dgAlg nu, 0 k its full subcategory of connected objects and by dgAlg nu, 0 k ⊂ dgAlg nu k the associated ∞-categories.
-For A ∈ dgAlg nu, 0 k , we denote by dgBiMod nu, 0 A the category of connective k-complexes with a left and a right action of A. We denote by dgBiMod nu, 0 A the associated ∞-category. (4) -Let cdga 0 k denote the category of connective commutative dg-algebras over k.We denote by cdga 0 k its ∞-category.-Let sSets be the ∞-category of spaces, Sp 0 the ∞-category of connective spectra, and Σ ∞ : sSets Sp 0 : Ω ∞ the adjunction between the infinite suspension and loop space functors.
Acknowledgements.-The question answered in this text naturally appeared while working on [FHK19] with G. Faonte and M. Kapranov.A discussion with G. Ginot and M. Zeinalian raised the problem dealt with in our last section.I thank them for the many discussions we had, that led to this question.I thank D. Calaque, P-G.Plamondon, J. Pridham and M. Robalo for useful discussions on the content of this article.
Finally, I thank J. Pridham for bringing [Pri16] to my attention when the first version of our work appeared online.Some arguments used in [Pri16] are fairly similar to those we use here.

Relative cyclic homology and K-theory
In this first section, we will introduce cyclic homology, K-theory and the relative Chern character between them.Most of the content has already appeared in the literature.The only original fragment is the extension of some of the statements and proofs to simplicial H-unital algebras.
1.1.Hochschild and cyclic homologies (A).Definitions.-Fix an associative dg-algebra A ∈ dgAlg nu k .Assuming (for a moment) that A is unital, its Hochschild homology is It comes with a natural action of the circle, and we define its cyclic homology HC k • (A) to be the (homotopy) coinvariants HH k • (A) hS 1 under this action.To define these homologies for a non-unital algebra A, we first formally add a unit to A and form • .These definitions turn out to agree with the former ones when A is already unital.
Unfortunately, we will need later down the road a construction of HC k • (A) for A non-unital that does not rely on the one for unital algebras.We will therefore work with the following explicit models.We will first define strict functors, and then invert the quasi-isomorphisms.
We fix A ∈ dgAlg nu k a (not necessarily unital) associative algebra in complexes over k.We also fix an A-bimodule M .Throughout this section, the tensor product ⊗ will always refer to the tensor product over k.
Definition 1.1.1.-We call the (augmented) Bar complex of A with coefficient in M and denote by B k • (A, M ) the ⊕-total complex of the bicomplex with M in degree 0 and with differential −b : where ε i = i + j<i |a j | (|a| standing for the degree of the homogeneous element a).
One easily checks that b squares to 0 and commutes with the internal differentials of A and M .We denote by Assuming A is unital and the right action of A on M is unital, we can build a nullhomotopy of B k • (A, M ) 0. This contractibility does not hold for general nonunital algebras and modules.
J.É.P. -M., 2021, tome 8 Definition 1.1.2(Wodzicki).-The dg-algebra A as above is called H-unital if Remark 1.1.3.-In the above definitions, we only used the right action of A on M .
In the original article of Wodzicki, such a module M would be called right H-unitary.
A similar notion exists for left modules.
Remark 1.1.4.-If A is H-unital, then the right module M ⊗ A is H-unitary, for any M .Indeed, we then have Definition 1.1.5.-We denote by H k • (A, M ) the ⊕-total complex of the bicomplex with M in degree 0 and with differential b : Here also, the differential squares to 0 and is compatible with the internal differentials of A and M .We denote by If A is (H-)unital, the complex H k • (A) is the usual Hochschild complex, that computes the Hochschild homology of A. For general A's, we need to compensate for the lack of contractibility of the Bar-complex with some extra-term.
Let t, N : A ⊗n+1 → A ⊗n+1 by the morphisms given on homogeneous elements by the formulas One easily checks that they define morphisms of complexes 1 − t : Moreover we have N (1 − t) = 0 and (1 − t)N = 0. We can therefore define Hochschild and cyclic homology as follows.
Definition 1.1.6.-We define the Hochschild homology HH k • (A) and the cyclic homology HC k • (A) of A as the ⊕-total complexes of the following bicomplexes The Connes exact sequence is the obvious fiber and cofiber sequence B induced by the above definitions.

B. Hennion
Remark 1.1.7.-The Hochschild homology of A is the homotopy cofiber of In particular, we have a fiber (and cofiber) sequence Moreover, under this assumption, using the reduced Bar complex as a resolution of A as a A ⊗ A o -dg-module, we easily show: Remark 1.1.8.-The normalization N can be seen as a morphism N : HH k which in turn is an action of (the k-valued homology of) the circle S 1 on HH k • (A).Choosing a suitable resolution of k as an (B).Relation to Chevalley-Eilenberg homology.-Cyclic homology is sometimes referred to as additive K-theory for the following reason: it is related to (the homology of) the Lie algebra gl ∞ (A) of finite matrices the same way K-theory is related to (the homology of) the group GL ∞ (A).More specifically, for A a dg-algebra, the generalized trace map is a morphism  and Tsygan [Tsy83] to prove (independently) the following statement for A a discrete algebra, and by Burghelea [Bur86] for general dg-algebras.
Theorem 1.1.9(Loday-Quillen, Tsygan, Burghelea).-When A is unital, the morphism Tr induces an equivalence of Hopf algebras ), where the product on the left-hand-side is given by the direct sum of matrices.
1.2.Filtrations, relative homologies and Wodzicki's excision theorem Definition 1.2.1.-Let f : A → B be a map of (possibly non-unital) connective (5)  dg-algebras.The relative Hochschild (resp.cyclic) homology of A over B is the homotopy fiber (5) For simplicity, we restrict ourselves to the connective case.The general case -unneeded for our purposes -would work similarly.
J.É.P. -M., 2021, tome 8 If I denotes the homotopy fiber of f endowed with its induced (non-unital) algebra structure, we have canonical morphisms

an extension of (possibly nonunital) connective dg-algebras. If I is H-unital, then the induced sequences
are fiber and cofiber sequences.In other words, the canonical morphisms η HH and η HC are equivalences.
Wodzicki proves in [Wod89] the above theorem in the case where A, B and I are concentrated in degree 0. Although his proof should be generalizable to connective dg-algebras, some computations seem to become tedious.Fortunately, Guccione and Guccione published in [GG96] another proof of this result, that is very easily generalizable to connective dg-algebras.There is also a more recent article of Donadze and Ladra [DL14] proving this result for simplicial algebras.
We will actually not need Theorem 1.2.2 in what follows.We will, however, need to reproduce some steps of its proof in a more complicated situation.In this subsection, we will give a short proof of Theorem 1.2.2,where we have isolated the statements to be used later.The proof closely follows the work of [GG96].
We fix f : A → B a degree-wise surjective morphism in dgAlg nu, 0 k .We denote by I the kernel of f .Let M ∈ dgBiMod nu, 0 ).These filtrations were originally found in [GG96].
Fix n ∈ N. Let p ∈ N. We set We denote by given as the total complex of the bicomplex In particular, we have J.É.P. -M., 2021, tome 8 Lemma 1.2.3.-The quotients Proof.-The first equivalence is an application of the above corollary.The equality follows from the trivial observation that the left action if I on N is trivial.The last equivalence is implied by the fact that I is H-unital, together with Remark 1.1.4.

(B). A filtration from B k
Lemma 1.2.6.-For any n ∈ N, we have: J.É.P. -M., 2021, tome 8 Proof.-We have The other terms, for 0 j n, cancel because of the vanishing of the composite Proof.-It follows from Lemma 1.2.6 and Corollary 1.2.5 that the filtration from -We start with an extension I → A → B of connective dg-algebras.Up to suitable replacements, we can assume that f : A → B is a degreewise surjective, and that I is its kernel.Consider the commutative diagram The functor H k • (A, −) preserves fiber sequences, and therefore the horizontal sequence is a fiber sequence.From Corollary 1.2.4 and Corollary 1.2.7, we deduce that α and β are quasi-isomorphisms.In particular the diagonal sequence is a fiber sequence.Similarly, we show that the sequence is a fiber sequence.The result then follows from the definitions of HH k • and HC k • .
(D).Invariance under quasi-isomorphisms.-Let us record for future use that all the constructions of the previous paragraphs are well-behaved with respect to quasiisomorphisms.
(1) Let A → A be a quasi-isomorphism in dgAlg nu, 0 k and let M 1 → M 2 be a quasi-isomorphism of connective A -bimodules.The induced morphisms In particular, we have quasi-isomorphisms (2) Let f : A → B and g : A → B be two fibrations in dgAlg nu, 0 k and let A → A and B → B be two quasi-isomorphisms commuting with f and g.Denote by I and I the kernels of f and g, respectively (so that the induced morphism I → I is a quasiisomorphism too).Then (a) The following induced morphisms are quasi-isomorphisms: Proof.-These are standard arguments: each of the complexes at hand is defined as the total space of a simplicial object.In particular, it comes with a canonical filtration whose graded parts are of one of the following forms (up to the obvious notational changes) Since k is a field, the induced morphisms between these graded parts are quasiisomorphisms and the result follows.
• descend to ∞-functors (that we will denote in the same way) between the appropriate ∞-categories localized along quasi-isomorphisms.
1.3.Relative cyclic homology and K-theory.-In this subsection, we recall the fundamental notions of K-theory and of the equivariant Chern character, at least in the relative setting.We consider the (connective) K-theory functor as an ∞-functor dgAlg 0 k → Sp 0 .Let (dgAlg 0 k ) ∆ 1 nil denote the ∞-category of morphisms A → B that are surjective with nilpotent kernel at the level of H 0 .Definition 1.3.1.-The relative K-theory functor is the ∞-functor

Since the map
-Using Bass' exact sequence, one can easily show (see for instance [Bei14, §2.8]) that relative connective K-theory and relative non-connective K-theory are equivalent: Remark 1.3.4.-Goodwillie's definition of relative K-theory and cyclic homology in [Goo86] differs from ours by a shift of 1.
The main result of [Goo86] states that the Chern character induces an equivalence ch : The construction of the relative Chern character from the absolute one is straightforward.We will however need a more hands-on construction in Section 4. The two constructions have been proven to coincide by Cortiñas and Weibel in [CW09] (see Remark 1.3.11below).
In order to construct our relative Chern character, we will need some explicit model for relative K-theory of connective dg-algebras over k (containing Q).For convenience, we will work with the equivalent model of simplicial k-algebras.We denote by sAlg k the category of simplicial (unital) algebras over k, endowed with its standard model structure.
For A ∈ sAlg k , we denote by M n (A) the simplicial set obtained by taking n × nmatrices level-wise.Finally, we define the group of invertible matrices as the pullback We have π 0 (GL n (A)) GL n (π 0 A) and π i (GL n (A)) M n (π i A) for i 1.In particular, this construction preserves homotopy equivalences.The simplicial set GL n (A) is a group-like simplicial monoid and we denote by BGL n (A) its classifying space.Finally, we denote by BGL ∞ (A) the colimit colim n BGL n (A).
Applying Quillen's plus construction to BGL ∞ (A) yields a model for K-theory, so that there is an equivalence Based on this equivalence, we will build in the next paragraph a model for relative K-theory using relative Volodin spaces.
(B).Relative Volodin construction.-The Volodin model for the K-theory of rings first appeared in [Vol71] in the absolute case.The relative version seems to originate from an unpublished work of Ogle and Weibel.It can also be found in [Lod92].We will need a version of that construction for simplicial algebras over k.
We fix a fibration f : A → B in sAlg k such that the induced morphism π 0 (A) → π 0 (B) is surjective.In particular, the morphism f is level-wise surjective.We denote by I its kernel and we assume that 1 and let σ be a partial order on the set {1, . . ., n}.We denote by T σ n (A, I) the sub simplicial set of M n (A) given in dimension p by the subset of M n (A p ) consisting of matrices of the form 1 + (a ij ) with a ij ∈ I p if i is not lower than j for the order σ.
As a simplicial set, T σ n (A, I) is isomorphic to a simplicial set of the form A α × I β with α and β are integers depending on σ, such that α + β = n 2 .In particular, this construction is homotopy invariant.Moreover, the map T σ n (A, I) → M n (A) factors through GL n (A).Actually, T σ n (A, I) is a simplicial subgroup of the simplicial monoid GL n (A).The group π 1 (X(A, I)) contains as a maximal perfect subgroup the group E(π 0 (A)) and we apply the plus construction to this pair.
In particular, we have a (functorial) equivalence Proof.-This is a classical argument, that can be found in [Lod92, §11.3] for rings.We simply extend it to simplicial algebras.We start by drawing the following commutative diagram (2) (3) J.É.P. -M., 2021, tome 8 The space X(A, 0) is acyclic (see [Sus81] for the discrete case, the simplicial case being deduced by colimits).It follows (by the properties of the plus construction), that row (2) is a fibration sequence.Obviously, so are row (3) and columns (a) and (c).It now suffices to show that column (b) is a fibration sequence.Consider the commutative diagram

(τ ) p
The square (τ ) is homotopy Cartesian.Indeed, the morphism p is induced by a pointwise surjective fibration of group-like simplicial monoids (recall that we assumed π 0 (I) to be nilpotent).It is therefore a fibration.We can now see that the diagram is cartesian on the nose by looking at its simplices.The bottom row is a fibration sequence.It follows that the top row (which coincides with column (b)) is also a fibration sequence.
As a consequence, row (1) induces a fibration sequence in homology.Since X(A, 0) is acyclic, the homologies of X(A, I) and of Ω ∞ K(f ) are isomorphic.It follows that X(A, I) + and Ω ∞ K(f ) are homotopy equivalent.
(C).Malcev's theory.-In order to construct our relative Chern character it is now enough to relate the homology of the relative Volodin spaces X(A, I) with cyclic homology.This step uses Malcev's theory, that relates homology of nilpotent uniquely divisible groups (such as T σ n (A, I) for A discrete) with the homology of an associated nilpotent Lie algebra.The original reference is [Mal49].
For simplicity, we will only work with Lie algebras of matrices.We assume for now that A is a discrete unital Q-algebra.Let n ⊂ gl n (A) be a nilpotent sub-Lie algebra (for some n).Denote by N the subgroup N := exp(n) ⊂ GL n (A).We have the following proposition (for a proof, we refer to [SW92, Th. 5.11]).
Proposition 1.3.8(Malcev, Suslin-Wodzicki).-There is a quasi-isomorphism where BN denotes the classifying space of N .Moreover, this quasiisomorphism is compatible with the standard filtrations on these complexes.
The construction of this quasi-isomorphism is based on two statements.First, the completion of the algebras U (n) and Q[N ] along their augmentation ideals are isomorphic and, second, the standard resolutions of Q as a trivial module on these algebras are compatible.It follows that this quasi-isomorphism is actually compatible with the standard filtrations on both sides.
Fix I ⊂ A a nilpotent ideal.For n ∈ N and σ a partial order on {1, . . ., n}, we denote by t σ n (A, I) ⊂ gl n (A) the (nilpotent) Lie algebra of matrices (a ij ) such that a ij ∈ I if i is not smaller than j for the partial order σ.We have then by definition T σ n (A, I) = exp(t σ n (A, I)) and therefore a quasi-isomorphism τ : Return now to the general case of a morphism of simplicial rings f : A → B such that π 0 A → π 0 B is surjective with nilpotent kernel.First, we replace A so that A → F is a filtration (i.e., is levelwise surjective) and ker(A 0 → B 0 ) is nilpotent.Using the monoidal Dold-Kan correspondence (see [SS03]), we can further assume A n (resp.B n ) to be a nilpotent extension of A 0 (resp.B 0 ).All in all, we have replaced f with an equivalent morphism, which is levelwise surjective and such that the kernel preserve geometric realizations and we therefore get a (functorial) quasi-isomorphism ) by applying τ level-wise.

(D). The relative Chern character. -The usual construction (used among others by Goodwillie) is based on the absolute Chern character
We will only need the induced relative (and Q-linear) version: Definition 1.3.9.-We will denote by ch the (Q-linear) relative Chern character Theorem 1.3.10(Goodwillie [Goo86]).-The morphism ch is a quasi-isomorphism.
In order to compare a tangent map to the generalized trace in Section 4, we will need another description of the relative Chern character.Using the natural inclusion t σ n (A, I) → gl ∞ (A) and the generalized trace map, we get a morphism Taking the colimit on n and σ, we find a version of the relative Chern character . This version a priori does not descend to a morphism of spectra.However, Cortiñas and Weibel proved that ch induces a morphism that is functorially homotopic to our ch Q (see [CW09]).
Remark 1.3.11.-Cortiñas and Weibel only consider the case of discrete algebras.If f : A → B is a surjective morphism of simplicial algebras, we can reduce to the discrete case.Indeed, writing f as a geometric realization of discrete f n : A n → B n and setting The relative Chern characters evaluated on I ⊂ A → B are then (both) determined by their value on discrete algebras, and therefore coincide using Cortiñas and Weibel's result.
J.É.P. -M., 2021, tome 8 (E).The case of H-unital algebras.-In the proof of our main result, we will use a Goodwillie theorem for H-unital algebras.To extend Theorem 1.3.10 to this context, we will need the following result of Suslin and Wodzicki [SW92] (see [Tam18] for its generalization to simplicial Q-algebras) Theorem 1.3.12(Suslin-Wodzicki, Tamme).-If I → A → B is an extension of (possibly non-unital) simplicial Q-algebras and I is H-unital, then the induced sequence is a fiber sequence of connective spectra.In particular, if either A → B admits a section or I is nilpotent, it is also a cofiber sequence.
We fix a nilpotent extension f : A B of H-unital simplicial k-algebras.We get a commutative diagram whose rows and columns are fiber and cofiber sequences where the functor k − formally adds a (k-linear) unit. (6)We get a natural equivalence Similarly, using Theorem 1.2.2, we have . Theorem 1.3.10thus leads to the following Corollary 1.3.13.-There is a functorial equivalence , in the more general case of f : A → B a nilpotent extension of H-unital simplicial Q-algebras (that coincides with the relative Chern character when both A and B are unital).

Formal deformation problems
Infinitesimal deformations of algebraic objects can be encoded by a tangential structure on the moduli space classifying these objects.In [Pri10] and [Lur11], Pridham and Lurie established an equivalence between so-called formal moduli problems and differential graded Lie algebras.This section starts by recalling Pridham and Lurie's works.We then establish some basic facts about abelian or linear formal moduli problems.k /k denote the full subcategory of augmented connective k-cdga's spanned by Artinian (7) ones.Let C be an ∞-category with finite limits.
We say that a functor F : Definition 2.1.2.-A pre-FMP (pre-formal moduli problem) is a functor dgArt k → sSets.We denote by PFMP k their category.A FMP (formal moduli problem) is a pre-FMP F satisfying the condition (S).We denote by FMP k the category of formal moduli problems, and by i : FMP k → PFMP k the inclusion functor.
Example 2.1.3.-Let X be an Artin (and possibly derived) stack and x ∈ X be a k-point.The functor B → {x} × X(k) X(B) defined on Artinian dg-algebras satisfies the Schlessinger condition (see [TV08] for instance).It thus defines a formal moduli problem.
The category PFMP k is presentable, and the full subcategory FMP k is strongly reflexive.In particular, the inclusion i : FMP k → PFMP k admits a left adjoint.Definition 2.1.4.-We denote by L : PFMP k → FMP k the left adjoint to the inclusion i.We call it the formalization functor.
Definition 2.1.5.-A shifted dg-Lie algebra over k is a complex V together with a dg-Lie algebra structure on V [−1].We denote by dgLie Ω k the ∞-category of shifted dg-Lie algebras.
Remark 2.1.6.-The notation Ω is here to remind us the shift in the Lie structure.Note that shifting a complex V by −1 amounts to computing its (pointed) loop space ΩV Theorem 2.1.7(Pridham, Lurie).-The functor T : FMP k → dgMod k (computing the tangent complex) factors through the forgetful functor dgLie Ω k → dgMod k .In other words, the tangent complex TF of a formal moduli problem admits a natural shifted Lie structure.Moreover, the functor F → TF induces an equivalence (7) Recall that A is called Artinian if its cohomology is finite dimensional over k, and if H 0 (A) is a local ring.
J.É.P. -M., 2021, tome 8 Example 2.1.9.-Let X be a smooth scheme and x ∈ X a k-point.Example 2.1.3yields an associated formal moduli problem whose tangent complex is T X,x , the tangent space of X at x.The shifted Lie structure is the trivial.More interestingly, if G is a smooth group scheme, then its classifying stack BG is an Artin stack.The tangent complex to the associated formal moduli problem is , the shift of the Lie algebra of G.The shifted Lie algebra structure on g[1] is the usual Lie algebra structure on g.This will be discussed in more detail in Section 4 below.
Let us recall briefly how the equivalence in Theorem 2.1.7 is constructed.Consider the Chevalley-Eilenberg cohomological functor It admits a left adjoint, denoted by D k , so that for any L and B, we have The equivalence e :

Proving this construction indeed defines an equivalence is based on the following key lemma
Lemma 2.1.10(see [Lur11,2.3.5]).-For B ∈ dgArt k , the adjunction morphism ) In what follows, we will need slightly more general versions of formal moduli problems, namely formal moduli problems with values in an ∞-category such as complexes of vectors spaces or connective spectra.
Definition 2.1.11.-Let C be an ∞-category with all finite limits.A C -valued pre-FMP is a functor dgArt k → C .A C -valued formal moduli problem (or FMP) is a C -valued pre-FMP satisfying the condition (S).
We conclude this section by recording a functoriality statement, whose proof is straightforward and left to the reader.Lemma 2.1.14.-Let f : C D : g be an adjunction between presentable ∞-categories.
(1) Composing with g induces a functor g * : PFMP D k → PFMP C k that maps formal moduli problems to formal moduli problems.
(2) The induced functor g * : k is the functor given by composing with f . ( 2.2.Abelian moduli problems.-The moduli problem of concern in this article is constructed from the (connective) K-theory functor.In particular, its values are endowed with an abelian group structure, or equivalently are connective spectra.We shall therefore establish a couple of basic properties of these abelian formal moduli problems.
Forgetting the abelian group structure and the free abelian group functor form an adjunction We also have a similar adjunction J.É.P. -M., 2021, tome 8 Note the categories PFMP Ab k and FMP Ab k identify with PFMP C k (resp.FMP C k ), when C = Sp 0 is the category of connective spectra (see [Lur16, Rem.5.2.6.26]).In particular, the above adjunction is given by applying pointwise the functors Σ ∞ : sSets Sp 0 : Ω ∞ .Finally, we denote by i Ab : FMP Ab k → PFMP Ab k the inclusion functor, and by L Ab its left adjoint.
It is in general not an equivalence.In particular, an abelian pre-FMPs, such as the K-theory functor, will have two different associated formal moduli problems, and two different tangent Lie algebras (one of which will automatically be abelian, see below).
(B).Abelian dg-Lie algebras.-It follows from Theorem 2.1.7 that the category FMP Ab k is equivalent to the category dgLie Ω,Ab k of abelian group objects in dgLie Ω k .We shall call objects of dgLie Ω,Ab k abelian dg-Lie algebras.Although the following statement is well-known to the community, we could not locate a proof in the literature.We therefore provide one. of the proposition.-For any pointed category, we denote by Ω its pointed loop space endofunctor.Denote by FMP n k the category of FMP in (n − 1)-connective spaces.In particular, we have FMP 0 k = FMP k .The inclusion of (n − 1)-connective spaces into all spaces induces a fully faithful functor FMP n k → PFMP k .We denote by u n the composite Lemma 2.2.4.-The functor u n is an equivalence.
Proof.-For any ∞-category C with finite products, denote by Mon gp En (C ) the ∞-category of group-like E n -monoids in C (see [Lur16, Def.5.2.6.6]).By [Lur16, Th. 5.2.6.10], the n-th loop space defines a (pointwise) equivalence En (FMP k ).It follows from [BKP18, Prop.2.15] that taking the n-th loop space also defines an equivalence FMP k → Mon gp En (FMP k ).Its inverse is homotopic to the composite functor L •B n , where B n is the n-fold delooping functor.The composition is then homotopic to u n , and is an equivalence.
J.É.P. -M., 2021, tome 8 We continue our proof of Proposition 2.2.2.The forgetful functor f : dgLie Ω k → dgMod k commutes with limits (and thus with Ω).We get a commutative diagram, where the leftmost column is obtained by taking the limit of the rows the category dgMod k is stable, the projection on the rightmost component is an equivalence C 4 dgMod k .The category C 1 identifies with the category of formal moduli problems in the limit category Remains to prove that f ∞ is an equivalence.Since f is conservative, so is f ∞ .The functor f ∞ is the limit of functors with left adjoints, and therefore it admits a left adjoint g ∞ .Denote by g the left adjoint of f .Up to a shift, the functor g identifies with the free Lie algebra functor: The full subcategory of dgMod k spanned by V 's such that φ V is an equivalence is stable under filtered colimits.We may thus assume V to be perfect, concentrated in (cohomological) degrees lower than some integer m.Fix i ∈ Z.For n > m + i, the cohomology group In particular, the map φ V is a quasi-isomorphism.
We denote by CE Ω • : dgLie Ω k → dgMod k the functor mapping a shifted dg-Lie algebra L to the reduced Chevalley-Eilenberg complex of its shift CE • (L[−1]).Lemma 2.2.7.-The functor θ : dgMod k → dgLie Ω k identifies with the functor mapping a complex to itself with the trivial bracket.As a consequence, CE Ω • is left adjoint to θ.
Proof.-Denote by h(V ) the (shifted) dg-Lie algebra with trivial bracket built on V ∈ dgMod k .Since g is given as a free (shifted) dg-Lie algebra, there is a canonical morphism g(V ) → h(V ) given by collapsing the free brackets.By construction, the functor θ is given by the formula In particular, we find a functorial morphism in dgLie We have already seen in the proof of Proposition 2.2.2 that the image by f of this morphism is an equivalence.The first result follows by conservativity of f .For the second statement, we simply observe that CE Ω • is the left derived functor of the abelianization functor L → L/ [L, L] , which is left adjoint to h.Proposition 2.2.9.-The functor e Ab is equivalent to the functor mapping V ∈ dgMod k to the abelian formal moduli problem where Aug computes the augmentation ideal of a given Artinian cdga and (−) 0 truncates the given complex (and considers it as a connective spectrum through the Dold-Kan equivalence).

B. Hennion
Proof.-The equivalence dgLie Ω k FMP k is constructed by identifying dgArt k with a full subcategory of dgLie Ω k of so-called good (shifted) dg-Lie algebras, that generates dgLie Ω k in a certain way.This identification is given by the Chevalley-Eilenberg functor CE Ω • .In particular, given V ∈ dgMod k , and B ∈ dgArt k , we have (D).Diagrammatic summary.-It follows from Proposition 2.2.2 that an abelian formal moduli problem is determined by its tangent complex (without any additional structure).We denote by Ab the composite functor Ab := T Ab • L Ab .We get the commutative diagrams of right adjoints and of left adjoints We denote by j Q the forgetful functor and by e Q the functor dgMod k → FMP Q k given by the formula As a consequence, we have the following factorization of the adjunction L Ab i Ab : Proof.-The functor j Q is given by post-composing with the limit preserving functor dgMod 0 Q → Sp 0 .In particular, it preserves the condition (S).Recall that Q is idempotent in spectra: Q ∧ Q Q.It follows that j Q is fully faithful.This proves Assertion (a).
Proposition 2.2.9 implies Assertion (b).All is left is Assertion (c), which follows from Assertion (a) and Assertion (b).
, where F is pointed (i.e., satisfies the condition (S1)) and F (k) is the constant functor.Since the inclusion i Q (resp.i Ab ) factors through the category of pointed functors, its left adjoint L Q (resp.L Ab ) can be decomposed into two functors.The first associates F to F , while the second forces (S2).In particular, we have resp.L Ab (F ) L Ab (F ) and Ab (F ) Ab (F ) .
(B).Generators.-In this paragraph, we will identify families of generators of the category PFMP Q k .
Definition 2.3.5.-For B ∈ dgArt k , we denote by the functor k is generated under colimits by the free Q-linear presheaves generated by these Map dgArt k (B, −), i.e., by the S Q (B)'s.The second statement follows.
We now compute explicitly the formal moduli problem associated to such a generator.
Proof.-From Remark 2.3.4,we have The result then follows from Lemma 2.2.8 in conjunction with the factorization from Proposition 2.3.2.
(C).Monoidality.-We will now consider monoidal structures on the adjunction We first observe that both sides admit a natural tensor structure: ⊗ k on the RHS, and the pointwise application of ⊗ Q on the LHS.Let I be the functor I : dgArt k → dgMod Q mapping an Artinian B to its augmentation ideal Aug(B).It is by construction an ideal in the commutative algebra object I : B → B and therefore inherits a non-unital commutative algebra structure.
In particular, the functor e Q : V → τ 0 V ⊗ k I is non-unitally lax symmetric monoidal.
As a direct consequence of this lemma, we get that the functor Q is non-unitally colax symmetric monoidal.
Proposition 2.3.9.-The functor Q is non-unitally symmetric monoidal once restricted to the full subcategory of pointed functors.
Proof.-We are to prove that for any pair is an equivalence.Fixing F , we denote by D F ⊂ PFMP Q k the full subcategory spanned by the G's such that γ F,G is an equivalence.Since the tensor products ⊗ k and ⊗ Q preserve colimits in each variable, and since Q preserves colimits, the category D F is stable under colimits.Using Lemma 2.3.6,we can therefore reduce the question to the case where G (and by symmetry, also F ) is of the form S Q (B).Let B 1 and B 2 be Artinian cdga's over k, and assume that F = S Q (B 1 ) and G = S Q (B 2 ).
J.É.P. -M., 2021, tome 8 The reduced Künneth formula provides a (functorial) equivalence Applying Q on that equivalence, we find using Lemma 2.3.7 Since B 1 and B 2 are perfect as k-dg-modules, the LHS identifies with The map γ S Q (B1),S Q (B2) is thus a retract of the equivalence and is therefore itself an equivalence.
Proof.-Since the involved functors preserves all colimits, we can reduce to the generating case V = C, which is trivial.
where G is pointed and G(k) is a constant functor.We can therefore assume that G is either pointed or constant.The first case follows from Proposition 2.3.9, while the second case follows from Lemma 2.3.10 (for C = Q).

Excision
In this section, we fix A an algebra object in PFMP Q k .We also denote by A the associated pointed functor Note that A inherits a non-unital algebra structure.Finally, we denote by A the (non-unital) algebra Q (A ) Q (A ) in dgMod k .Remark 3.1.2.-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.
Remark 3.1.5.-The equivalence of Corollary 3.1.3is 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 3.1.3is also compatible with the λ-operations.
Proof.-The augmented Bar complex B Q • (A , M ) identifies as the homotopy cofiber of the augmentation B Q (A , M ) → M , where B Q (−, −) 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 Taking the homotopy cofiber of the augmentation on both sides, we find the first claimed equivalence.Similarly, the functor H Q • (A , M ) is again the homotopy colimit of a standard semi-simplicial diagram.For later use, we will work in a slightly bigger generality.Let R be a (possibly non-unital) discrete k-algebra.We denote by g R the functor is representable by the non-unital algebra k[t] 1 of polynomials P over k such that P (0) = 0.By the Yoneda lemma, the natural transformation ζ is determined by P ∈ k[t] 1 .The fact that ζ is k-linear implies that P is of degree 1, and the fact that it preserves the Lie brackets implies that P = at with a 2 = a.In particular, the natural transformation ζ (and therefore also ξ) is either the identity or the 0 transformation.Since ξ is the image by Q of the Malcev equivalence This concludes the proof of Theorem 4.2.1.
Proposition 2.2.2.-The forgetful functor dgLie Ω,Ab k −→ dgMod k is an equivalence.In particular FMP Ab k dgLie Ω,Ab k dgMod k .Definition 2.2.3.-We denote by T Ab : FMP Ab k ∼ −→ dgMod k the equivalence of the above proposition.We denote by e Ab : dgMod k ∼ −→ FMP Ab k its inverse.
and therefore C 3 dgLie Ω,Ab k .Moreover, the equivalence T ∞ is homotopic to the equivalence FMP Ab k dgLie Ω,Ab k induced directly from T : FMP k dgLie Ω k by taking abelian group objects on both sides.The projections on the rightmost component C 1 → FMP 0 k = FMP k and C 3 → dgLie Ω k identify with the functors forgetting the abelian group structure, while the functor f ∞ : dgLie Ω,Ab k C 3 → C 4 dgMod k is the forgetful functor.

(
C). Description of the equivalence FMP Ab k dgMod k .-We will give a more explicit description of the equivalence of Proposition 2.2.2.The definition implies that T Ab : FMP Ab k ∼ −→ dgMod k simply computes the tangent complex (at the only k-point).Let us also describe its inverse e Ab .Lemma 2.2.8.-Let B ∈ dgArt k and X = Map(B, −) ∈ FMP k .We haveT F Ab (X) Aug(B) ∨ ,where Aug computes the augmentation ideal of a given Artinian, and (−) ∨ computes the k-linear dual.Proof.-Since B is Artinian, we deduce from Lemma 2.1.10that B is canonically equivalent to the Chevalley-Eilenberg cohomology of it (shifted) tangent Lie algebra TX.It follows that the tangent Lie algebra of F Ab X is the dg-module CE Ω • (TX) Aug(B)∨ .
Q-linear moduli problems (A).Definitions Definition 2.3.1.-A Q-linear (pre-)FMP is a (pre-)FMP with values in the ∞-category C = dgMod 0 Q .We shorten the notations by setting Ab denotes the inverse of the equivalence T : FMP Ab k → dgMod k .Proposition 2.3.2.-The following assertions hold.(a) The functor j Q : PFMP Q k → PFMP Ab k preserves formal moduli problems and is fully faithful.(b) The functor e Ab factors as Lemma 2.3.8.-The functor e Q is non-unitally lax symmetric monoidal.Proof.-We consider the constant moduli problem functor dgMod k → PFMP Q k mapping V to the constant functor V .It is right adjoint to the symmetric monoidal functor F → F (k) ⊗ Q k and therefore inherits a lax monoidal structure.
Remark 3.1.6(Goodwillie derivative).-Denote by f the functor Perf 0 k → dgArt k mapping a connective perfect k-complex M to the split square zero extension k ⊕ M .Restricting along f defines a functor from PFMP Sp 0 k to the category of functors J.É.P. -M., 2021, tome 8F : Perf 0 k → Sp 0 .Such functors F satisfying some Schlessinger-like condition form a category equivalent to that of k-complexes.We find a commutative diagramFct(Perf 0 k , Sp 0 ) Ab φwhere φ (as well as e Ab ) is fully faithful.Now, the left adjoints Ab and (say) ψ of e Ab and φ respectively, do not commute with − • f .The functor ψ can actually be interpreted in terms of Goodwillie derivative.For instance, it can be proved to mapK(A k ) • f to k ∈ dgMod k .Note that k[1], seen as a k ⊗ k-module, corepresents the Goodwillie derivative of the K-theory of k:∂ K HH • (k, −)[1] : dgMod k⊗k −→ Sp.More generally, one can proveψ(K(A C ) • f ) HH k • (C)[1].Moreover, the Beck-Chevalley transformation ψ • (− • f ) → Ab is identified with the usual morphism HH k • (C)[1] → HC k • (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 putative circle action) may give us a less computational (and more conceptual) proof of Corollary 3.1.3,based directly on Goodwillie calculus.Lemma 3.1.7.-Let M ∈ PFMP Q k be an A -bimodule.Assume that M (k) 0. We set M := Q (M ) as an A-bimodule.The canonical morphisms

g
R : dgAlg 0,nu k −→ dgLie 0 k mapping a connective dg-algebra C to the Lie algebra underlying the associative algebra C ⊗ k R. Examples include the case where R is the algebra of n × n-matrices with coefficients in k, where we find g R = gl n .Obviously, the construction R → g R is functorial.For an algebra R, we also denote by BG R (A A ) ∈ PFMP k the functorBG R (A A ) : B −→ B exp g R A A (B) = B exp g R (A ⊗ k Aug(B)) ,where exp(gR (A ⊗ k Aug(B))) is the subgroup of (A⊗ k B⊗ k R) × consisting of elements of the form 1 + M with M ∈ A ⊗ k Aug(B) ⊗ k R. Lemma 4.1.4.-There is a functorial (in R) equivalence (BG R (A A )) g R (A)[1] ∈ dgLie Ω k .Proof.-Denote by R + = k R the unital k-algebra obtained by formally adding a unit to R. Consider the functor F :dgArt k → sSets mapping B to the maximal ∞-groupoid in Perf R + ⊗ k A⊗ k B .Deformations of the perfect module R ⊗ k A are then controlled by the functorDef(R ⊗ k A) : B −→ F (B) ⊗ F (k) {R ⊗ A}.It follows from [Lur11, Cor.5.2.15 and Th.3.3.1]that the deformations of R ⊗ k A are controlled by the dg-Lie algebra g R (A) = End(R ⊗ k A).Moreover, we have a natural transformation BG R (A A ) → Def(R ⊗ k A) 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 ofBGL n (A A ) is gl n (A)[1].4.2.The generalized trace.-Let A : dgArt k → dgAlg 0 Q be any functor.It comes with a canonical natural transformation BGL ∞ (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 byA A : dgArt k → dgAlg 0 Q the functor B → B ⊗ k A. The natural transformation BGL ∞ (A A ) → Ω ∞ K(A A ) induces,by taking the tangent Lie algebras, a morphismT : gl ∞ (A)[1] (BGL ∞ (A A )) −→ (Ω ∞ K(A A )) −→ θ Ab (K(A A )) θ(HC k • (A)[1]) in dgLie Ω k .The morphism T is homotopic to the generalized trace map Tr.J.É.P. -M., 2021, tome 8 category, then PFMP C k and FMP C k are presentable categories, and i C admits a left adjoint.In the above situation, we will denote by L C the left adjoint to i C .
computes the (reduced) rational homology of a given simplicial set.Note that PFMP Q k is equivalent to the ∞-category of Q-linear objects in PFMP k .As a category of presheaves, PFMP k is generated under colimits by the representable functors Map dgArt k (B, −) (for B ∈ dgArt k ).It follows that Using the notations of Definition 2.1.11forC = Sp the category of spectra and the proof of Proposition 2.2.2, we get that FMP Sp k ( C 3 ) is equivalent to dgMod k .We get an adjunction Sp .It follows from Remark 1.3.2 and our main theorem that for any algebra object A in PFMP Sp : PFMP Sp k FMP Sp k dgMod k : e