Formal deformation problems

'

where the functorkn−formally adds a (k-linear) unit.⁽⁶⁾We get a natural equivalence K(f)∧Q'K(knf)∧Q.

Similarly, using Theorem 1.2.2, we haveHC^Q_•(f)[1]'HC^Q_•(knf)[1]. Theorem 1.3.10 thus leads to the following

Corollary1.3.13. — There is a functorial equivalence ch : K(f)∧Q−→^∼ HC^Q_•(f)[1],

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 andB are unital).

2. Formal deformation problems

Infinitesimal deformations of algebraic objects can be encoded by a tangential struc-ture 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.

(6)Sokn−is the left adjoint to the forgetful functorsAlg_k→sAlg^nu_k

2.1. Formal moduli problems and dg-Lie algebras

Definition2.1.1. — Let dgArt_k ⊂ cdga⁶_k⁰/k denote the full subcategory of aug-mented connective k-cdga’s spanned by Artinian⁽⁷⁾ ones. Let C be an ∞-category with finite limits.

We say that a functorF:dgArt_k→C satisfies the condition (S) if

(S)







(S1) The objectF(k)is final inC.

(S2) For any mapA→B∈dgArt_k that is surjective onH⁰, the induced morphismF(k×BA)→F(k)×_F(B)F(A)is an equivalence.

Definition2.1.2. — A pre-FMP (pre-formal moduli problem) is a functordgArt_k→ sSets. We denote byPFMP_k their category. A FMP (formal moduli problem) is a pre-FMPF satisfying the condition (S). We denote byFMP_kthe category of formal moduli problems, and byi :FMPk→PFMPk the inclusion functor.

Example2.1.3. — Let X be an Artin (and possibly derived) stack and x∈X be a k-point. The functor B 7→ {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 categoryPFMP_k is presentable, and the full subcategoryFMP_k is strongly reflexive. In particular, the inclusioni :FMPk→PFMPk admits a left adjoint.

Definition 2.1.4. — We denote by L : PFMPk → FMPk the left adjoint to the inclusioni. We call it the formalization functor.

Definition2.1.5. — A shifted dg-Lie algebra overk is a complexV together with a dg-Lie algebra structure on V[−1]. We denote bydgLie^Ω_k the∞-category of shifted dg-Lie algebras.

Remark2.1.6. — The notationΩis here to remind us the shift in the Lie structure.

Note that shifting a complexV by−1amounts to computing its (pointed) loop space ΩV 'V[−1].

Theorem2.1.7(Pridham, Lurie). — The functorT:FMPk→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 functorF 7→TF induces an equivalence

FMPk'dgLie^Ω_k.

Definition2.1.8. — We denote by ` the composite functor`:=T◦L :PFMPk → dgLie^Ω_k, and bye : dgLie^Ω_k −→^∼ FMP_k the inverse functorT⁻¹.

(7)Recall thatAis called Artinian if its cohomology is finite dimensional overk, and ifH⁰(A)is a local ring.

Example2.1.9. — LetX be a smooth scheme and x∈X a k-point. Example 2.1.3 yields an associated formal moduli problem whose tangent complex isTX,x, the tan-gent space ofX at x. The shifted Lie structure is the trivial.

More interestingly, if Gis a smooth group scheme, then its classifying stack BG is an Artin stack. The tangent complex to the associated formal moduli problem is g[1] = TG,1[1], the shift of the Lie algebra ofG. The shifted Lie algebra structure on g[1]is the usual Lie algebra structure ong. 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

CE^•_k:dgLie^op_k −→cdga_k k.

It admits a left adjoint, denoted byDk, so that for anyL andB, we have Map_dgLie

k(L,Dk(B))'Map_cdga^aug

k (B,CE^•_k(L)).

The equivalencee :dgLie^Ω_k −→^∼ FMPk is then given onV ∈dgLie^Ω_k by the formula e(V)(B) := Map_dgLie

k(Dk(B), V[−1]).

Proving this construction indeed defines an equivalence is based on the following key lemma

Lemma2.1.10(see [Lur11, 2.3.5]). — ForB∈dgArt_k, the adjunction morphism B −→CE^•_k(Dk(B))

is an equivalence.

In what follows, we will need slightly more general versions of formal moduli prob-lems, namely formal moduli problems with values in an∞-category such as complexes of vectors spaces or connective spectra.

Definition2.1.11. — LetC be an∞-category with all finite limits. AC-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 denote byPFMP^C_k the category of C-valued pre-FMPs, by FMP^C_k the cat-egory of C-valued FMPs. We also denote by i_C: FMP^C_k →PFMP^C_k the inclusion functor.

Lemma2.1.12. — If C is a presentable ∞-category, thenPFMP^C_k andFMP^C_k are presentable categories, andi_C admits a left adjoint.

Definition2.1.13. — In the above situation, we will denote byL^C the left adjoint toi_C.

Proof of Lemma2.1.12. — The presentability ofPFMP^C_k is [Lur09, 5.5.3.6]. Remains to prove thatFMP^C_k ⊂PFMP^C_k is a strongly reflexive category (see [Lur09, p. 482]).

Fix a cartesian square(σ)in dgArt_k

C A

k B

(σ) f

where f is surjective on H⁰. We denote by D(σ) ⊂ PFMP^C_k the full subcategory spanned by functorsF mapping(σ)to a pullback square. SinceFMP^C_k =T

(σ)D(σ)

and because of [Lur09, 5.5.4.18], it suffices to prove thatD_(σ) is strongly reflexive in PFMP^C_k.

Restriction along (σ) defines a functor σ^∗: PFMP^C_k → Fct(K,C) where K =

∆¹×∆¹ is the square. The functor σ^∗ admits a left adjoint σ_! (namely the left Kan extension functor, see [Lur09, 4.3.3.7]). A functorF belongs toD_(σ)if and only if σ^∗(F) is a pullback square. By [Lur09, 5.5.4.19], the full subcategory Fct(K,C) spanned by pullback squares is strongly reflexive. It thus follows from [Lur09, 5.5.4.17]

thatD_(σ)is strongly reflexive inPFMP^C_k.

We conclude this section by recording a functoriality statement, whose proof is straightforward and left to the reader.

Lemma2.1.14. — Let f: C D:g be an adjunction between presentable ∞-cate-gories.

(1) Composing withg induces a functor g_∗:PFMP^D_k →PFMP^C_k that maps for-mal moduli problems to forfor-mal moduli problems.

(2) The induced functor g∗: FMP^D_k → FMP^C_k admits a left adjoint f! given by the composition f_! = L^D◦f_∗◦i_C, where f_∗: PFMP^C_k → PFMP^D_k is the functor given by composing with f.

(3) We havef!◦L^C 'L^D◦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.

(A). Abelian formal moduli problems. — Denote by FMP^Ab_k (resp.PFMP^Ab_k ) the category of abelian group objects (i.e., group-likeE_∞-monoids) in (pre-)formal moduli problems. We call their objects abelian (pre-)formal moduli problems.

Forgetting the abelian group structure and the free abelian group functor form an adjunction

F^Ab:FMPkFMP^Ab_k : GAb. We also have a similar adjunction

Σ^∞: PFMP_kPFMP^Ab_k : Ω^∞.

Note the categoriesPFMP^Ab_k andFMP^Ab_k identify withPFMP^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 byL^Ab its left adjoint.

Remark 2.2.1. — We have a Beck-Chevalley transformation L◦Ω^∞ → G_Ab◦L^Ab. 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 categorydgLie^Ω,Ab_k of abelian group objects indgLie^Ω_k. We shall call objects ofdgLie^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.

Proposition2.2.2. — The forgetful functor

dgLie^Ω,Ab_k −→dgMod_k

is an equivalence. In particularFMP^Ab_k 'dgLie^Ω,Ab_k 'dgMod_k.

Definition2.2.3. — We denote by T^Ab:FMP^Ab_k −→^∼ dgMod_k the equivalence of the above proposition. We denote by eAb:dgMod_k−→^∼ FMP^Ab_k its inverse.

of the proposition. — For any pointed category, we denote byΩits pointed loop space endofunctor. Denote byFMP^>_kⁿ the category of FMP in(n−1)-connective spaces.

In particular, we haveFMP^>_k⁰=FMP_k. The inclusion of(n−1)-connective spaces into all spaces induces a fully faithful functorFMP^>_kⁿ→PFMPk. We denote byun

the composite

u_n:FMP^>_kⁿ −→PFMP_k L

−−→FMP_k.

Lemma2.2.4. — The functor un is an equivalence.

Proof. — For any ∞-category C with finite products, denote by Mon^gp_E

n(C) the

∞-category of group-like En-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 Ωⁿ:FMP^>_kⁿ → Mon^gp_E

n(FMPk).

It follows from [BKP18, Prop. 2.15] that taking the n-th loop space also defines an equivalenceFMP_k→Mon^gp_E

n(FMP_k). Its inverse is homotopic to the composite functorL◦Bⁿ, whereBⁿ is then-fold delooping functor. The composition

FMP^>_kⁿ Ωⁿ

−−−→Mon^gp_E

n(FMP_k) L◦Bⁿ

−−−−−−→FMP_k

is then homotopic toun, and is an equivalence.

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

C1:= limFMP^>_kⁿ · · · FMP^>_k² FMP^>_k¹ FMP^>_k⁰

Since the category dgMod_k is stable, the projection on the rightmost component is an equivalenceC4'dgMod_k. The categoryC1identifies with the category of formal moduli problems in the limit category

lim(· · · −→sSets_>1−→sSets_>0)'Sp_>0. By [Lur16, Rem. 5.2.6.26],C1 is equivalent toFMP^Ab_k , and therefore

C3'dgLie^Ω,Ab_k .

Moreover, the equivalenceT∞is homotopic to the equivalenceFMP^Ab_k 'dgLie^Ω,Ab_k induced directly fromT: FMP_k 'dgLie^Ω_k by taking abelian group objects on both sides. The projections on the rightmost component C1 → FMP^>_k⁰ = FMP_k and C3→dgLie^Ω_k identify with the functors forgetting the abelian group structure, while the functorf_∞: dgLie^Ω,Ab_k 'C3→C4'dgMod_k is the forgetful functor.

Remains to prove thatf_∞is an equivalence. Sincef is conservative, so isf_∞. The functor f_∞ is the limit of functors with left adjoints, and therefore it admits a left adjointg_∞. Denote by g the left adjoint of f. Up to a shift, the functor g identifies with the free Lie algebra functor:

g(V)'FreeLie(V[−1])[1].

For a fixedV ∈dgMod_k'C4, the adjunction unitV →f_∞◦g_∞(V)identifies with the canonical map

φV:V −→colimnΩⁿ(g(V[n])).

The full subcategory of dgMod_k spanned byV’s such that φV 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 groupHⁱ(Ωⁿ(g(V[n])))is independent ofnand isomorphic toHⁱ(V).

In particular, the mapφ_V is a quasi-isomorphism.

Definition2.2.5. — We denote byCE^Ω_•: dgLie^Ω_k →dgMod_k the functor mapping a shifted dg-Lie algebraLto the Chevalley-Eilenberg complex of its shiftCE•(L[−1]).

We denote by CE^Ω_•: dgLie^Ω_k → dgMod_k the functor mapping a shifted dg-Lie algebraL to the reduced Chevalley-Eilenberg complex of its shiftCE•(L[−1]).

Definition2.2.6. — We denote byθ:dgMod_k'dgLie^Ω,Ab_k →dgLie^Ω_k the functor forgetting the abelian group structure.

Lemma2.2.7. — The functorθ:dgMod_k→dgLie^Ω_k identifies with the functor map-ping a complex to itself with the trivial bracket. As a consequence,CE^Ω_• is left adjoint toθ.

Proof. — Denote by h(V) the (shifted) dg-Lie algebra with trivial bracket built on V ∈dgMod_k. Sinceg 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 functorθis given by the formula

θ(V)'colimnΩⁿ⁻¹(g(V[n−1]))∈dgLie^Ω_k. In particular, we find a functorial morphism indgLie^Ω_k

θ(V)'colimnΩⁿ⁻¹(g(V[n−1]))−→colimnΩⁿ⁻¹(h(V[n−1]))'h(V).

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 off.

For the second statement, we simply observe thatCE^Ω_• is the left derived functor of the abelianization functorL7→L/[L, L], which is left adjoint toh.

(C). Description of the equivalenceFMP^Ab_k 'dgMod_k. — We will give a more expli-cit 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 inverseeAb.

Lemma2.2.8. — Let B∈dgArt_k andX = Map(B,−)∈FMPk. We have TF^Ab(X)'Aug(B)^∨,

whereAug computes the augmentation ideal of a given Artinian, and(−)^∨ computes the k-linear dual.

Proof. — Since B is Artinian, we deduce from Lemma 2.1.10 thatB is canonically equivalent to the Chevalley-Eilenberg cohomology of it (shifted) tangent Lie algebra TX. It follows that the tangent Lie algebra ofF^AbX is the dg-moduleCE^Ω_•(TX)'

Aug(B)^∨.

Proposition 2.2.9. — The functor e_Ab is equivalent to the functor mapping V ∈ dgMod_k to the abelian formal moduli problem

B7−→ Aug(B)⊗_kV60

,

where Aug computes the augmentation ideal of a given Artinian cdga and (−)⁶⁰ truncates the given complex (and considers it as a connective spectrum through the Dold-Kan equivalence).

Proof. — The equivalencedgLie^Ω_k 'FMPk 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 (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`<sup>Ab</sup>the composite functor`^Ab :=T^Ab◦L^Ab. We get the commutative diagrams of right adjoints and of left adjoints

dgLie^Ω_k FMPk PFMPk

Definition2.3.1. — AQ-linear (pre-)FMP is a (pre-)FMP with values in the ∞-cate-goryC =dgMod⁶⁰

Q . We shorten the notations by setting

FMP^Q_k :=FMP^C_k, PFMP^Q_k :=PFMP^C_k, i_Q:= i_C and L^Q:= L^C. We denote byj_Q the forgetful functorPFMP^Q_k →PFMP^Ab_k and bye_Q the functor dgMod_k→FMP^Q_k given by the formula

e_Q(V) :B7−→ Aug(B)⊗kV60

∈dgMod⁶_Q⁰.

Recall thate_Ab denotes the inverse of the equivalenceT:FMP^Ab_k →dgMod_k. Proposition2.3.2. — The following assertions hold.

(a) The functorj_Q:PFMP^Q_k →PFMP^Ab_k preserves formal moduli problems and is fully faithful.

(b) The functor e_Ab factors as

e_Ab'j_Q◦e_Q:dgMod_k e_Q

−−−→FMP^Q_k j_Q

−−−→FMP^Ab_k .

(c) The functorse_Q:dgMod_k→FMP^Q_k andj_Q:FMP^Q_k →FMP^Ab_k are equiva-lences.

As a consequence, we have the following factorization of the adjunctionL^Ab aiAb:

dgMod_k'FMP^Ab_k 'FMP^Q_k PFMP^Q_k PFMP^Ab_k .

i_Ab i_Q L^Q

j_Q

− ∧Q L^Ab

Proof. — The functorj_Qis given by post-composing with the limit preserving functor dgMod⁶_Q⁰ → 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).

Definition2.3.3. — We denote byT^Q:FMP^Q_k →dgMod_kthe inverse ofe_Q. We de-note by`^Q the compositeT^Q◦L^Q.

Remark 2.3.4. — Since k is both initial and final in dgArt_k, any F ∈ PFMP^Q_k (resp. in PFMP^Ab_k ) splits as F ' F⊕F(k), where F is pointed (i.e., satisfies the condition (S1)) and F(k) is the constant functor. Since the inclusioni_Q (resp.iAb) factors through the category of pointed functors, its left adjointL^Q (resp.L^Ab) can be decomposed into two functors. The first associatesF toF, while the second forces (S2). In particular, we have

L^Q(F)'L^Q(F) and `<sup>Q</sup>(F)'`^Q(F) resp. L^Ab(F)'L^Ab(F) and `<sup>Ab</sup>(F)'`^Ab(F)

.

(B). Generators. — In this paragraph, we will identify families of generators of the categoryPFMP^Q_k.

Definition2.3.5. — ForB ∈dgArt_k, we denote byS^Q(B)∈PFMP^Q_k (resp.S^Q(B)) the functor

S^Q(B) := C• Map_dgArt

k(B,−),Q

resp. S^Q(B) := C• Map_dgArt

k(B,−),Q ,

whereC•(−,Q)(resp.C•(−,Q)) computes the (reduced) rational homology of a given simplicial set.

Lemma2.3.6. — The category PFMP^Q_k is generated under colimits by functors of the form S^Q(B). The full subcategory ofPFMP^Q_k spanned by pointed functors (i.e., satisfying the condition (S1)) is generated under colimits by functors of the form S^Q(B).

Proof. — 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

PFMP^Q_k is generated under colimits by the free Q-linear presheaves generated by theseMap_dgArt

k(B,−), i.e., by theS^Q(B)’s. The second statement follows.

We now compute explicitly the formal moduli problem associated to such a gener-ator.

Lemma2.3.7. — Let B∈dgArt_k. There are functorial equivalences

`<sup>Q</sup>(S<sup>Q</sup>(B))'`^Q(S^Q(B))'Aug(B)^∨, where(−)^∨ computes thek-linear dual andAugthe augmentation.

Proof. — From Remark 2.3.4, we have`<sup>Q</sup>(S<sup>Q</sup>(B))'`^Q(S^Q(B)). The result then fol-lows 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

`^Q:PFMP^Q_k FMP^Q_k 'dgMod_k : e_Q.

We first observe that both sides admit a natural tensor structure: ⊗k on the RHS, and the pointwise application of⊗_Q on the LHS.

Lemma2.3.8. — 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 functorV. It is right adjoint to the symmetric monoidal functorF 7→F(k)⊗_Qkand therefore inherits a lax monoidal structure.

Let I be the functorI : dgArt_k → dgMod_Q mapping an Artinian B to its aug-mentation ideal Aug(B). It is by construction an ideal in the commutative algebra objectI :B7→B and therefore inherits a non-unital commutative algebra structure.

In particular, the functor e_Q: V 7→ τ⁶⁰ V ⊗_kI

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 pairF, G∈PFMP^Q_k of pointed functors (i.e., F(k)'G(k)'0), the natural morphism

γF,G: `^Q F⊗_QG

−→`<sup>Q</sup>(F)⊗k`^Q(G)

is an equivalence. FixingF, we denote byDF ⊂PFMP^Q_kthe full subcategory spanned by the G’s such that γF,G is an equivalence. Since the tensor products ⊗k and ⊗_Q preserve colimits in each variable, and since `^Q preserves colimits, the categoryDF

is stable under colimits. Using Lemma 2.3.6, we can therefore reduce the question to the case whereG(and by symmetry, alsoF) is of the formS^Q(B). LetB1andB2be Artinian cdga’s overk, and assume thatF = S^Q(B1)andG= S^Q(B2).

The reduced Künneth formula provides a (functorial) equivalence S^Q B1⊗kB2

'C•

Map_dgArt

k(B1,−)×Map_dgArt

k(B2,−),Q

'S^Q(B1)⊗_QS^Q(B2)⊕S^Q(B1)⊕S^Q(B2).

Applying`^Q on that equivalence, we find using Lemma 2.3.7 Aug B1⊗kB2^∨

'`^Q S^Q(B1)⊗_QS^Q(B2)

⊕Aug(B1)^∨⊕Aug(B2)^∨. SinceB1 andB2are perfect ask-dg-modules, the LHS identifies with

Aug(B1)^∨⊗kAug(B2)^∨⊕Aug(B1)^∨⊕Aug(B2)^∨. The mapγ_SQ

(B₁),S^Q(B₂) is thus a retract of the equivalence

`^Q(S^Q(B1⊗kB2))'Aug(B1⊗kB2)^∨

and is therefore itself an equivalence.

Lemma 2.3.10. — Let C ∈ cdga⁶_Q⁰ and V ∈ dgMod⁶_C⁰. Let F: dgArt_k → dgMod⁶_C⁰ be a pre-FMP. Tensoring pointwise byV defines a new pre-FMPF⊗CV. The canonical morphism

`<sup>Q</sup>(F⊗CV)−→(`^QF)⊗CV is an equivalence indgMod_k.

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 ∈PFMP^Q_k. IfF is pointed (i.e., F(k)'0)) and if

`<sup>Q</sup>(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).

Dans le document Benjamin Hennion The tangent complex of K-theory Tome 8 (2021), p. 895-932. < (Page 18-28)