Comparison of Commutative and Associative Deformation Theories

B⁰(n)_∗

A^<α_∗ //A^≤α_∗ , hence a pushout diagram diagram of formalE1moduli problems

Spec(k⊕k[nα+ 1]) //

Spec(k)

X^<α //X^≤α.

It follows that the mapX(i)→X(i+1) satisfies the criterion of Lemma 1.5.9. SinceX(i) is prorepresentable, we conclude thatX(i+ 1) is prorepresentable.

3.3 Comparison of Commutative and Associative Deformation Theories

Letkbe a field of characteristic zero and let X : Alg^sm_k →S be a formalE1-moduli problem. The forgetful functor CAlg_k →Alg_k carries smallE∞-algebras overk to smallE1-algebras overk, and therefore induces a forgetful functorθ : CAlg^sm_k →Alg^sm_k . The composite functor (X ◦θ) : CAlg^sm_k →S is a formal moduli problem overk. Consequently, composition withθdetermines a functorφ: Moduli⁽¹⁾_k →Moduli_k. Theorems 3.0.4 and 2.0.2 supply equivalences of∞-categories

Alg^aug_k 'Moduli⁽¹⁾_k Liek 'Modulik,

so that we can identifyφ with a functorφ⁰ : Alg^aug_k →Liek. Our goal in this section is to give an explicit description of the functorφ⁰.

Recall that the∞-category Alg_k ofE1-algebras overkcan be identified with the underlying∞-category of the model category Alg^dg_k of differential graded algebras over k (Proposition A.7.1.4.6). Let (Alg^dg_k )_/k denote the the category of augmenteddifferential graded algebras over k. Then (Alg^dg_k )_/k inherits a model structure, and (becausek∈Alg^dg_k is fibrant) the underlying∞-category of (Alg^dg_k )_/k can be identified with Alg^aug_k . For every object :A_∗→kof (Alg^dg_k )_/k, we letm_A∗ = ker() denote the augmentation ideal ofA_∗. Thenm_A∗ inherits the structure of a nonunital differential graded algebra overk. In particular, we can view mA_∗ as a differential graded Lie algebra overk(see Example 2.1.6). The constructionA_∗7→mA_∗determines a functor (Alg^dg_k )_/k →Lie^dg_k , which carries quasi-isomorphisms to quasi-isomorphisms. We therefore obtain an induced functor of∞-categoriesψ: Alg^aug_k →Liek. We will prove that the functorsψ, φ⁰ : Alg^aug_k →Liek

are equivalent to one another. We can state this result more precisely as follows:

Theorem 3.3.1. Letk be a field of characteristic zero. The diagram of∞-categories

Alg^aug_k ^ψ //

Liek

Moduli⁽¹⁾_k ^φ //Modulik

commutes (up to canonical homotopy). Hereφandψare the functors described above, and the vertical maps are the equivalences provided by Theorems 2.0.2 and 3.0.4.

To prove Theorem 3.3.1, we need to construct a homotopy between two functors Alg^aug_k →Moduli_k ⊆ Fun(CAlg^sm_k ,S). Equivalently, we must construct a homotopy between the functors

F, F⁰: Alg^aug_k ×CAlg^sm_k →S given by

F(A, R) = Map_Liek(D(R), ψ(A)) F⁰(A, R) = Map_Alg^aug

k (D⁽¹⁾(R), A).

Composing the Koszul duality functor D : (CAlg^aug_k )^op → Lie_k with the equivalence of ∞-categories Lie_k 'Moduli_k, we obtain the functor Spec : (CAlg^aug_k )^op → Moduli_k of Example 1.1.16. It follows from Yoneda’s lemma that this functor is fully faithful when restricted to (CAlg^sm_k )^op, so that D induces an equivalence from (CAlg^aug_k )^op onto its essential imageC⊆Liek. The inverse of this equivalence is given by g_∗7→C^∗(g_∗). It follows that we can identifyF andF⁰ with functorsG, G⁰ : Alg^aug_k ×C^op→S, given by the formulas

G(A,g_∗) = Map_Liek(g_∗, ψ(A)) G⁰(A,g_∗) = Map_Alg^aug

k (D⁽¹⁾C^∗(g_∗), A).

Note that the forgetful functor (Alg^dg_k )_/k →Lie^dg_k is a right Quillen functor, with left adjoint given by the universal enveloping algebra construction g_∗ 7→ U(g_∗) of Remark 2.1.7. It follows that the functor ψ admits a left adjoint Liek→Alg^aug_k , which we will also denote byU. Then the functorG: Alg^aug_k ×C^op→S can be described by the formulaG(A,g_∗) = Map_Alg^aug

k (U(g_∗), A). Theorem 3.3.1 is therefore a consequence of the following assertion:

Proposition 3.3.2. Let k be a field of characteristic zero. Then the diagram of∞-categories

(Liek)^{op C}

∗ //

U

CAlg^aug_k

(Alg^aug_k )^{op D}

(1) //Alg^aug_k commutes up to canonical homotopy.

The proof of Proposition 3.3.2 will require a brief digression.

Definition 3.3.3. Letλ:M→C×D andλ⁰ :M⁰ →C⁰×D⁰ be pairings of∞-categories. A morphism of pairingsfrom λto λ⁰ is a triple of maps

α:M→M⁰ β :C→C⁰ γ:D→D⁰ such that the diagram

M

λ

α //M⁰

λ⁰

C×D ^(β,γ)//C⁰×D⁰

commutes up to homotopy. Assume thatλ and λ⁰ are left representable. We will say that a morphism of pairings (α, β, γ) isleft representableif it carries left universal objects ofMto left universal objects ofM⁰. Proposition 3.3.4. Letλ:M→C×Dandλ⁰:M⁰→C⁰×D⁰be left representable pairings of∞-categories, which induce functorsDλ:C^op→D andDλ⁰ :C^0op→D⁰. Let(α, β, γ) fromλtoλ⁰. Then the diagram

C^{op Dλ} //

β

D

γ

C^{0op Dλ0} //D⁰ commutes up to canonical homotopy.

Proof. The right fibrationsλandλ⁰ are classified by functors

C^op×D^op→S C^0op×D^0op→S,

which we can identify with mapsχ:C^op→Fun(D^op,S) andχ⁰ :C^0op→Fun(D^0op,S). LetG: Fun(D^0op,S)→ Fun(C^0op,S) be the functor given by composition with β. Then α induces a natural transformation χ → G◦χ⁰◦β. Let F denote a left adjoint toG, so that we obtain a natural transformationu:F ◦χ→χ⁰◦β of functors from C^op to Fun(D^0op,S). Let j_D : D →Fun(D^op,S) and j_D⁰ : D⁰ → Fun(D^0op,S) denote the Yoneda embeddings. Thenχ'j_D◦Dλ and χ⁰'j_D⁰◦Dλ⁰, and Proposition T.5.2.6.3 gives an equivalence F◦j_D'j_D⁰◦γ. Thenudetermines a natural transformation

j_D⁰ ◦γ◦Dλ'F◦j_D◦Dλ'F◦χ→^u χ⁰◦β'j_D⁰ ◦Dλ⁰◦β.

Since j_D⁰ is fully faithful, this is the image of the a natural transformation of functors γ◦Dλ →Dλ⁰◦β.

Our assumption thatα carries left universal objects of Mto left universal objects of M⁰ implies that this natural transformation is an equivalence.

Construction 3.3.5. LetCbe a category. We define a new category TwArr(C) as follows:

(a) An object of TwArr(C) is given by a triple (C, D, φ), where C ∈ C, D ∈ D, and φ : C → D is a morphism in C.

(b) Given a pair of objects (C, D, φ),(C⁰, D⁰, φ⁰)∈ TwArr(C), a morphism from (C, D, φ) to (C⁰, D⁰, φ⁰) consists of a pair of morphismsα:C→C⁰, β:D⁰→D for which the diagram

C ^φ //

α

D

C^{0 φ}

0 //D⁰

β

OO

commutes.

(c) Given a pair of morphisms

(C, D, φ)^(α,β)−→ (C⁰, D⁰, φ⁰)^(α

0,β⁰)

−→ (C⁰⁰, D⁰⁰, φ⁰⁰) in TwArr(C), the composition of (α⁰, β⁰) with (α, β) is given by (α⁰◦α, β◦β⁰).

We will refer to TwArr(C) as the twisted arrow category of C. The construction (C, D, φ) 7→ (C, D) determines a forgetful functor λ : TwArr(C) → C×C^op which exhibits TwArr(C) as fibered in sets over C×C^op(the fiber ofλover an object (C, D)∈C×C^opcan be identified with the set Hom_C(C, D)). It follows that the induced map

λ: N(TwArr(C))→N(C)×N(C)^op

is a pairing of ∞-categories. This pairing is both left and right representable, and the associated duality functors

Dλ: N(C)^op→N(C)^op D⁰_λ: N(C)→N(C) are equivalent to the identity.

Remark 3.3.6. We will discuss an∞-categorical version of Construction 3.3.5 in§4.2.

We will deduce Proposition 3.3.2 from the following:

Proposition 3.3.7. Let k be a field of characteristic zero and letM⁽¹⁾→Alg^aug_k ×Alg^aug_k be the pairing of

∞-categories of Construction 3.1.4. There exists a left representable map of pairings

N(TwArr(LieT OO ^dg_k ))

λ

M⁽¹⁾

N(Lie^dg_k )×N(Lie^dg_k )^{op U×C}

∗//Alg^aug_k ×Alg^aug_k .

Here U and C^∗ denote the (covariant and contravariant) functors fromN(Lie^dg_k ) toAlg^aug_k induced by the universal enveloping algebra and cohomological Chevalley-Eilenberg constructions, respectively.

Proof of Proposition 3.3.2. As noted in Construction 3.3.5, the pairing of∞-categories N(TwArr(Lie^dg_k ))→ N(Lie^dg_k )×N(Lie^dg_k )^op induces the identity functor id : N(Lie^dg_k )^op → N(Lie^dg_k )^op. Applying Proposition 3.3.4 to the morphism of pairings T of Proposition 3.3.7, we obtain an equivalence between the functors C^∗◦id,D⁽¹⁾◦U : N(Lie^dg_k )^op→Alg^aug_k . Since the canonical map

Fun(Lie^op_k ,Alg^op_k )→Fun(N(Lie^dg_k )^op,Alg^aug_k )

is fully faithful, we obtain an equivalence between the functorsC^∗,D⁽¹⁾◦U : Lie^op_k →Alg^aug_k .

Proof of Proposition 3.3.7. Letg∗ be a differential graded Lie algebra and let Cn(g)∗be as in Construction 2.2.1. The universal enveloping algebraU(Cn(g)∗) has the structure of a (differential graded) Hopf algebra, where the comultiplication is determined by the requirement that the image of Cn(g)∗ consists of primitive elements. In particular, we have a counit map: Cn(g)_∗→k. Let End(UCn(g)_∗) denote the chain complex ofU(Cn(g)_∗)-comodule maps fromU(Cn(g)_∗) to itself. SinceU(Cn(g)_∗) is cofree as a comodule over itself, composition with the counit map :U(Cn(g)_∗)→kinduces an isomorphismθ from End(UCn(g)_∗) to the k-linear dual ofUCn(g)_∗. We regard End(UCn(g)_∗) as endowed with theoppositeof the evident differential graded Lie algebra structure, so thatUCn(g)_∗ has the structure of a right module over End(UCn(g)_∗). Let Endg(UCn(g)_∗) denote the subcomplex of End(UCn(g)_∗) consisting of rightU(g_∗)-module maps, so thatθ restricts to an isomorphism from Endg(UCn(g)_∗) to thek-linear dualC^∗(g_∗) ofC_∗(g_∗)'UCn(g)_∗⊗_U(g∗)k.

It is not difficult to verify that this isomorphism is compatible with the multiplication onC^∗(g) described in Construction 2.2.13. It follows thatUCn(g)_∗ is equipped with a right action ofC^∗(g_∗), which is compatible

with the right action of U(g_∗) on UCn(g)_∗. Let M_∗(g_∗) denote the k-linear dual of UCn(g)_∗. Then M_∗ is a contravariant functor, which carries a differential graded Lie algebra g_∗ to a chain complex equipped with commuting right actions of U(g∗) and C^∗(g∗). Moreover, the unit map k → U E(g∗) determines a quasi-isomorphismg∗ :M∗(g∗)→k.

Note that the initial objectk ∈Alg⁽¹⁾_k can be identified with a classifying object for endomorphisms of the unit object k ∈ Modk. Using Theorem A.6.1.2.34 and Proposition A.6.1.2.39, we can identify Alg^aug_k with the fiber product LMod(Modk)×Mod_k{k}. LetX⊆(Modk)_/k denote the full subcategory spanned by the final objects, so that we have an equivalence of∞-categories

α:M⁽¹⁾'(Alg_k×Alg_k)×Alg_kLMod(Mod_k)×ModkX.

We define a more rigid analogue of M⁽¹⁾ as follows: letY⊆(Vect^dg_k )_k/ be the full subcategory spanned by the quasi-isomorphisms of chain complexesV_∗→kand letCdenote the category

Alg^dg_k ×Alg^dg_k ×_Algdg k

LMod(Vect^dg_k )×_Vectdg k Y,

so thatαdetermines a functorT⁰⁰: N(C)→M⁽¹⁾. We will defineT as a composition TwArr(N(Lie^dg_k ))^T

0

→N(C)^T

00

→M⁽¹⁾.

Here the functor T⁰ assigns to each map γ : h_∗ → g_∗ of differential graded Lie algebras the object of C given by (U(h_∗), C^∗(g_∗), M_∗(g_∗), _g∗), where M_∗(g_∗) is regarded as a left module over U(h_∗)⊗_kC^∗(g_∗) by combining the commuting left actions ofU(g_∗) andC^∗(g_∗) onM_∗(g_∗) (and composing with the mapγ).

We now claim that the diagramσ:

N(TwArr(Lie^dg_k )) ^T //

λ

M⁽¹⁾

N(Lie^dg_k )×N(Lie^dg_k )^{op U×C}

∗//Alg^aug_k ×Alg^aug_k

commutes up to canonical homotopy. Consider first the composition of T with the map M⁽¹⁾ → Alg^aug_k given by projection onto the first factor. Unwinding the definitions, we see that this map is given by the composing the equivalence ξ : LMod(Modk)×Mod_kX'Alg^aug_k with the functor T₀⁰ : N(TwArr(Lie^dg_k ))→ LMod(Modk)×Mod_kXgiven by

T₀⁰(γ:h_∗→g_∗) = (U(h_∗), M_∗(g_∗), g_∗).

The counit map U(g∗)→ k determines a quasi-isomorphism ofU(h∗)-modules k→ M∗(g∗), so that T₀⁰ is equivalent to the functor T⁰₀ given by T⁰₀(γ : h_∗ → g_∗) = (U(h_∗), k,id_k), which (after composing with ξ) can be identified with the map N(TwArr(Lie^dg_k )) → N(Lie^dg_k ) →^U Alg^aug_k . Now consider the composition of T with the map M⁽¹⁾ → Alg^aug_k given by projection onto the second factor. This functor is given by composing the equivalence ξ with the functor T₁⁰ : N(TwArr(Lie^dg_k )) → LMod(Modk)×Mod_k X given by T₁⁰(γ :h∗ →g∗) = (C^∗(g), M∗(g∗), g∗). Note that g is a map of C^∗(g)-modules and therefore determines an equivalence of T₁⁰ with the functor T⁰₁ given by T⁰₁(γ : h∗ → g∗) = (C^∗(g), k,idk). It follows that the composition ofT₁⁰ withξcan be identified with the composition

N(TwArr(Lie^dg_k ))→N(Lie^dg_k )^{op C}

∗

→Alg^aug_k .

This proves the homotopy commutativity of the diagramσ. After replacingT by an equivalent functor, we can assume that the diagramσis commutative.

It remains to show thatσdetermines a left representable map between pairings of∞-categories. Letg_∗ be a differential graded Lie algebra, and let End(M_∗(g_∗)) denote the differential graded algebra of endomor-phisms of the chain complexM∗(g∗). SinceM∗(g∗) is quasi-isomorphic tok, the unit mapk→End(M∗(g∗)) is a quasi-isomorphism of differential graded algebras. Unwinding the definitions, we must show that the mapθ:U(g∗)⊗kC^∗(g∗)→End(M∗(g)) exhibitsC^∗(g∗) as Koszul dual (as anE1-algebra) toU(g∗). LetA∗

denote the differential graded algebra of endomorphisms ofUCn(g)∗ (as a chain complex). Then θfactors as a composition

U(g_∗)⊗kC^∗(g_∗) ^θ

0

→A_∗ ^θ

00

→End(M_∗(g_∗))

where θ⁰⁰ is a quasi-isomorphism. It will therefore suffice to show thatθ⁰ exhibits C^∗(g_∗) as Koszul dual to U(g_∗). Since U E_∗(g_∗) is a free U(g_∗)-module, this is equivalent to the requirement that θ⁰ induces a quasi-isomorphism φ : C^∗(g_∗) → W_∗, where W_∗ is the differential graded algebra of right U(g_∗)-module maps from UCn(g)_∗ to itself. This is clear, since φ admits a left inverse given by composition with the quasi-isomorphismUCn(g)_∗→k.

Dans le document Derived Algebraic Geometry X: Formal Moduli Problems (Page 72-77)