Koszul Duality for Modules

Our goal in this section is to prove the following non-commutative analogue of Theorem 2.4.1:

Theorem 3.5.1. Let k be a field, letA∈Alg^aug_k , and let X : Alg^sm_k →S be the formalE1 moduli problem associated to A(see Theorem 3.0.4). Then there are canonical equivalences of ∞-categories

QCoh^!_L(X)'RModA QCoh^!_R(X)'LModA. In particular, we have fully faithful embeddings

QCoh_L(X),→RMod_A QCoh_R(X),→LMod_A. The main ingredient in the proof of Theorem 3.5.1 is the following result.

Proposition 3.5.2. Let k be a field, and let χ! : Alg^sm_k → dCat∞ be as in Construction 3.4.20 (given on objects by χ!(R) = RMod^!_R), and let χ⁰ : Alg^op_k be the functor which classifies the Cartesian fibration LMod(Modk)→Alg_k (given by the formulaχ⁰(A) = LModA). Then χ! is homotopic to the composition

Alg^sm_k →Alg^aug_k ^D

(1)

−→(Alg^aug_k )^op→Alg^op_k ^χ

0

→Cdat∞.

In particular, for everyR∈Alg^sm_k , there is a canonical equivalence of∞-categoriesRMod^!_R'LMod_D(1)(R). Before giving the proof of Proposition 3.5.2, let us explain how it leads to a proof of Theorem 3.5.1.

Proof of Theorem 3.5.1. We will construct the equivalence QCoh^!_R(X)' LModA; the construction of the equivalence QCoh^!_L(X)'RMod_Ais similar. Letkbe a field and let QCoh^!_R: Fun(Alg^sm_k ,S)^op→dCat_∞be as in Construction 3.4.20. Let Ψ : Alg^aug_k →Moduli⁽¹⁾_k be the equivalence of∞-categories provided by Theorem 3.0.4, and let Ψ⁻¹denote a homotopy inverse to Ψ. LetL: Fun(Alg^sm_k ,S)→Moduli⁽¹⁾_k denote a left adjoint to

the inclusion functor Moduli⁽¹⁾_k ⊆Fun(Alg^sm_k ,S) (see Remark 1.1.17), and letDb⁽¹⁾: Fun(Alg^sm_k ,S)→Alg^aug_k be the composition of Ψ⁻¹◦L. The functor Db⁽¹⁾ preserves small colimits, and the composition of Db⁽¹⁾ with the Yoneda embedding (Alg^sm_k )^op → Fun(Alg^sm_k ,S) can be identified with the Koszul duality functor D⁽¹⁾ : (Alg^sm_k )^op → Alg^aug_k . Let χ⁰ : Alg^op_k → dCat∞ be as in Proposition 3.5.2 (given on objects by χ⁰(A) = LModA), and letF : Fun(Alg^sm_k ,S)^op→dCat_∞ denote the composite functor

Fun(Alg^sm_k ,S)^opDb

(1)

−→(Alg^aug_k )^op→Alg^op_k ^χ

0

→dCat_∞.

Let C denote the full subcategory of Fun(Alg^sm_k ,S) spanned by the corepresentable functors. Proposition 3.5.2 implies that there is an equivalence of functorsα0:F|C^op→QCoh^!_R|C^op. Since QCoh^!_Ris a right Kan extension of its restriction toC^op, the equivalenceα0extends to a natural transformationF→QCoh^!_R. We will prove:

(∗) If X : Alg^sm_k → S is a formal E1 moduli problem, then α induces an equivalence of ∞-categories F(X)→QCoh^!_R(X).

TakingX = Ψ(A) forA∈Alg^aug_k , we see that (∗) guarantees an equivalence of∞-categories β_A: LMod_A'χ⁰(Ψ⁻¹Ψ(A))'F(Ψ(A))→QCoh^!_R(X).

It remains to prove (∗). Let E ⊆ Fun(Alg^sm_k ,S) be the full subcategory spanned by those functors X for which α induces an equivalence of ∞-categories F(X) → QCoh^!_R(X). The localization functor L : Fun(Alg^sm_k ,S)→Moduli⁽¹⁾_k , the equivalence Ψ⁻¹: Moduli⁽¹⁾_k →Alg^aug_k , and the forgetful functor Alg^aug_k → Alg_k preserve small colimits. Lemma 2.4.32 implies that the functor χ⁰ : Alg^op_k → dCat_∞ preserves sifted limits. It follows that the functor F preserves sifted limits. Since QCoh^!_R preserves small limits, the ∞-category E is closed under sifted colimits in Fun(Alg^sm_k ,S). Since E contains all corepresentable functors and is closed under filtered colimits, it contains it contains all prorepresentable formal moduli problems (see Definition 1.5.3). Proposition 1.5.8 implies that every formalE1 moduli problemX can be obtained as the geometric realization of a simplicial objectX_•of Fun(Alg^sm_k ,S), where eachX_n is prorepresentable. SinceE is closed under geometric realizations in Fun(Alg^sm_k ,S), we conclude thatX ∈Eas desired.

We now turn to the proof of Proposition 3.5.2. Consider first the functor χ⁰ : Alg^op_k →Cdat∞classifying the Cartesian fibration p : LMod(Modk) → Alg_k. Using Remark 3.4.9, we see that χ⁰ also classifies the coCartesian fibration Dl⁰(p^op)→Alg^op_k . Let LMod^perf(Mod_k) denote the full subcategory of LMod(Mod_k) spanned by those pairs (A, M), where A ∈Alg_k and M is a perfect left module over A. Let p_perf denote the restriction ofpto LMod(Modk). Proposition 3.4.10 supplies an equivalence of Dl⁰(p^op)'Dl^lex(p^op_perf) of coCartesian fibrations over Alg^op_k . By construction,χ_! : Alg^sm_k →dCat_∞ classifies the coCartesian fibration RMod^!(Mod_k) = Dl^lex(q) → Alg^sm_k , where q denotes the Cartesian fibration LMod^sm(Mod_k) → Alg^sm_k . Consequently, Proposition 3.5.2 is a consequence of the following:

Proposition 3.5.3. Let k be a field and letD: Alg^sm_k →Alg^op_k denote the composition Alg^sm_k ,→Alg^aug_k ^D

(1)

−→(Alg^aug_k )^op→Alg^op_k ,

where D⁽¹⁾ denotes the Koszul duality functor of Definition 3.1.6. Then there is a pullback diagram of

∞-categories

LMod^sm(Mod_k)

D⁰ //LMod^perf(Mod_k)^op

Alg^sm_k ^D //Alg^op_k .

We now proceed to construct the diagram appearing in the statement of Proposition 3.5.3.

Construction 3.5.4. Fix a fieldk. We letM⁽¹⁾ →Alg^aug_k ×Alg^aug_k be the pairing of∞-categories defined in Construction 3.1.4. The objects ofM⁽¹⁾ are given by triple (A, B, ), whereA, B∈Alg_k and:A⊗kB →k is an augmentation onA⊗_kB (which then determines augmentations onAandB). Set

M= LMod(Mod_k)×_AlgkM⁽¹⁾×_AlgkLMod(Mod_k),

so that Mis an ∞-category whose objects can be identified with quintuples (A, B, , M, N), whereA, B ∈ Alg_k, : A⊗kB → k is an augmentation, M ∈ LModA, and N ∈ LModB. There is an evident functor χ:M^op→S, given on objects by the formula

χ(A, B, , M, N) = Map_LModA⊗

k B(M ⊗kN, k)

Thenχclassifies a right fibration M^LM →M. Let LMod^aug(Modk) denote the∞-category LMod(Modk)×Alg_kAlg^aug_k ,

so that the forgetful functor M^LM →LMod^aug(Mod_k)×LMod^aug(Mod_k) is a right fibration and therefore determines a pairing of LMod^aug(Modk) with itself.

Proposition 3.5.5. Let k be a field, let λ: M⁽¹⁾ → Alg^aug_k ×Alg^aug_k be the pairing of Construction 3.1.4 andλ⁰:M^LM→LMod^aug(Modk)×LMod^aug(Modk)the pairing of Construction 3.5.4. Thenλ⁰ is both left and right representable. Moreover, the forgetful functor M^LM →M⁽¹⁾ is both left and right representable.

Proof. We will show thatλ⁰ is left representable andM^LM →M⁽¹⁾ is left representable; the corresponding assertions for right representability will follow by symmetry. Fix an objectA∈Alg^aug_k and a leftA-module M. Let B =D⁽¹⁾(A) be the Koszul dual of Aand :A⊗kB →k the canonical map. Proposition 3.1.10 implies that determines a duality functorD: LMod^op_A →LModB. We letN =D(M), so that there is a canonical map of left A⊗kB-modulesµ:M ⊗kN →k. The quintuple (A, B, , M, N) is an object of the

∞-category Mof Construction 3.5.4, andµdetermines a lifting to an objectX ∈M^LM. We complete the proof by observing thatX is left universal and has left universal image inM⁽¹⁾.

It follows from Proposition 3.5.5 that the pairingM^LM →LMod^aug(Mod_k)×LMod^aug(Mod_k) determines a duality functorD^LM : LMod^aug(Mod_k)→LMod^aug(Mod_k)^op. LetD⁰ denote the composite map

LMod^sm(Mod_k)→LMod^aug(Mod_k)^D−→^LMLMod^aug(Mod_k)^op→LMod(Mod_k)^op. By construction, we have a commutative diagramσ:

LMod^sm(Modk)

p

D⁰ //LMod(Modk)^op

q

Alg^sm_k ^D //Alg^op_k .

We next claim that the functorD⁰ carriesp-Cartesian morphisms toq-Cartesian morphisms. Unwinding the definitions, we must show that iff :R→R⁰ is a morphism in Alg^sm_k andM is a small leftR⁰-module, then the canonical map

θM :D⁽¹⁾(R)⊗_D(1)(R⁰)Dµ⁰(M)→Dµ(M)

is an equivalence, where Dµ : LMod^op_R → LMod_D(1)(R) and Dµ⁰ : LMod^op_R → LMod_D(1)(R) are the duality functors determined by the pairings µ : R⊗kD⁽¹⁾(R) → k and µ⁰ : R⁰⊗kD⁽¹⁾(R⁰) → k. The modules

M ∈ LModR⁰ for which θM is an equivalence span a stable subcategory of LModR⁰ which includes k, and therefore contains all smallR⁰-modules (Lemma 3.4.2).

To complete the proof of Proposition 3.5.3, it suffices to show that the functorD⁰ carries LMod^sm(Modk) into LMod^perf(Modk)^opand induces an equivalence of∞-categories

LMod^sm(Mod_k)→LMod^perf(Mod_k)^op×_Alg^op

k Alg^sm_k .

Using Corollary T.2.4.4.4, we are reduced to proving thatD⁰ induces an equivalence of∞-categories LMod^sm_R →(LMod^perf_D(1)(R))^op

for every R ∈ Alg^sm_k . This is a consequence of Remark 3.4.2 together with the following more general assertion:

Proposition 3.5.6. Let kbe a field and let µ:A⊗kB→k be a map ofE1-algebras over kwhich exhibits B as a Koszul dual of A. Then the duality functor D_µ : LMod^op_A → LMod_B restricts to an equivalence C→LMod^perf_B , where C denotes the smallest stable subcategory ofLModA which contains k (regarded as a left A-module via the augmentationA→A⊗kB→^µ k) and is closed under retracts.

Proof. Let D⁰_µ : LMod^op_B → LModA be as in Notation 3.1.11, and let D denote the full subcategory of LMod_A spanned by those objectsM for which the unit mapM →D⁰_µDµ(M) is an equivalence in LMod_A. It is clear thatD is a stable subcategory of LMod_A which is closed under retracts. Since µ exhibits B as a Koszul dual ofA, the subcategoryD containsk so thatC⊆D. It follows that the functor D_µ|C is fully faithful. Moreover, the essential image of D_µ|C is the smallest stable full subcategory of LMod_B which containsDµ(k)'B and is closed under retracts: this is the full subcategory LMod^perf_B ⊆LMod_B.

Remark 3.5.7. Let k be a field of characteristic zero and let θ : CAlg^sm_k → Alg^sm_k denote the forgetful functor. LetX : Alg^sm_k →Sbe a formalE1moduli problem overk, so thatX◦θis a formal moduli problem overk. For eachR∈CAlg^sm_k , we have a canonical equivalence of∞-categories ModR'RMod_θ(R). Passing to the inverse limit over points η ∈X(θ(R)), we obtain a functor QCoh_R(X)→ QCoh(X◦θ). According to Theorem 3.0.4, there exists an augmented E1-algebra A over k such that X is given by the formula X(R) = Map_Alg^aug

k (D⁽¹⁾(R), A). Let mA denote the augmentation ideal of A. Regard mA as an object of Liek, so thatX◦θis given by the formula

(X◦θ)(R) = Map_Lie

k(D(R),m_A)

(see Theorem 3.3.1). Theorems 2.4.1 and 3.5.1 determine fully faithful embeddings QCoh_R(X),→LMod_A QCoh(X◦θ),→Rep_m

A.

We have an evident map of E1-algebras U(mA) → A, which determines a forgetful functor LModA → LMod_U(mA)'Rep_mA. With some additional effort, one can show that the diagram

QCoh_R(X) //

LModA

QCoh(X◦θ) //Rep_m

A

commutes up to canonical homotopy. That is, the algebraic models for quasi-coherent sheaves provided by Theorems 2.4.1 and 3.5.1 in the commutative and noncommutative settings are compatible with one another.

We conclude this section with a discussion of the exactness properties of equivalences QCoh^!_L(X)'RModA QCoh^!_R(X)'LModA

appearing in Theorem 3.5.1. Let RMod^cn_A and LMod^cn_A denote the full subcategories spanned by those right and leftA-modules whose underlying spectra are connective. Note that the equivalences of Theorem 3.5.1 depend functorially onA, and whenA=kthey are equivalent to the identity functor from the∞-category Modk to itself. Let∗denote the final object of Moduli⁽¹⁾_k , so that we have a canonical map of formal moduli problems ∗ → X (induced by the map of augmented E1-algebras k → A). It follows that Theorem 3.5.1 gives an equivalence

QCoh^!_L(X)^cn ' QCoh^!_L(X)×_QCoh!

L(∗)QCoh^!_L(∗)^cn ' RModA×RMod_kRMod^cn_k

' RMod^cn_A

and, by symmetry, an equivalence QCoh^!_R(X)^cn 'LMod^cn_A. Combining this observation with Proposition 3.4.27, we obtain the following result:

Proposition 3.5.8. Letkbe a field, letA∈Alg^aug_k , and letX : Alg^sm_k →Sbe the formalE1 moduli problem associated to A(see Theorem 3.0.4). Then the fully faithful embeddings

QCoh_L(X),→RModA QCoh_R(X),→LModA. of Theorem 3.5.1 restrict to equivalences of ∞-categories

QCoh_L(X)^cn'RMod^cn_A QCoh_R(X)^cn'LMod^cn_A . Warning 3.5.9. IfAis an arbitraryE1-ring, then the full subcategories

LMod^cn_A = LMod_A×SpSp^cn⊆LMod_A RMod^cn_A = RMod_A×SpSp^cn⊆RMod_A

are presentable, closed under small colimits, and closed under extensions. It follows from Proposition A.1.4.5.11 that LModAand RModA admit t-structures with

(LModA)_≥0= LMod^cn_A (RModA)_≥0= RMod^cn_A .

However, it is often difficult to describe the subcategories (LModA)_≤0⊆LModAand (RModA)_≤0⊆RModA. In particular, they generally do not coincide with the subcategories

LModA×SpSp_≤0⊆LModA RModA×SpSp_≤0⊆RModA

unless theE1-ringA is connective.

4 Moduli Problems for E

ⁿ

-Algebras

Letkbe a field. In§2 and§3 we studied the∞-categories Modulikand Moduli⁽¹⁾_k consisting of formal moduli problems defined for commutative and associative algebras overk, respectively. In the∞-categorical context, there is a whole hierarchy of algebraic notions in between these two extremes. Recall that the commutative

∞-operad can be identified with the colimit of a sequence

Ass^⊗ 'E^⊗1 →E^⊗2 →E^⊗3 → · · ·,

where E^⊗n denotes the Boardman-Vogt ∞-operad of little n-cubes (see Corollary A.5.1.1.5). Consequently, the∞-category CAlg_k ofE∞-algebras overkcan be identified with the limit of a tower of∞-categories

· · · →Alg⁽³⁾_k →Alg⁽²⁾_k →Alg⁽¹⁾_k 'Alg_k,

where Alg⁽ⁿ⁾_k denotes the∞-category ofEn-algebras overk. Our goal in this section is to prove a generaliza-tion of Theorem 3.0.4 in the setting ofEn-algebras, for an arbitrary integern≥0. To formulate our result, we need a bit of terminology.

Definition 4.0.1. Letkbe a field, letn≥1, and letAbe anEn-algebra overk. We will say thatAissmall if its image in Alg_k is small, in the sense of Definition 3.0.1. We let Alg^(n),sm_k denote the full subcategory of Alg^sm_k spanned by the smallEn-algebras overk.

Remark 4.0.2. Letn≥1 and letAbe anEn-algebra overk. ThenAis small if and only if it is connective, π∗A is finite-dimensional overk, and the unit map k→(π0A)/m is an isomorphism, where m denotes the radical ofπ0A.

Remark 4.0.3. Letk be a field and let A be anEn-algebra over k, for n≥0. Anaugmentation onA is a map of En-algebras :A→k. We let Alg^(n),aug_k = (Alg⁽ⁿ⁾_k )_/k denote the∞-category of augmented En -algebras overk. Note that ifn≥1 andA∈Alg⁽ⁿ⁾_k is small, then the space Map_Alg(n)

k

(A, k) of augmentations onAis contractible. It follows that the projection map

Alg^(n),aug_k ×_Alg(n) k

Alg^(n),sm_k →Alg^(n),sm_k

is an equivalence of∞-categories. We will henceforth abuse notation by identifying Alg^(n),sm_k with its inverse image in Alg^(n),aug_k .

Remark 4.0.4. It will be convenient to have a version of Definition 3.0.1 also in the case n = 0. We therefore adopt the following convention: we will say that an augmentedE0-algebraAoverkissmallifAis connective andπ_∗Ais a finite dimensional vector space overk. We let Alg^(0),sm_k denote the full subcategory of Alg^(0),aug_k spanned by the small augmentedE0-algebras overk.

Notation 4.0.5. Letkbe a field, let n≥0, and let:A→kbe an augmented En-algebra overk. We let mA denote the fiber of the map in the stable∞-category Modk. We will refer tomA as theaugmentation idealofA. The construction (:A→k)7→mAdetermines a functor

m: Alg^(n),aug_k →Mod_k. In the casen= 0, this functor is an equivalence of∞-categories.

Definition 4.0.6. Letkbe a field, letn≥0 be an integer and letX : Alg^(n),sm_k →Sbe a functor. We will say thatX is aformal En moduli problemif it satisfies the following conditions:

(1) The spaceX(k) is contractible.

(2) For every pullback diagram

R //

R0

R₁ //R₀₁

in Alg^sm_k for which the underlying mapsπ0R0→π0R01←π0R1 are surjective, the diagram X(R) //

X(R₀)

X(R₁) //X(R₀₁) is a pullback square.

We let Moduli⁽ⁿ⁾_k denote the full subcategory of Fun(Alg^(n),sm_k ,S) spanned by the formalEnmoduli problems.

Example 4.0.7. It is not difficult to show that a functorX: Alg^(0),sm_k →Sis a formalE0moduli problem if and only if it is strongly excisive (see Definition A.1.4.4.4): that is, if and only ifX carries the initial object of Alg^(0),sm_k to a final object ofS, and carries pushout squares to pullback squares.

We are now ready to formulate our main result.

Theorem 4.0.8. Let k be a field and letn≥0 be an integer. Then there is an equivalence of ∞-categories Ψ : Alg^(n),aug_k →Moduli⁽ⁿ⁾_k . Moreover, the diagram

Alg^(n),aug_k ^Ψ //

m

Moduli⁽ⁿ⁾_k

T[−n]

Modk //Sp

commutes up to homotopy, where T : Moduli⁽ⁿ⁾_k → Sp denotes the tangent complex functor (so that Ω^∞−mTX ' X(k⊕k[m]) for m ≥ 0) and m : Alg^(n),aug_k → Modk is the augmentation ideal functor of Notation 4.0.5.

Example 4.0.9. Whenn= 1, Theorem 4.0.8 follows from Theorem 3.0.4 and Remark 3.2.6.

Remark 4.0.10. Suppose that k is a field of characteristic zero. For each n ≥ 0, there is an evident forgetful functor CAlg^sm_k →Alg^(n),sm_k , which induces a forgetful functorθ: Moduli⁽ⁿ⁾_k →Moduli_k. Using the equivalences

Lie_k 'Moduli_k Moduli⁽ⁿ⁾_k 'Alg^(n),aug_k

of Theorems 2.0.2 and 4.0.8, we can identifyθwith a map Alg^(n),aug_k →Liek. We can summarize the situation informally as follows: ifAis an augmentedEn-algebra overk, then the shifted augmentation idealmA[n−1]

inherits the structure of a differential graded Lie algebra over k. In particular, at the level of homotopy groups we obtain a Lie bracket operation

[,] :πpmA×πqmA→πp+q+n−1mA.

One can show that this Lie bracket is given by theBrowder operationonmA. If Free : Modk →Alg⁽ⁿ⁾_k denotes the free algebra functor (left adjoint to the forgetful functor Alg⁽ⁿ⁾_k →Modk), then the Browder operation is universally represented by the the mapφappearing in the cofiber sequence of augmentedEn-algebras

Free(k[p+q+n−1])→^φ Free(k[p]⊕k[q])→Free(k[p])⊗kFree(k[q]) supplied by Theorem A.5.1.5.1.

The appearance of the theory ofEn-algebras on both sides of the equivalence Alg^(n),aug_k 'Moduli⁽ⁿ⁾_k ⊆Fun(Alg^(n),sm_k ,S)

is somewhat striking: it is a reflection of the Koszul self-duality of the little cubes operadsE^⊗n (see [17]). In particular, there is a Koszul duality functor

D⁽ⁿ⁾: (Alg^(n),aug_k )^op→Alg^(n),aug_k .

This functor is not difficult to define directly : if A is an augmented Eⁿ-algebra over k, then D⁽ⁿ⁾(A) is universal among En-algebras over k such that the tensor product A⊗k D⁽ⁿ⁾(A) is equipped with an

augmentation extending the augmentation onA. The equivalence Ψ appearing in the statement of Theorem 4.0.8 carries an augmentedEn-algebraA to the functorX given by the formula

X(R) = Map_Alg(n),aug k

(D⁽ⁿ⁾(R), A).

In §4.5, we will prove that the Koszul duality functor D⁽ⁿ⁾ is a deformation theory (in the sense of Definition 1.3.9), so that Theorem 4.0.8 is a consequence of Theorem 1.3.12. The main point is to produce a full subcategory Ξ₀⊆Alg^(n),aug_k which satisfies axiom (D3) of Definition 1.3.1. We will define Ξ₀to be the full subcategory of Alg^(n),aug_k spanned by those augmented En algebras A which satisfy suitable finiteness and coconnectivity conditions. We will then need to prove two things:

(a) The full subcategory Ξ0⊆Alg^(n),aug_k has good closure properties.

(b) For every objectA∈Ξ0, the biduality mapA→D⁽ⁿ⁾D⁽ⁿ⁾(A) is an equivalence.

The verification of (a) comes down to connectivity properties of free algebras over theEn-operad. We will establish these properties in§4.1, using topological properties of configuration spaces of points in Euclidean space.

The proof of (b) is more involved, and requires us to have a good understanding of the Koszul duality functor D⁽ⁿ⁾. Let us begin with the case n = 1, which we have already studied in §3.1. Let A be an E1-algebra over a fieldk. Then the Koszul dualD⁽¹⁾(A) can be described as a classifying object forA-linear maps fromk to itself, or equivalently as the k-linear dual of the objectk⊗Ak. In §4.3, we will show that the algebra structure onD⁽¹⁾(A) can be obtained by dualizing acoalgebrastructure on Bar(A) =k⊗Ak: in particular, we have a comultiplication given by

Bar(A) =k⊗Ak'k⊗AA⊗Ak→k⊗Ak⊗Ak'Bar(A)⊗kBar(A).

The proof will require an∞-categorical generalization of the twisted arrow category introduced in Construc-tion 3.3.5, which we will study in§4.2.

The bar construction A 7→Bar(A) = k⊗Ak is in some ways better behaved than the Koszul duality functor D⁽¹⁾: for example, it is a symmetric monoidal functor, while D⁽¹⁾ is not (see Warning 3.1.20). In

§4.4, we will use this observation to analyze the Koszul duality functorD⁽ⁿ⁾ for a general integern, using induction on n. Using Theorem A.5.1.2.2, we can identify the ∞-category Alg⁽ⁿ⁺¹⁾_k with Alg(Alg⁽ⁿ⁾_k ), the

∞-category of associative algebra objects of Alg⁽ⁿ⁾_k . IfAis an augmentedEⁿ⁺¹-algebra, then we can apply the bar construction to obtain a coalgebra object of Alg^(n),aug_k . It is not difficult to show that the Koszul duality functor

D⁽ⁿ⁻¹⁾: (Alg^(n),aug_k )^op→Alg^(n),aug_k is lax monoidal. We will show that the composite map

(Alg^(n+1),aug_k )^op ' Alg(Alg^(n),aug_k )^op

Bar→ Alg((Alg^(n),aug_k )^op)

D⁽ⁿ⁾

→ Alg(Alg^(n),aug_k ) ' Alg^(n+1),aug_k

can be identified with the Koszul duality functor D⁽ⁿ⁺¹⁾. This will allow us to deduce results about the Koszul duality functorsD⁽ⁿ⁾ from analogous facts about the Koszul duality functorD⁽¹⁾, and in particular to deduce (a) from Corollary 3.1.15 (see Theorem 4.4.5).

Remark 4.0.11. Let X : Alg^(n),sm_k → S be a formal En-moduli problem for n ≥ 1. Using the ideas introduced in§3.4, we can define∞-categories QCoh_L(X) and QCoh^!_L(X) of quasi-coherent and Ind-coherent sheaves onX, respectively. According to Theorem 4.0.8, the functorX is given by the formula

X(R) = Map_Alg(n),aug k

(D⁽ⁿ⁾(R), A)

for some (essentially unique) augmented En-algebra A overk. Since the bar constructionB 7→ Bar(B) is symmetric monoidal, it carries augmentedEm-algebras overkto augmentedEm−1-algebras overk. It follows that the iterated bar construction Barⁿ⁻¹(A) admits the structure of anE1-algebra. In this context, we have the following version of Theorem 2.4.1: there is an equivalence of∞-categories QCoh^!_L(X)'RMod_Barⁿ⁻¹_(A) (and therefore also a fully faithful embedding QCoh_L(X),→RMod_Barⁿ⁻¹_(A)). Moreover, one can show that this is an equivalence ofEn−1-monoidal∞-categories (here theEn−1-monoidal structure on RMod_Barⁿ⁻¹_(A) arises from the fact that Barⁿ⁻¹(A) can be regarded as anEn−1-algebra object of Alg^op_k ).

4.1 Coconnective E

n

-Algebras

Let k be a field and let n ≥ 0 be an integer. Our goal in this section is to study some finiteness and coconnectivity conditions onEn-algebras overk which will play a role in our proof of Theorem 4.0.8.

Definition 4.1.1. Letk be a field and let A be anE0-algebra overk: that is, a k-module equipped with a unit mape:k→A. Letm be an integer. We will say thatAis m-coconnectiveif the homotopy groups π_icofib(e) vanish fori >−m.

More generally, if A is an En-ring equipped with a map ofEn-rings k → A, we will say that A is m-coconnectiveif it ism-coconnective when regarded as an E0-algebra overk (here we do not require that A is anEⁿ-algebra overk, though this will always be satisfied in cases of interest to us).

Remark 4.1.2. IfAis anE1-algebra over a fieldk, thenAis coconnective (in the sense of Definition 3.1.13) if and only if it is 1-coconnective (in the sense of Definition 4.1.1).

Remark 4.1.3. Ifm >0, then anEn-algebraAoverkism-coconnective if and only if the unit mapk→A induces an isomorphismk→π0A, and the homotopy groupsπiAvanish fori >0 and−m < i <0.

Notation 4.1.4. Letkbe a field and letn≥0 be an integer. We let Free⁽ⁿ⁾: Modk→Alg⁽ⁿ⁾_k denote a left adjoint to the forgetful functor Alg⁽ⁿ⁾_k →Mod_k. For any object V ∈Mod_k, the free algebra Free⁽ⁿ⁾(V) is equipped with a canonical augmentation: Free⁽ⁿ⁾(V)→k, corresponding to the zero morphism. V →kin Modk.

Our main result can be stated as follows:

Theorem 4.1.5. Let k be a field, letA be anEn-algebra over k, and let m≥n be an integer. Suppose we are given a mapφ:V →Ain Mod_k, whereπ_iV '0fori≥ −m, and form a pushout diagram

Free⁽ⁿ⁾(V) ^φ

0 //

A

k //A⁰

where φ⁰ is the map of En-algebras determined by φ and is the augmentation of Notation 4.1.4. If A is m-coconnective, thenA⁰ is also m-coconnective.

Our proof of Theorem 4.1.5 is somewhat indirect. We will first show that the conclusion of Theorem 4.1.5 is valid under an additional hypothesis onA(Proposition 4.1.13). We will then use this variant of Theorem 4.1.5 to show that the additional hypothesis is automatically satisfied (Proposition 4.1.14). First, we need to introduce a bit of terminology.

Notation 4.1.6. Let k be a field and let A be an En-algebra over k. We let Mod^E_Aⁿ = Mod^E_Aⁿ(Mod_k)

Notation 4.1.6. Let k be a field and let A be an En-algebra over k. We let Mod^E_Aⁿ = Mod^E_Aⁿ(Mod_k)

Dans le document Derived Algebraic Geometry X: Formal Moduli Problems (Page 85-102)