DERIVED DEFORMATIONS OF SCHEMES

J.P.PRIDHAM

Abstract. We introduce a new approach to constructing derived deformation groupoids, by considering them as parameter spaces for strong homotopy bialgebras. This allows them to be constructed for all classical deformation problems, such as deformations of an arbitrary scheme, in any characteristic. We also give a general approach for studying deformations of diagrams.

Introduction

In [Pri2], the theory of simplicial deformation complexes (SDCs) was expounded as a means of governing deformation problems, giving an alternative to the theory of differential graded Lie algebras (DGLAs). The main advantages of SDCs over DGLAs are that they can be constructed canonically (and thus for a wider range of problems), and are valid in all characteristics.

In [Man2], Manetti showed that given a DGLA, or even an SHLA, governing a deformation problem, it is possible to define an extended deformation functor. The approach in this paper can almost be regarded as opposite to this — we try, for any deformation problem, to define an extended deformation functor with a geometric interpretation, meaning that the functor still parametrises geometric objects. We then see how this functor can be recovered from the SDC governing the problem.

Since almost all examples of SDCs come from monadic and comonadic adjunctions, in Section 4 we look at how to extend deformation groupoids in these scenarios. For a monad ⊤, the solution is to look at the strong homotopy ⊤-algebras, as defined by Lada in [CLM]. The idea is that the monadic axioms are only satisfied up to homotopy, with the homotopies satisfying further conditions up to homotopy, and so on. This approach allows us to define a quasi-smooth extended deformation functor associated to any SDC, with the same cohomology.

Using the constructions of §§3.2 and 3.3, we describe extended deformations of morphisms and diagrams (giving new results even for the problems in [Pri2]). This defines cohomology of a morphism in any such category, giving a variant of Van Osdol’s bicohomology ([Van]). One consequence is that the space describing extended deformations of the identity morphism on an object D is just the loop space of the space of extended deformations of D.

The structure of the paper is as follows. Sections 1 and 2 are introductory, summarising results from [Pri5] and properties of monads and comonads, respectively. Section 3 reprises material from [Pri2] on SDCs, and includes new results constructing SDCs associated to diagrams in §§3.2 and 3.3. The key motivating examples of deformations of a scheme are described in Examples 3.8 and 3.19.

Section 4 then gives the construction of the derived deformation functor (Definition 4.4), together with a simplified description of derived deformations of a morphism (Proposition 4.9), and the characterisation of derived deformations of an identity morphism as a loop space (Proposition 4.10).

This work was supported by Trinity College, Cambridge; and by the Engineering and Physical Sciences Research Council [grant number EP/F043570/1].

In [Pri2], it was shown that SDCs are equivalent to ℕ_{0}-graded DGLAs in characteristic 0, in such a way that the associated deformation groupoids are equivalent. In §5, we show that the associated extended deformation functors are also equivalent. Corollary 5.16 then shows how our functor of derived deformations of a smooth scheme coincides with existing constructions.

Contents

Introduction 1 1. Derived deformation functors 2 1.1. Simplicial Artinian rings 2 1.2. Properties of morphisms 3 1.3. Derived deformation functors 3 1.4. Quotient spaces 5 1.5. Cohomology and obstructions 5 1.6. Model structures 6 2. Monads and comonads 8 2.1. Algebras and coalgebras 8 2.2. Bialgebras 9 3. Constructing SDCs 9 3.1. SDCs from bialgebraic structures 10 3.2. SDCs from diagrams 11 3.3. Constrained deformations 13 4. Extended deformation functors from SDCs 14 4.1. Deformations of morphisms 16 5. Comparison with [Pri2] 17 5.1. DGLAs 18 5.2. Cosimplicial groups 18 5.3. The final comparison 20 5.4. An application 21 References 21

1. Derived deformation functors

With the exception of §1.4, the definitions and results in this section can all be found in [Pri5]. Fix a complete local Noetherian ring Λ, with maximal ideal μ and residue field k.

1.1. Simplicial Artinian rings.

Definition 1.1. Let C_{Λ} denote the category of local Artinian Λ-algebras with residue field k. We define sC_{Λ} to be the category of Artinian simplicial local Λ-algebras, with residue field k.

Definition 1.2. Given a simplicial complex V_{•}, recall that the normalised chain complex N^{s}(V)_{•} is given by N^{s}(V)_{n} := ⋂_{i>0} ker(∂_{i} : V_{n} → V_{n−1}), with differential ∂_{0}. The simplicial Dold-Kan correspondence says that N^{s} gives an equivalence of categories between simplicial complexes and non-negatively graded chain complexes in any abelian category. Where no ambiguity results, we will denote N^{s} by N.

Lemma 1.3. A simplicial complex A_{•} of local Λ-algebras with residue field k and maximal ideal 𝔪(A)_{•} is Artinian if and only if:

(1) the normalisation N(cot A) of the cotangent space cot A := 𝔪(A)/(𝔪(A)^{2}+μ𝔪(A)) is finite-dimensional (i.e. concentrated in finitely many degrees, and finitedimensional in each degree).

(2) For some n > 0, 𝔪(A)^{n} = 0.

Proof. [Pri5] Lemma 1.16 □

As in [Gro], we say that a functor is left exact if it preserves all finite limits. This is equivalent to saying that it preserves final objects and fibre products.

Definition 1.4. Define Sp to be the category of left-exact functors from C_{Λ} to Set. Define cSp to be the category of left-exact functors from sC_{Λ} to Set.

Definition 1.5. Given a functor F : C_{Λ} → Set, we write F : sC_{Λ} → Set to mean A ↦ F (A_{0}) (corresponding to the inclusion Sp ↪ cSp).

1.2. Properties of morphisms.

Definition 1.6. As in [Man1], we say that a functor F : C_{Λ} → Set is smooth if for all surjections A → B in C_{Λ}, the map F (A) → F (B) is surjective.

Definition 1.7. We say that a map f : A → B in C_{Λ} is acyclic if π_{i}(f) : π_{i}(A) → π_{i}(B) is an isomorphism of Artinian Λ-modules for all i. f is said to be surjective if each f_{n} : A_{n} → B_{n} is surjective.

Note that for any simplicial abelian group A, the homotopy groups can be calculated by π_{i}A ≅ H_{i}(NA), the homology groups of the normalised chain complex. These in turn are isomorphic to the homology groups of the unnormalised chain complex associated to A.

Definition 1.8. We define a small extension e : I → A → B in sC_{Λ} to consist of a surjection A → B in sC_{Λ} with kernel I, such that 𝔪(A) · I = 0. Note that this implies that I is a simplicial complex of k-vector spaces.

Lemma 1.9. Every surjection in sC_{Λ} can be factorised as a composition of small extensions. Every acyclic surjection in sC_{Λ} can be factorised as a composition of acyclic small extensions.

Proof. [Pri5] Lemma 1.23. □

Definition 1.10. We say that a morphism α : F → G in cSp is smooth if for all small extensions A ↠ B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is surjective.

Similarly, we call α quasi-smooth if for all acyclic small extensions A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is surjective.

Lemma 1.11. A morphism α : F → G in Sp is smooth if and only if the induced morphism between the objects F, G ∈ cSp is quasi-smooth, if and only if it is smooth.

Proof. [Pri5] Lemma 1.31. □

1.3. Derived deformation functors.

Definition 1.12. Define the scSp to be the category of left-exact functors from sC_{Λ} to the category 𝕊 of simplicial sets.

Definition 1.13. A morphism α : F → G in scSp is said to be smooth if

(S1) for every acyclic surjection A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is

a trivial fibration in 𝕊; (S2) for every surjection A → B in sC_{Λ}, the map F (A) → F (B) ×_{G(B)} G(A) is a

surjective fibration in 𝕊.

A morphism α : F → G in scSp is said to be quasi-smooth if it satisfies (S1) and (Q2) for every surjection A → B in sC_{Λ}, the map F (A) → F (B)×_{G(B)}G(A) is a fibration

in 𝕊.

Definition 1.14. Given A ∈ sC_{Λ} and a finite simplicial set K, define A^{K} ∈ C_{Λ} by

(A^{K})_{i} := Hom_{𝕊}(K × Δ^{i}, A) ×_{Hom_{Set}(π_{0}K,k)} k.

Definition 1.15. Given F ∈ scSp, define \underline{F} : sC_{Λ} → 𝕊 by

\underline{F} (A)_{n} := F_{n}(A^{Δ^{n}}).

For F ∈ cSp, we may regard F as an object of scSp (with the constant simplicial structure), and then define \underline{F} as above.

Lemma 1.16. A map α : F → G in cSp is smooth (resp. quasi-smooth) if and only if the induced map of functors \underline{α} : \underline{F} → \underline{G} is smooth (resp. quasi-smooth) in scSp.

Proof. [Pri5] Lemma 1.36. □

The following Lemma will provide many examples of functors which are quasi-smooth but not smooth.

Lemma 1.17. If F → G is a quasi-smooth map of functors F, G : sC_{Λ} → 𝕊, and K → L is a cofibration in 𝕊, then

F^{L} → F^{K} ×_{G^{K}} G^{L}

is quasi-smooth.

Proof. This is an immediate consequence of the fact that 𝕊 is a simplicial model category, following from axiom SM7, as given in [GJ] §II.3. □

The following lemma is a consequence of standard properties of fibrations and trivial fibrations in 𝕊.

Lemma 1.18. If F → G is a quasi-smooth map of functors F, G : sC_{Λ} → 𝕊, and H → G is any map of functors, then F ×_{G} H → H is quasi-smooth.

Definition 1.19. A map α : F → G of functors F, G : C_{Λ} → 𝕊 is said to be smooth (resp. quasi-smooth, resp. trivially smooth) if for all surjections A ↠ B in C_{Λ}, the maps

F (A) → F (B) ×_{G(B)} G(A)

are surjective fibrations (resp. fibrations, resp. trivial fibrations).

Proposition 1.20. A map α : F → G of left-exact functors F, G : C_{Λ} → 𝕊 is smooth if and only if the maps F_{n} ⟶^{α_{n}} G_{n} of functors F_{n}, G_{n} : C_{Λ} → Set are all smooth.

Proof. [Pri5] Proposition 1.39. □

Proposition 1.21. If a morphism F ⟶^{α} G of left-exact functors F, G : sC_{Λ} → 𝕊 is such that the maps

θ : F (A) → F (B) ×_{G(B)} G(A)

are surjective fibrations for all acyclic small extensions A → B, then \underline{α} : \underline{F} → \underline{G} is quasismooth (resp. smooth) if and only if θ is a fibration (resp. surjective fibration) for all small extensions A → B.

Proof. [Pri5] Proposition 1.63. □

Definition 1.22. We will say that a morphism α : F → G of quasi-smooth objects of scSp is a weak equivalence if, for all A ∈ sC_{Λ}, the maps π_{i}F (A) → π_{i}G(A) are isomorphisms for all i.

1.4. Quotient spaces.

Definition 1.23. Given functors X : sC_{Λ} → 𝕊 and G : sC_{Λ} → sGp, together with a right action of G on X, define the quotient space by

[X/G]_{n} = (X ×^{G} W G)_{n} = X_{n} × G_{n−1} × G_{n−2} × … G_{0},

with operations as standard for universal bundles (see [GJ] Ch. V). Explicitly:

∂_{i}(x, g_{n−1}, g_{n−2}, …, g_{0}) = \begin{cases}(∂_{0}x ∗ g_{n−1}, g_{n−2}, …, g_{0}) & i = 0; \\ (∂_{i}x, ∂_{i−1}g_{n−1}, …, (∂_{0}g_{n−i})g_{n−i−1}, g_{n−i−2}, …, g_{0}) & 0 < i < n; \\ (∂_{n}x, ∂_{n−1}g_{n−1}, …, ∂_{1}g_{1}) & i = n; \end{cases}
σ_{i}(x, g_{n−1}, g_{n−2}, …, g_{0}) = (σ_{i}x, σ_{i−1}g_{n−1}, …, σ_{0}g_{n−i}, e, g_{n−i−1}, g_{n−i−2}, …, g_{0}).

The space [•/G] is also denoted W̄ G, and is a model for the classifying space BG of G. Note replacing W G with any other fibrant cofibrant contractible G-space EG will give the same properties.

Lemma 1.24. If G : sC_{Λ} → sGp is smooth, then W̄ G is smooth.

Proof. For any surjection A → B, we have G(A) → G(B) fibrant and surjective on π_{0}, which by [GJ] Corollary V.6.9 implies that W̄ G(A) → W̄ G(B) is a fibration. If A → B is also acyclic, then everything is trivial by properties of W̄ and G. □

Remark 1.25. Observe that this is our first example of a quasi-smooth functor which is not a right Quillen functor for the simplicial model structure. The definitions of smoothness and quasi-smoothness were designed with W̄ G in mind.

Lemma 1.26. If X is quasi-smooth, then so is [X/G] → W̄ G.

Proof. This follows from the observation that for any fibration (resp. trivial fibration) Z → Y of G-spaces, [Z/G] → [Y/G] is a fibration (resp. trivial fibration). □

Corollary 1.27. If X is quasi-smooth and G smooth, then [X/G] is quasi-smooth.

Proof. Consider the fibration X → [X/G] → W̄ G. □

1.5. Cohomology and obstructions. Given a quasi-smooth morphism α : F → G in scSp, there exist k-vector spaces H^{i}(F/G) for all i ∈ ℤ.

By [Pri5] Corollary 1.46, these have the property that for any simplicial k-vector space V with finite-dimensional normalisation,

π_{m}(F (k ⊕ V) ×_{G(k⊕V)} {0}) ≅ H^{−m}(F/G ⊗ V),

where V^{2} = 0 and

H^{i}(F/G ⊗ V) := ⨁_{n≥0} H^{i+n}(F/G) ⊗ π_{n}(V).

If G = • (the one-point set), we write H^{j}(F) := H^{j}(F/•).

We now have the following characterisation of obstruction theory:

Theorem 1.28. If α : F → G in scSp is quasi-smooth, then for any small extension e : I → A ⟶^{f} B in sC_{Λ}, there is a sequence of sets

π_{0}(F A) ⟶^{f_{∗}} π_{0}(F B ×_{GB} GA) ⟶^{o_{e}} H^{1}(F/G ⊗ I)

exact in the sense that the fibre of o_{e} over 0 is the image of f_{∗}. Moreover, there is a group action of H^{0}(F/G ⊗ I) on π_{0}(F A) whose orbits are precisely the fibres of f_{∗}.

For any y ∈ F_{0}A, with x = f_{∗}y, the fibre of F A → F B ×_{GB} GA over x is isomorphic to ker(α : F I → GI), and the sequence above extends to a long exact sequence

\xymatrix{⋯ \ar[r]^{f_{∗}} & π_{n}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & H^{1−n}(F/G ⊗ I) \ar[r]^{∂_{e}} & π_{n−1}(F A, y) \ar[r]^{f_{∗}} & ⋯ \\ ⋯ \ar[r]^{f_{∗}} & π_{1}(F B ×_{GB} GA, x) \ar[r]^{o_{e}} & H^{0}(F/G ⊗ I) \ar[r]^{−∗y} & π_{0}(F A).}

Proof. [Pri5] Theorem 1.45. □

Corollary 1.29. A map α : F → G of quasi-smooth F, G ∈ scSp is a weak equivalence if and only if the maps H^{j}(α) : H^{j}(F) → H^{j}(G) are all isomorphisms.

Corollary 1.30. If α : F → G is quasi-smooth in scSp, then α is smooth if and only if H^{i}(F/G) = 0 for all i > 0.

Proposition 1.31. Let X, Y, Z : sC_{Λ} → 𝕊 be left-exact functors, with X ⟶^{α} Y and Y ⟶^{β} Z quasi-smooth. There is then a long exact sequence

… ⟶^{∂} H^{j}(X/Y) → H^{j}(X/Z) → H^{j}(Y/Z) ⟶^{∂} H^{j+1}(X/Y) → H^{j+1}(X/Z) → …

Proof. [Pri5] Proposition 1.61. □

1.6. Model structures.

Theorem 1.32. There is a simplicial model structure on scSp (called the geometric model structure), for which the fibrations are quasi-smooth morphisms, and weak equivalences between quasi-smooth objects are those given in Definition 1.22.

Proof. This is [Pri5] Theorem 2.14. □

Thus the homotopy category Ho(scSp) is equivalent to the category of quasi-smooth objects in scSp, localised at the weak equivalences of Definition 1.22.

Definition 1.33. Given any morphism f : X → Z, we define H^{n}(X/Z) := H^{n}(X̂/Z), for X ⟶^{i} X̂ ⟶^{p} Z a factorisation of f with i a geometric trivial cofibration, and p a geometric fibration.

1.6.1. Homotopy representability.

Definition 1.34. Define the category S to consist of functors F : sC_{Λ} → 𝕊 satisfying the following conditions:

(A0) F (k) is contractible.

(A1) For all small extensions A ↠ B in sC_{Λ}, and maps C → B in sC_{Λ}, the map F (A×_{B} C) → F (A)×_{F (B)}^{h}F (C) is a weak equivalence, where ×^{h} denotes homotopy fibre product.

(A2) For all acyclic small extensions A ↠ B in sC_{Λ}, the map F (A) → F (B) is a weak

equivalence. Say that a natural transformation η : F → G between such functors is a weak equiva-

lence if the maps F (A) → G(A) are weak equivalences for all A ∈ sC_{Λ}, and let Ho(S) be the category obtained by formally inverting all weak equivalences in S.

Remark 1.35. We may apply the long exact sequence of homotopy to describe the homotopy groups of homotopy fibre products. If f : X → Z, g : Y → Z in 𝕊 and P = X ×_{Z}^{h} Y, the map θ : π_{0}(P) → π_{0}(X)×_{π_{0}(Z)} π_{0}(Y) is surjective. Moreover, π_{1}(Z, ∗) acts transitively on the fibres of θ over ∗ ∈ π_{0}Z.

Take v ∈ π_{0}(P) over ∗. Then there is a connecting homomorphism ∂ : π_{n}(Z, ∗) → π_{n−1}(P, v) for all n ≥ 1, giving a long exact sequence

… ⟶^{∂} π_{n}(P, v) → π_{n}(X, v) × π_{n}(Y, v) ⟶^{f·g^{−1}} π_{n}(Z, ∗) ⟶^{∂} π_{n−1}(P, v) ….

Theorem 1.36. There is a canonical equivalence between the geometric homotopy category Ho(scSp) and the category Ho(S).

Proof. This is [Pri5] Theorem 2.30. □

1.6.2. Equivalent formulations. If k is a field of characteristic 0, then we may work with dg algebras rather than simplicial algebras.

Definition 1.37. Define dgC_{Λ} to be the category of Artinian local differential ℕ_{0}-graded graded-commutative Λ-algebras with residue field k.

Definition 1.38. Define a map A → B in dgC_{Λ} to be a small extension if it is surjective and the kernel I satisfies I · 𝔪(A) = 0.

Definition 1.39. Define sDGSp to be the category of left exact functors from dgC_{Λ} to 𝕊.

Definition 1.40. Say a map X → Y in sDGSp is quasi-smooth if for all small extensions f : A → B in dgC_{Λ}, the morphism

X(A) → Y (A) ×_{Y (B)} X(B)

is a fibration in 𝕊, which is moreover a trivial fibration if f is acyclic.

Definition 1.41. We will say that a morphism α : F → G of quasi-smooth objects of sDGSp is a weak equivalence if, for all A ∈ sC_{Λ}, the maps π_{i}F (A) → π_{i}G(A) are isomorphisms for all i.

Proposition 1.42. There is a model structure on sDGSp, for which the fibrations are quasi-smooth morphisms, and weak equivalences between quasi-smooth objects are those given in Definition 1.41.

Proof. This is [Pri5] Proposition 4.12. □

Most of the constructions from sC_{Λ} carry over to dgC_{Λ}. However, there is no straightforward analogue of Definition 1.15.

Definition 1.43. Define the normalisation functor N : sC_{Λ} → dgC_{Λ} by mapping A to its associated normalised complex NA, equipped with the Eilenberg-Zilber shuffle product (as in [Qui]).

Definition 1.44. Define Spf N^{∗} : sDGSp → scSp by mapping X : dgC_{Λ} → 𝕊 to the composition X ∘ N : sC_{Λ} → 𝕊. Note that this is well-defined, since N is left exact.

Theorem 1.45. Spf N^{∗} : sDGSp → scSp is a right Quillen equivalence.

Proof. This is [Pri5] Theorem 4.18. □

In particular, this means that Spf N^{∗} maps quasi-smooth morphisms to quasi-smooth morphisms, and induces an equivalence RSpf N^{∗} : Ho(sDGSp) → Ho(scSp).

Now assume that Λ = k.

Theorem 1.46. Ho(sDGSp) is equivalent to the category of L_{∞}-algebras localised at tangent quasi-isomorphisms (as considered in [Kon]). This is also equivalent to the category of DGLAs (see §5.1) localised at quasi-isomorphisms.

Proof. Combine [Pri5] Proposition 4.42 and Corollary 4.57. □

2. Monads and comonads

2.1. Algebras and coalgebras.

Definition 2.1. A monad (or triple) on a category B is a monoid in the category of endofunctors of B (with the monoidal structure given by composition of functors). A comonoid (or cotriple) is a comonoid in the category of endofunctors of B.

The following is standard:

Lemma 2.2. Take an adjunction

\xymatrix{A \ar[r]^{G}_{⊤} & E \ar[l]^{F}}

with unit η : id → GF and co-unit ε : F G → id. Then ⊤ := GF is a monad with unit η and multiplication μ := GεF, while ⊥ := F G is a comonad, with co-unit ε and comultiplication Δ := F ηG.

Definition 2.3. Given a monad (⊤, μ, η) on a category E, define the category E^{⊤} of ⊤-algebras to have objects

⊤E ⟶^{θ} E,

such that θ ∘ η_{E} = id and θ ∘ ⊤θ = θ ∘ μ_{E}.

A morphism

g : (⊤E_{1} ⟶^{θ} E_{1}) → (⊤E_{2} ⟶^{ϕ} E_{2})

of ⊤-algebras is a morphism g : E_{1} → E_{2} in E such that ϕ ∘ ⊤g = g ∘ θ.

We define the comparison functor K : A → E^{⊤} by

B ↦ (UF UB ⟶^{Uε_{B}} UB)

on objects, and K(g) = U(g) on morphisms.

Definition 2.4. The adjunction

\xymatrix{A \ar[r]^{U}_{⊤} & E \ar[l]^{F}},

is said to be monadic (or tripleable) if K : A → E^{⊤} is an equivalence.

Examples 2.5. Intuitively, monadic adjunctions correspond to algebraic theories, such as the adjunction

\xymatrix{Ring \ar[r]^{U}_{⊤} & Set, \ar[l]^{ℤ[−]}}

between rings and sets, U being the forgetful functor. Other examples are k-algebras over k-vector spaces, or groups over sets.

Definition 2.6. Dually, given a comonad (⊥, Δ, ε) on a category A, we define the category A_{⊥} of ⊥-coalgebras by

(A_{⊥})^{opp} := (A^{opp})^{⊥^{opp}},

noting that ⊥^{opp} is a monad on the opposite category A^{opp}. The adjunction of Lemma 2.2 is said to be comonadic (or cotripleable) if the adjunction on opposite categories is monadic.

Example 2.7. If X is a topological space (or any site with enough points) and X′ is the set of points of X, let u : X′ → X be the associated morphism. Then the adjunction u^{−1} ⊣ u_{∗} on sheaves is comonadic, so the category of sheaves on X is equivalent to u^{−1}u_{∗}-coalgebras in the category of sheaves (or equivalently presheaves) on X′

A more prosaic example is that for any ring A, the category of A-coalgebras is comonadic over the category of A-modules.

2.2. Bialgebras. As in [Van] §IV, take a category B equipped with both a monad (⊤, μ, η) and a comonad (⊥, Δ, γ), together with a distributivity transformation λ : ⊤⊥ ⟹ ⊥⊤ for which the following diagrams commute:

\xymatrix{⊤⊥ \ar@{=>}[rr]^{λ} \ar@{=>}[d]_{⊤Δ} &  & ⊥⊤ \ar@{=>}[d]_{Δ⊤} & ⊤⊥ \ar@{=>}[rr]^{λ} &  & ⊥⊤ \\ ⊤⊥^{2} \ar@{=>}[r]^{λ⊥} & ⊥⊤⊥ \ar@{=>}[r]^{⊥λ} & ⊥^{2}⊤ & ⊤^{2}⊥ \ar@{=>}[u]_{μ⊥} \ar@{=>}[r]^{⊤λ} & ⊤⊥⊤ \ar@{=>}[r]^{λ⊤} & ⊥⊤^{2} \ar@{=>}[u]_{⊥μ}}

[figure: ⊤⊥; λ; ⊤γ; ⊥⊤; γ⊤; ⊤; ⊤⊥; λ; ⊥⊤; ⊥.; η⊥; ⊥η]

Definition 2.8. Given a distributive monad-comonad pair (⊤, ⊥) on a category B, define the category B_{⊥}^{⊤} of bialgebras as follows. The objects of B_{⊥}^{⊤} are triples (α, B, β) with

(⊤B ⟶^{α} B) an object of B^{⊤} and B ⟶^{β} ⊥B an object of B_{⊥}, such that the composition (β ∘ α) : ⊤B → ⊥B agrees with the composition

⊤B ⟶^{⊤β} ⊤⊥B ⟶^{λ} ⊥⊤B ⟶^{⊥α} ⊥B.

A morphism f : (α, B, β) → (α′, B′, β′) is a morphism f : B → B′ in B such that α′ ∘ ⊤f = f ∘ α and β′ ∘ f = ⊥f ∘ β.

To understand how the data (⊤, ⊥, η, μ, γ, Δ, λ) above occur naturally, note that by [Van] §IV or [Pri2] §2, these data are equivalent to a diagram

\xymatrix{D \ar[r]^{U}_{⊤} \ar[d]_{V} & E \ar[l]^{F} \ar[d]_{V} \\ A \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & B, \ar[l]^{F} \ar[u]_{G}^{⊣}}

with F ⊣ U monadic, G ⊢ V comonadic and U, V commuting with everything (although G and F need not commute). The associated monad is ⊤ = UF, and the comonad ⊥ = V G. Distributivity ensures that D ≃ E^{⊤} ≃ (B_{⊥})^{⊤} and D ≃ A_{⊥} ≃ (B^{⊤})_{⊥}. In other words, D ≃ B_{⊥}^{⊤}.

Example 2.9. If X is a topological space (or any site with enough points) and X′ is the set of points of X, let D be the category of sheaves of rings on X. If B is the category of sheaves (or equivalently presheaves) of sets on X′, then the description above characterises D as a category of bialgebras over B, with the comonad being u^{−1}u_{∗} for u : X′ → X, and the monad being the free polynomial functor.

3. Constructing SDCs

Recall the definition of an SDC:

Definition 3.1. A simplicial deformation complex E^{•} consists of smooth left-exact functors E^{n} : C_{Λ} → Set for each n ≥ 0, together with maps

\begin{matrix}∂^{i} : E^{n} → E^{n+1} & 1 ≤ i ≤ n \\ σ^{i} : E^{n} → E^{n−1} & 0 ≤ i < n, \end{matrix}

an associative product ∗ : E^{m} × E^{n} → E^{m+n}, with identity 1 : • → E^{0}, where • is the constant functor •(A) = • (the one-point set) on C_{Λ}, such that:

(1) ∂^{j}∂^{i} = ∂^{i}∂^{j−1} i < j.

(2) σ^{j}σ^{i} = σ^{i}σ^{j+1} i ≤ j.

(3) σ^{j}∂^{i} = \begin{cases}∂^{i}σ^{j−1} & i < j \\ id & i = j, i = j + 1 \\ ∂^{i−1}σ^{j} & i > j + 1 \end{cases} .

(4) ∂^{i}(e) ∗ f = ∂^{i}(e ∗ f).

(5) e ∗ ∂^{i}(f) = ∂^{i+m}(e ∗ f), for e ∈ E^{m}.

(6) σ^{i}(e) ∗ f = σ^{i}(e ∗ f).

(7) e ∗ σ^{i}(f) = σ^{i+m}(e ∗ f), for e ∈ E^{m}.

From the viewpoint of homotopical algebra, there is a more natural way of characterising the smoothness criterion for E^{•}. Analogously to [GJ] Lemma VII.4.9, we define matching objects by M^{−1}E := •, M^{0}E := E^{0}, and for n > 0

M^{n}E = {(e_{0}, e_{1}, …, e_{n}) ∈ (E^{n})^{n+1} | σ^{i}e_{j} = σ^{j−1}e_{i} ∀i < j}.

Proposition 3.2. The canonical maps \underline{σ} : E^{n+1} → M^{n}E, given by e ↦ (σ^{0}e, σ^{1}e, …, σ^{n}e), are all smooth, for n ≥ 0.

Proof. Since E^{n} is smooth, by the Standard Smoothness Criterion (e.g. [Man1] Proposition 2.17) it suffices to show that this is surjective on tangent spaces. The tangent space of M^{n}E consists of (n + 1)-tuples γ_{i} ∈ C^{n}(E) satisfying σ^{i}γ_{j} = σ^{j−1}γ_{i}, for i < j. For any cosimplicial complex C^{•}, there is a decomposition of the associated cochain complex as C^{n} = N_{c}^{n}(C)⊕D^{n}(C), where N_{c}^{n}(C) = ∩_{i=0}^{n−1} ker σ^{i}, and D^{n}(C) = ∑_{i=1}^{n} ∂^{i}C^{n−1}. Moreover \underline{σ} : D^{n} → M^{n−1}C is an isomorphism, giving the required surjectivity. □

Definition 3.3. Given an SDC E, recall from [Pri2] that the Maurer-Cartan functor MC_{E} : C_{Λ} → Set is defined by

MC_{E}(A) = {ω ∈ E^{1}(A): ω ∗ ω = ∂^{1}(ω)}.

The group E^{0}(A) acts on this by conjugation, and we define 𝔇𝔢𝔣_{E}(A) to be the groupoid with objects MC_{E}(A) and morphisms given by E^{0}(A) via this action. We say that an SDC governs a deformation problem if 𝔇𝔢𝔣_{E} is equivalent to the associated deformation functor.

Definition 3.4. Recall that C^{•}(E) denotes the tangent space of E^{•}, i.e. C^{n}(E) = E^{n}(k[ϵ]) for ϵ^{2} = 0. This has the natural structure of a cosimplicial complex, by [Pri2], and we set H^{i}(E) := H^{i}(C^{•}(E)).

3.1. SDCs from bialgebraic structures.

Definition 3.5. Recall from [Pri4] that Δ_{∗∗} is defined to be the subcategory of the ordinal number category Δ containing only those non-decreasing morphisms f : m → n with f(0) = 0, f(m) = n. We define a monoidal structure on this category by setting m ⊗ n = m + n, with

(f ⊗ g)(i) = \begin{cases}f(i) & i ≤ m \\ g(i − m) + n & i ≥ m, \end{cases}

for f : m → n.

Definition 3.6. As in [Pri4], define monoidal structures on Set^{Δ_{∗∗}} and 𝕊^{Δ_{∗∗}} by setting

(X ⊗ Y)^{n} := ∐_{a+b=n} X^{a} ⊗ Y^{b},

with operations given by

∂^{i}(x ⊗ y) = \begin{cases}(∂^{i}x) ⊗ y & i ≤ a \\ x ⊗ (∂^{i−a}y) & i > a; \end{cases}
σ^{i}(x ⊗ y) = \begin{cases}(σ^{i}x) ⊗ y & i < a \\ x ⊗ (σ^{i−a}y) & i ≥ a. \end{cases}

The identity I is given by I^{0} = • and I^{n} = ∅ for n > 0.

Note that an SDC over Λ is a smooth left-exact functor from C_{Λ} to the category of monoids in Set^{Δ_{∗∗}}.

Assume that we have a diagram

\xymatrix{D \ar[r]^{U}_{⊤} \ar[d]_{V} & E \ar[l]^{F} \ar[d]_{V} \\ A \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & B, \ar[l]^{F} \ar[u]_{G}^{⊣}}

of homogeneous (i.e. preserving fibre products, but not the final object) functors from C_{Λ} to Cat as in [Pri2] §2 (i.e. B has uniformly trivial deformation theory, with the diagram satisfying the conditions of §2.2). Recall that we write ⊤_{h} = UF, ⊥_{h} = F U, ⊥_{v} = V G, and ⊤_{v} = GV.

Proposition 3.7. For the diagram above and A ∈ C_{Λ}, B(A) has the structure of category enriched in Set^{Δ_{∗∗}}, with

Hom_{B}(B, B′)^{n} = Hom_{B}(⊤_{h}^{n}B, ⊥_{v}^{n}B′).

Proof. [Pri4] Proposition 2.12. □

Examples 3.8. (1) If X is a topological space (or any site with enough points) and X′ is the set of points of X, let D(A) be the category of sheaves of flat A-algebras on X′. If B is the category of sheaves (or equivalently presheaves) of flat A-modules on X′, then the description above characterises D as B_{⊥_{v}}^{⊤_{h}}, with ⊥_{v} = u^{−1}u_{∗} for u : X′ → X, and ⊤_{h} being the free A-algebra functor for module. This example arises when considering deformations of a scheme X in [Pri2] §3.2, since deformations of X are equivalent to deformations of the sheaf 𝒪_{X} of algebras.

(2) Another important example (considered in [Pri2] §3.1) is when D(A) is the category of flat Hopf algebras over A, with A(A), E(A) and B(A) the categories of flat algebras, coalgebras and modules, respectively.

(3) In order to make the first example functorial, we could let B be the category of pairs ({M_{x}}_{x∈X}, X), for X a topological space and {M_{x}}_{x∈X} a presheaf of flat A-modules on X′, with a morphism f^{♯} : ({N_{y}}_{y∈Y} , Y) → ({M_{x}}_{x∈X}, X) given by a map f : X → Y of topological spaces, together with maps f_{x}^{♯} : N_{f(x)} → M_{x} for all x ∈ X.

We may define ⊥_{v} and ⊤_{h} as before, and then B_{⊥_{v}}^{⊤_{h}} will be the category of pairs (𝒪_{X}, X), where X is a topological space and 𝒪_{X} a sheaf of flat A-algebras on X.

3.2. SDCs from diagrams.

Definition 3.9. Given a morphism f : D → D′ in D(k), choose lifts B, B′ ∈ B(Λ) of UV D, UV D′ ∈ B(k) (which exist since the deformation theory of B is uniformly trivial). Then define

E_{D/B}^{n}(f) := ℋ om^{n}(B, B′)_{UV (α_{D′}^{n}∘f∘ε_{D}^{n})} : C_{Λ} → Set,

where α : 1 → ⊤_{v} and ε : ⊥_{h} → 1 are the unit and counit of the respective adjunctions.

Write E_{D/B}^{∗}(D) := E_{D/B}^{∗}(id_{D}). Note that uniform triviality of B ensures that these constructions are independent of the choices of lift, since any other choice is isomorphic.

Lemma 3.10. E_{D/B}(f)(A) has the natural structure of a cosimplicial complex.

For every pair of composable morphisms f, g in D(k) between such objects, there is a product

E_{D/B}(f)(A) ⊗ E_{D/B}(g)(A) → E_{D/B}(f ∘ g)(A)

in Set^{Δ_{∗∗}}, functorial in A.

Proof. It follows from Proposition 3.7 that E_{D/B}(f)(A) ∈ Set^{Δ_{∗∗}}, with operations

∂^{i}(x) = ⊥_{v}^{i−1}V α_{G⊥_{v}^{n−i}B} ∘ x ∘ ⊤_{h}^{i−1}Uε_{F ⊤_{h}^{n−i}B}, 0 < i ≤ n
σ^{i}(x) = ⊥_{v}^{i}γ_{⊥_{v}^{n−i−1}B} ∘ x ∘ ⊤_{h}^{i}η_{⊤_{h}^{n−i−1}B}, 0 ≤ i < n,

for η : 1 → ⊤_{h} and γ : ⊥_{v} → 1 the respective unit and co-unit. The multiplication also follows from Proposition 3.7.

The canonical object of MC(E_{D/B}(id_{D})(Λ)) corresponding to D gives an element ω_{D} ∈ E_{D/B}(id_{D})(Λ)^{1} and we then enhance the structure above to give a cosimplicial structure by setting

∂^{0}x := ω_{D′} ∗ x ∂^{n+1}x = x ∗ ω_{D},

for x ∈ E_{D/B}(f)(A)^{n}. □

Definition 3.11. Given an SDC E, and a simplicial set X, define an SDC E^{X} by

(E^{X})^{n} = (E^{n})^{X_{n}}.

For x ∈ X_{n+1}, y ∈ Y_{n+1}, z ∈ X_{m+n}, 1 ≤ i ≤ n, 0 ≤ j < n, e ∈ (E^{X})^{n} and f ∈ (E^{X})^{m}, we define the operations by

∂^{i}(e)(x) := ∂^{i}(e(∂_{i}x))
σ^{j}(e)(y) := σ^{j}(e(σ_{i}y)),
(f ∗ e)(z) := f((∂_{m+1})^{n}z) ∗ e((∂_{0})^{m}z).

Definition 3.12. Let C_{D/B}^{•}(f) be the the tangent space of E_{D/B}(f). This is a vector space over k, and we define

Ext_{D/B}^{∗}(f) := H^{∗}(C_{D/B}^{•}(f));

this construction is closely related to Van Osdol’s bicohomology ([Van]).

Lemma 3.13. If X is a finite simplicial set, then

H^{n}(E^{X}) ≅ ⨁_{i+j=n} H^{i}(E) ⊗ H^{j}(X, k).

Proof. Since X is finite, C^{•}(E^{X}) ≅ C^{•}(E) ⊗ k^{X}, and the result now follows from the Künneth formula. □

Definition 3.14. Given a small category 𝕀 and an 𝕀-diagram 𝔻 : 𝕀 → D(k), define the SDC E_{D/B}^{•}(𝔻) by

E_{D/B}^{n}(𝔻) = ∏_{\begin{matrix}i_{0}⟶^{f_{1}}i_{1}⟶^{f_{2}}...⟶^{fn}i_{n} \\ in 𝕀 \end{matrix}} E^{n}(𝔻(f_{n} ∘ f_{n−1} ∘ … f_{0})) = ∏_{x∈B𝕀_{n}} E^{n}(𝔻(∂_{1}^{n−1}x)),

where B𝕀 is the nerve of 𝕀 (so B𝕀_{0} = Ob (𝕀), B𝕀_{1} = Mor (𝕀)), and ∂_{1}^{−1} := σ_{0}. We define the operations by the formulae of Definition 3.11.

Theorem 3.15. The SDC E_{D/B}^{•}(𝔻) governs deformations of the diagram 𝔻 : 𝕀 → D(k).

Proof. This follows immediately from [Pri4] Lemma 1.36, which characterises objects of 𝔇𝔢𝔣_{E} as diagrams from 𝕀 to a category equivalent to D(A). □

Lemma 3.16. Given a diagram 𝔻 : 𝕀 → D(k), the cohomology groups H^{∗}(E_{D/B}^{•}(𝔻)) are given by hypercohomology of the bicomplex

∏_{i∈Ob 𝕀} C_{D/B}^{•}(𝔻(i)) ⟶^{f_{∗}−f^{∗}} … → ∏_{\begin{matrix}i_{0}⟶^{f_{1}}i_{1}⟶^{f_{2}}...⟶^{fn}i_{n} \\ in N_{n}𝕀 \end{matrix}} C_{D/B}^{•}(𝔻(f_{n} ∘ f_{n−1} ∘ … f_{0}))) → …,

where N_{n}𝕀 ⊂ B_{n}𝕀 consists of non-degenerate simplices, or equivalently strings of nonidentity morphisms.

Proof. Since C^{•}(E_{D/B}^{•}(𝔻)) is the diagonal of the bicosimplicial complex

[m, n] ↦ ∏_{\begin{matrix}i_{0}⟶^{f_{1}}i_{1}⟶^{f_{2}}...⟶^{fn}i_{n} \\ in 𝕀 \end{matrix}} C_{D/B}^{m}(𝔻(f_{n} ∘ f_{n−1} ∘ … f_{0})),

the Eilenberg-Zilber theorem implies that it is homotopy equivalent to the total complex of the associated binormalised complex. The vertical normalisation is just given by replacing B_{n}𝕀 with N_{n}𝕀. □

Example 3.17. If we define D and B as in Example 3.8.3, then the category of flat schemes over A is a full subcategory of D(A), closed under deformations. Therefore Theorem 3.15 constructs an SDC governing deformations of a diagram of schemes.

For a morphism f : (X, 𝒪_{X}) → (Y, 𝒪_{Y}) in D(k)^{opp}, the reasoning of [Pri2] §3.2 adapts to show that

Ext_{D/B}^{∗}(f) = 𝔼xt_{𝒪_{Y}}^{∗} (L_{•}^{Y/k}, Rf_{∗}𝒪_{X}) = 𝔼xt_{𝒪_{X}}^{∗} (f^{∗}L_{•}^{Y/k}, 𝒪_{X}),

where L_{•}^{Y/k} is the cotangent complex of [Ill].

3.3. Constrained deformations. We now consider a generalisation of §3.2, by taking a small diagram

𝔻 : 𝕀 → D(k),

a subcategory 𝕁 ⊂ 𝕀, and 𝔻̃|_{𝕁} : 𝕁 → D(Λ) lifting 𝔻|_{𝕁}. We wish to describe deformations of 𝔻 which agree with 𝔻̃|_{𝕁} on 𝕁. Note that when 𝕀 = (0 → 1) and 𝕁 = {1}, this is the type of problem considered in [FM] and [Ran].

Proposition 3.18. Given a 𝕀-diagram 𝔻 : 𝕀 → D(k), with 𝔻̃|_{𝕁} as above, the groupoid of deformations of 𝔻 fixing 𝔻̃|_{𝕁} is governed by the SDC

E_{D/B}^{•}(𝔻) ×_{E_{D/B}^{•}(𝔻|_{𝕁})} •,

where • → E_{D/B}^{•}(𝔻|_{𝕁}) is defined by the object of MC(E_{D/B}^{•}(𝔻|_{𝕁})) corresponding to 𝔻̃|_{𝕁}.

Proof. By Theorem 3.15, it suffices to show that 𝔇𝔢𝔣(E_{D/B}^{•}(𝔻) ×_{E_{D/B}^{•}(𝔻|_{𝕁})} •) is equivalent to the 2-fibre product

𝔇𝔢𝔣(E_{D/B}^{•}(𝔻)) ×_{𝔇𝔢𝔣(E^{•}(𝔻|_{𝕁}))}^{h} ({𝔻̃|_{𝕁}}, id).

We know that the functor MC preserves inverse limits, so

MC(E^{•}(𝔻) ×_{E^{•}(𝔻|_{𝕁})} {𝔻̃|_{𝕁}}) = MC(E_{D/B}^{•}(𝔻)) ×_{MC(E^{•}(𝔻|_{𝕁}))} {𝔻̃|_{𝕁}}.

Since E^{0}(𝔻)(A) → E^{0}(𝔻|_{𝕁})(A) is also surjective (by smoothness), we see that

𝔇𝔢𝔣(E^{•}(𝔻) ×_{E^{•}(𝔻|_{𝕁})} {𝔻̃|_{𝕁}}) ≃ 𝔇𝔢𝔣(E_{D/B}^{•}(𝔻)) ×_{𝔇𝔢𝔣(E^{•}(𝔻|_{𝕁}))}^{h} ({𝔻̃|_{𝕁}}, id).
□

Example 3.19. Given a morphism f : X → Y of schemes over k, and a flat formal deformation 𝔜 of Y over Λ, we may consider deformations of X over 𝔜, or equivalently deformations of this diagram fixing 𝔜. Define the diagram 𝔻 to be f^{♯} : (𝒪_{Y} , Y) → (𝒪_{X}, X) in the category D(k) of Example 3.8.3, and let 𝔻̃|_{𝕁} be the object (𝒪_{𝔜}, 𝔜) of D(Λ). Proposition 3.18 then gives an SDC

E := E_{D/B}^{•}(𝔻) ×_{E_{D/B}^{•}(𝔻|_{𝕁})} •

governing this problem.

Lemma 3.16 implies that the tangent complex C^{•}(E) is the mapping cone of C_{D/B}^{•}(𝒪_{X}, X) → C_{D/B}^{•}(f^{♯}), so by Example 3.17, the cohomology of this SDC is given by

H^{∗}(E) ≅ 𝔼xt_{𝒪_{X}}^{∗} (cone(f^{∗}L_{•}^{Y/k} → L_{•}^{X/k}), 𝒪_{X}) ≅ 𝔼xt_{𝒪_{X}}^{∗} (L_{•}^{X/Y} , 𝒪_{X}).

Example 3.20. We could go further, and let 𝔼 be a diagram Z ⟶^{g} X ⟶^{f} Y over k, with a fixed formal deformation g̃f : ℨ → 𝔜 of fg over Λ. Governing this deformation problem, we get another SDC

F := E_{D/B}^{•}(Z ⟶^{g} X ⟶^{f} Y) ×_{E_{D/B}^{•}(Z⟶^{gf}Y)} \{ℨ \underrightarrow{g̃f} 𝔜\}.

Now, C^{•}(F) = ker(C^{•}(E_{D/B}(𝔼)) → C^{•}(E_{D/B}(𝔼|_{𝕁}))), and Lemma 3.16 implies that C^{•}(E_{D/B}(𝔼)) is homotopy equivalent to the total complex of

C_{D/B}^{•}(Z) × C_{D/B}^{•}(X) × C_{D/B}^{•}(Y) → C_{D/B}^{•}(g) × C_{D/B}^{•}(fg) × C_{D/B}^{•}(f) → C_{D/B}^{•}(fg),

while C^{•}(E_{D/B}(𝔼|_{𝕁})) is homotopy equivalent to the total complex of

C_{D/B}^{•}(Z) × C_{D/B}^{•}(Y) → C_{D/B}^{•}(fg),

so C^{•}(F) is homotopy equivalent to the total complex of

C_{D/B}^{•}(X) → C_{D/B}^{•}(g) × C_{D/B}^{•}(f) → C_{D/B}^{•}(fg).

By Example 3.17,

H^{∗} ker(C_{D/B}^{•}(X) → C_{D/B}^{•}(f)) ≅ 𝔼xt_{𝒪_{X}}^{∗} (L_{•}^{X/Y} , 𝒪_{X})
H^{∗} ker(C_{D/B}^{•}(g) → C_{D/B}^{•}(fg)) ≅ 𝔼xt_{𝒪_{X}}^{∗} (L_{•}^{X/Y} , Rg_{∗}𝒪_{Z}),

and these isomorphisms combine to give

H^{∗}(F) ≅ 𝔼xt_{𝒪_{X}}^{∗} (L_{•}^{X/Y} , cone(𝒪_{X} → Rg_{∗}𝒪_{Z})[−1]).

Note that this more accurately captures the higher structure than the SDC of [Pri2] §3.3, whose cohomology had g_{∗}𝒪_{Z} in place of Rg_{∗}𝒪_{Z} above.

4. Extended deformation functors from SDCs

Given an SDC E, the aim of this section is to extend the classical deformation groupoid 𝔇𝔢𝔣_{E} : C_{Λ} → Grpd of [Pri2] from C_{Λ} to the whole of sC_{Λ}. Groupoids turn out to be too restrictive for our purposes, so we will define a simplicial set-valued functor functor Def_{E} : sC_{Λ} → 𝕊 extending the classifying space B𝔇𝔢𝔣_{E} of the deformation groupoid.

For a monad ⊤, the obvious extension of the functor describing deformations of a ⊤- algebra is the functor of deformations of a strong homotopy ⊤-algebra. Strong homotopy algebras were defined by Lada in [CLM] to characterise the structures arising on deformation retracts of ⊤-algebras in topological spaces, but the description works over any simplicial category. This motivates the following definition:

Definition 4.1. Given an SDC E, define the Maurer-Cartan functor MC_{E} : sC_{Λ} → Set by

MC_{E}(A) ⊂ ∏_{n≥0} E^{n+1}(A^{I^{n}}),

consisting of those \underline{ω} satisfying:

ω_{m}(s_{1}, …, s_{m}) ∗ ω_{n}(t_{1}, …, t_{n}) = ω_{m+n+1}(s_{1}, …, s_{m}, 0, t_{1}, …, t_{n});
∂^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n+1}(t_{1}, …, t_{i−1}, 1, t_{i}, …, t_{n});
σ^{i}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{i−1}, min{t_{i}, t_{i+1}}, t_{i+2}, …, t_{n});
σ^{0}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{2}, …, t_{n});
σ^{n−1}ω_{n}(t_{1}, …, t_{n}) = ω_{n−1}(t_{1}, …, t_{n−1}),
σ^{0}ω_{0} = 1,

where I := Δ^{1}.

Remarks 4.2. (1) One way to think of this construction is that, if we start with an element ω ∈ E^{1} such that σ^{0}ω = 1, then there are 2^{n} elements generated by ω in each E^{n+1}. To see this correspondence, take a vector in {0, 1}^{n}, then substitute “ω∗” for each 0, and “∂^{1}” for each 1, adding a final ω. These elements will be at the vertices of an n-cube, and ω_{n} is then a homotopy between them.

(2) Lada’s definition of a strong homotopy algebra differs slightly in that it omits all of the degeneracy conditions except σ^{0}ω_{0} = 1. Our choices are made so that we work with normalised, rather than unnormalised, cochain complexes associated to a cosimplicial complex. Since these are homotopy equivalent, both constructions will yield weakly equivalent deformation functors, even if we remove all degeneracy conditions.

(3) In [Pri4] Proposition 3.11 it is shown that \underline{MC} has a precise homotopy-theoretical interpretation as the derived functor associated to the functor sending an SDC E and A ∈ C_{Λ} to the set MC_{E}(A) from Definition 3.3. In the scenario of §3, it follows from the results of [Pri4] that for A ∈ sC_{Λ}, MC_{E}(A) is the set of objects of the Segal space of strong homotopy bialgebras over the object being deformed.

Proposition 4.3. MC_{E} : sC_{Λ} → Set is quasi-smooth. Moreover, if f : E → F is a morphism of SDCs such that the maps f^{n} : E^{n} → F^{n} are smooth for all n, then MC_{E} → MC_{F} is quasi-smooth.

Proof. This follows immediately from [Pri4] Lemma 3.9. □

Definition 4.4. By [Pri1] Lemma 1.5, E^{0} is a group, which we denote by G_{E}. Observe that G_{E} acts on MC_{E} by (g, ω) ↦ g ∗ ω ∗ g^{−1}. We now define the deformation functor Def_{E} : sC_{Λ} → 𝕊 by Def_{E} := [\underline{MC}_{E}/\underline{G}_{E}], for \underline{X} as in Definition 1.15, and [−, −] the homotopy quotient of §1.4.

Proposition 4.5. If A ∈ C_{Λ}, then Def_{E}(A) is just the classifying space B𝔇𝔢𝔣_{E}(A) ∈ 𝕊 of the deformation groupoid 𝔇𝔢𝔣_{E}(A) from Definition 3.3.

Proof. Take \underline{ω} ∈ MC_{E}(A). Since A ∈ C_{Λ}, A^{K} = A for all connected simplicial sets K, so E^{n+1}(A^{I^{n}}) = E^{n+1}(A), and ω_{n} = ω_{0}^{∗(n+1)}, with the Maurer-Cartan relations reducing to

∂^{1}ω_{0} = ω_{0} ∗ ω_{0} σ^{0}ω_{0} = 1.

These are precisely the conditions defining the Maurer-Cartan space MC_{E}(A) of Definition 3.3 , and 𝔇𝔢𝔣_{E}(A) is the groupoid given by the action of E^{0}(A) on MC_{E}(A), as required.

□

Proposition 4.6. The functor Def_{E} is quasi-smooth. More generally, if f : E → F is a morphism of SDCs, such that f^{n} : E^{n} → F^{n} is smooth for all n, then Def_{E} → Def_{F} is quasi-smooth.

Proof. This follows immediately from Corollary 1.27. □

Proposition 4.7. The cohomology groups H^{j}(Def_{E}) are isomorphic to the groups H^{j+1}(E) from Definition 3.4.

Proof. This follows immediately from [Pri4] Corollary 4.13. □

4.1. Deformations of morphisms. The problem which we now wish to consider is that of deforming a morphism with fixed endpoints. Assume that we have a category-valued functor D : C_{Λ} → Cat. Fix objects D, D′ in D(Λ), and a morphism f in D(k) from D to D′. The deformation problem which we wish to consider is to describe, for each A ∈ C_{Λ}, the set of morphisms f_{A} : D → D′ in D(A) deforming f. This amounts to taking the special case 𝕀 = (0 → 1) and 𝕁 = {0, 1} in §3.3.

Now assume that we have a diagram of functors from C_{Λ} to Cat as in §3, and consider the cosimplicial complex F^{•} in Sp given by F^{•} := E_{D/B}^{•}(f) from Lemma 3.10.

On sC_{Λ}, we now define a deformation functor

Def_{F} (A) ⊂ ∏_{n≥0} \underline{F}^{n}(A)^{Δ^{n}},

associated to F, to consist of those \underline{θ} satisfying:

∂^{i}θ_{n} = ϵ_{n+1−i}^{∗}θ_{n+1}
σ^{i}θ_{n} = η_{n−1−i}^{∗}θ_{n−1},

for face maps ϵ_{i} : Δ^{n} → Δ^{n+1} and degeneracy maps η_{i} : Δ^{n} → Δ^{n−1} defined as in [Wei] Ch.8.

Proposition 4.8. Def_{F} is quasi-smooth, and H^{i}(Def_{F}) ≅ H^{i}(F).

Proof. The first statement follows from [GJ] §VII.5, which shows that the total space functor Tot from cosimplicial simplicial sets to simplicial sets is right Quillen. The description of cohomology is straightforward. □

Proposition 4.9. If 𝕀 is the category (0 ⟶^{m} 1), let 𝔻 : 𝕀 → D(k) be the functor given by 𝔻(0) = D, 𝔻(1) = D′ and 𝔻(m) = f, then there is a canonical weak equivalence

Def_{F} ≃ Def(E_{D/B}(𝔻) ×_{E_{D/B}(D)×E_{D/B}(D′)} •),

where • → E_{D/B}(D) × E_{D/B}(D′) is defined by the object (D, D′) ∈ MC(E^{•}(D) × E^{•}(D′))(Λ).

Thus Def_{F} governs deformations of f which fix D, D′.

Proof. Let C := E_{D/B}(𝔻) ×_{E_{D/B}(D)×E_{D/B}(D′)} •. By [Pri4] Lemma 5.10, there are canonical equivalences \underline{MC}(C)(A) ≃ Def(F)(A), so we need only observe that C^{0} = 1, so Def(C) = \underline{MC}(C). The final statement then follows from Proposition 3.18. □

4.1.1. Deforming identity morphisms. If we now consider deformations of the morphism id_{D} : D → D, write F for the cosimplicial complex E_{D/B}^{•}(id_{D}) governing deformations of id_{D}, and E for the SDC describing deformations of D, as defined in [Pri2] §2 (or just by taking the special case 𝕀 = • of Definition 3.14). Note that E^{n} = F^{n}, with the operations agreeing whenever they are defined on both. If we write e := ∂^{0}1 ∈ F^{1}, note that we also have ∂^{0}f = e ∗ f and ∂^{n+1}f = f ∗ e for f ∈ F^{n}.

This gives us an isomorphism C^{•}(E) ≅ C^{•}(F), and hence H^{n}(Def_{E}) = H^{n+1}(E) ≅ H^{n+1}(Def_{F}).

Proposition 4.10. Under the scenario above, the simplicial set Def_{F} (A) is weakly equivalent to the loop space ΩDef_{E}(A) of Def_{E}(A) over the point ω_{D} ∈ Def_{D}(Λ). This equivalence is functorial in A ∈ sC_{Λ}

Proof. Define the SDC P E to be the fibre of ev_{0} : E^{I} → E over the constants {e^{n}}. It follows from Lemma 3.13 that the cohomology groups of P E are all 0. Now define the SDC ΩE to be the fibre of ev_{1} : P E → E over {e^{n}}.

By Proposition 4.9, Def_{F} is weakly equivalent to Def(ΩE). By Proposition 4.6, Def_{P E} → Def_{E} is quasi-smooth, and the fibre is Def_{ΩE}. Since Def_{P E} is contractible, this means that Def_{ΩE} is homotopic to the loop space of Def_{E}. □

Remark 4.11. Note that we can describe ΩE entirely in terms of the structure on F, since

(ΩE)^{n} = (F^{n})^{n},

with

∂^{i}(f_{1}, …, f_{n}) = (∂^{i}f_{1}, ∂^{i}f_{2}, …, ∂^{i}f_{i}, ∂^{i}f_{i}, …, ∂^{i}f_{n})
σ^{i}(f_{1}, …, f_{n}) = (σ^{i}f_{1}, σ^{i}f_{2}, …, σ^{i}f_{i}, σ^{i}f_{i+2}, …, σ^{i}f_{n})
(g_{1}, …, g_{m}) ∗ (f_{1}, … f_{n}) = (g_{1} ∗ e^{n}, …, g_{m} ∗ e^{n}, e^{m} ∗ f_{1}, … e^{m} ∗ f_{n}).

Now, given any smooth object F ∈ cSp, we may regard F as a cosimplicial complex of smooth objects in Sp (as in [Pri5] Definition 1.20), and then

\underline{F} = Def(F) ≃ Def(ΩE).

This means that we cannot expect derived deformation functors coming from SDCs to have any more structure than arbitrary deformation functors.

Remark 4.12. In the case of Hochschild cohomology, the deformation functor of a morphism R ⟶^{f} S of associative algebras can be defined over the category of Artinian associative algebras, rather than just C_{Λ}. This means that the Lie bracket H^{i}(f)×H^{j}(f) → H^{i+j+1}(f) defined in [Pri5] §5.2 extends to an associative cup product. If f = id_{R} is an identity, then we know that the Lie bracket vanishes (since Def_{f} is a loop space, by Proposition 4.9), which is why the cup product becomes commutative. Of course, we also have the bracket H^{i}(id_{R}) × H^{j}(id_{R}) → H^{i+j}(id_{R}) associated to the deformation functor of the object R.

5. Comparison with [Pri2]

Now assume that Λ = k, a field of characteristic 0. In [Pri5], an equivalence was given between the homotopy category of ℤ-graded DGLAs and Ho(scSp). Under the equivalences of Theorems 1.46 and 1.36, this equivalence sends a DGLA to its associated deformation functor in the sense of [Man2] (by [Pri5] Remark 4.46). By [Pri5] Proposition 3.27, the corresponding functor from dgC_{k} to 𝕊 is equivalent to Hinich’s simplicial nerve:

Definition 5.1. Given a DGLA L, recall from [Hin] Definition 8.1.1 that the simplicial nerve Σ(L) : dgC_{k} → 𝕊 is defined by

Σ(L)(A)_{n} := MC(L ⊗ 𝒜_{n})(A),

where 𝒜_{n} is is the algebra of polynomial differential forms on the standard n-simplex Δ^{n} (denoted Ω_{n} in [Hin]).

However, in [Pri2], a functor E was constructed from ℕ_{0}-graded DGLAs to SDCs, and Definition 4.4 then gives us an associated object of scSp. The purpose of this section is to show that the two constructions are consistent with each other.

5.1. DGLAs.

Definition 5.2. A differential graded Lie algebra (DGLA) is a graded vector space L = ⨁_{i} L^{i} over k, equipped with operators [,] : L × L → L bilinear and d : L → L linear, satisfying:

(1) [L^{i}, L^{j}] ⊂ L^{i+j}.

(2) [a, b] + (−1)^{āb̄}[b, a] = 0.

(3) (−1)^{c̄ā}[a, [b, c]] + (−1)^{āb̄}[b, [c, a]] + (−1)^{b̄c̄}[c, [a, b]] = 0.

(4) d(L^{i}) ⊂ L^{i+1}.

(5) d ∘ d = 0.

(6) d[a, b] = [da, b] + (−1)ā[a, db]

Here ā denotes the degree of a, mod 2, for a homogeneous.

A DGLA is said to be nilpotent if the lower central series Γ_{n}L (given by Γ_{1}L = L, Γ_{n+1}L = [L, Γ_{n}L]) vanishes for n ≫ 0.

Definition 5.3. Given a nilpotent Lie algebra 𝔤, define Û(𝔤) to be the pro-unipotent completion of the universal enveloping algebra of 𝔤, regarded as a pro-object in the category of algebras. As in [Qui] Appendix A, this is a pro-Hopf algebra, and we define exp(𝔤) ⊂ Û(𝔤) to consist of elements g with ε(g) = 1 and Δ(g) = g ⊗ g, for ε : Û(𝔤) → k the augmentation (sending 𝔤 to 0), and Δ : Û(𝔤) → Û(𝔤) ⊗ Û(𝔤) the comultiplication.

Since k is assumed to have characteristic 0, exponentiation gives an isomorphism from 𝔤 to exp(𝔤), so we may regard exp(𝔤) as having the same elements as 𝔤, but with multiplication given by the Campbell–Baker–Hausdorff formula.

Definition 5.4. Given a nilpotent DGLA L^{•}, define the Maurer-Cartan set by

MC(L) := \{ω ∈ L^{1} | dω + \frac{1}{2}[ω, ω] = 0 ∈ L^{2}\}

Define the gauge group Gg(L) by Gg(L) := exp(L^{0}), which acts on MC(L) by the gauge action

g(ω) = g · ω · g^{−1} − dg · g^{−1},

where · denotes multiplication in the universal enveloping algebra of L. That g(ω) ∈ MC(L) is a standard calculation (see [Kon] or [Man1]).

Definition 5.5. A morphism f : L → M of DGLAs is said to be a quasi-isomorphism if H^{∗}(f) : H^{∗}(L) → H^{∗}(M) is an isomorphism.

Proposition 5.6. There is a model structure on the category of ℤ-graded DGLAs, in which weak equivalences are quasi-isomorphisms, and fibrations are surjections. This category is Quillen-equivalent to the model category sDGSp of Definition 1.39.

Proof. [Pri5] Lemma 3.24 and Corollary 4.57. □

5.2. Cosimplicial groups.

Definition 5.7. Given an ℕ_{0}-graded DGLA L, let DL be its denormalisation. This becomes a cosimplicial Lie algebra via the Eilenberg-Zilber shuffle product. Explicitly:

D^{n}L := ⨁_{\begin{matrix}m+s=n \\ 1≤j_{1}<...<j_{s}≤n \end{matrix}} ∂^{j_{s}} … ∂^{j_{1}}L^{m},

where we define the ∂^{j} and σ^{i} using the simplicial identities, subject to the conditions that σ^{i}L = 0 and ∂^{0}v = dv − ∑_{i=1}^{n+1}(−1)^{i}∂^{i}v for all v ∈ L^{n}.

We now have to define the Lie bracket ⟦−, −⟧ from D^{n}L ⊗ D^{n}L to D^{n}L. Given a finite set I of strictly positive integers, write ∂^{I} = ∂^{i_{s}} … ∂^{i_{1}}, for I = {i_{1}, … i_{s}}, with 1 ≤ i_{1} < … < i_{s}. The Lie bracket is then defined on the basis by

⟦∂^{I}v, ∂^{J}w⟧ := \begin{cases}∂^{I∩J}(−1)^{(J\I,I\J)}[v, w] & |v| = |J\I|, |v| = |I\J|, \\ 0 & otherwise, \end{cases}

where for disjoint sets S, T of integers, (−1)^{(S,T)} is the sign of the shuffle permutation of S ⊔ T which sends the first |S| elements to S (in order), and the remaining |T | elements to T (in order). Note that this description only works for 0 ∉ I ∪ J. Definition 5.8. Now recall from [Pri2] §4.2, that the functor E : DGLA → SDC from ℕ_{0}-graded DGLAs to SDCs is defined by

E(L)^{n}(A) = exp(D^{n}(L) ⊗ 𝔪_{A}),

making E(L) into a cosimplicial complex of group-valued functors. To make it an SDC, we must define a ∗ product. We do this as the Alexander-Whitney cup product

g ∗ h = (∂^{m+n} … ∂^{m+2}∂^{m+1}g) · (∂^{0})^{m}h,

for g ∈ E(L)^{m}, h ∈ E(L)^{n}. Definition 5.9. Given a cosimplicial simplicial group G, define \underline{MC}(G) ∈ 𝕊 by \underline{MC}(G) ⊂ ∏_{n≥0}(G^{n+1})^{Δ^{n}}, satisfying the conditions of [Pri3] Lemma 3.3, i.e. the elements ω_{n} ∈ (G^{n+1})^{Δ^{n}} satisfy

∂_{i}ω_{n} = \begin{cases}∂^{i+1}ω_{n−1} & i > 0 \\ (∂^{1}ω_{n−1}) · (∂^{0}ω_{n−1})^{−1} & i = 0, \end{cases}
σ_{i}ω_{n} = σ^{i+1}ω_{n+1},
σ^{0}ω_{n} = 1.

Define MC : scGp → Set by MC(G) = \underline{MC}(G)_{0}.

There is an adjoint action of G^{0} on \underline{MC}(G), given by

(g ∗ ω)_{n} = (∂_{0}(∂^{1})^{n+1}(σ_{0})^{n+1}g) · ω_{n} · (∂^{0}(∂^{1})^{n}(σ_{0})^{n}g^{−1}),

as in [Pri3] Definition 3.8.

We then define Del(G) to be the homotopy quotient Del(G) = [\underline{MC}(G)/G^{0}] ∈ 𝕊.

Let exp denote exponentiation of a nilpotent Lie algebra (giving a unipotent group). Corollary 5.10. Given an ℕ_{0}-graded DGLA L, the deformation functor Def(E(L)) ∈ scSp is weakly equivalent to the functor

A ↦ Del(exp(DL ⊗ 𝔪(A))).

Proof. The SDC E(L) corresponds to A ↦ E(exp(DL ⊗ 𝔪(A)) in the notation of [Pri4] §3.1, so the result is an immediate consequence of [Pri4] Proposition 6.11. □ Corollary 5.11. For L as above, Def(E(L)) is weakly equivalent in the model category scSp to the functor A ↦ [MC(exp(DL ⊗ 𝔪(A)))/ exp(L^{0} ⊗ 𝔪(A_{0}))]. Proof. [Pri5] Lemma 2.26 implies that MC(exp(DL ⊗ 𝔪(A))) → \underline{MC}(exp(DL ⊗ 𝔪(A))) defines a weak equivalence in scSp (although the former is not fibrant), and similarly for L^{0} ⊗ 𝔪(A_{0}) → L^{0} ⊗ 𝔪(A), so we get a weak equivalence on passing to the homotopy quotient. □

5.3. The final comparison.

Definition 5.12. Given an ℕ_{0}-graded DGLA L, define Del(L) ∈ sDGSp to be the functor Del(L) : dgC_{k} → 𝕊 given by the homotopy quotient

A ↦ [MC(Tot^{Π}(L ⊗ N𝔪(A)))/ exp(L^{0} ⊗ 𝔪(A_{0})]

with respect to the gauge action of Definition 5.4.

Corollary 5.13. Given an ℕ_{0}-graded DGLA L, the deformation functor Def(E(L)) ∈ scSp is weakly equivalent to RSpf N^{∗}Del(L), for RSpf N^{∗} as in Theorem 1.45.

Proof. By Corollary 5.10, it suffices to show that the functors Del(L) and A ↦ Del(exp(DL ⊗ 𝔪(A))) are weakly equivalent in scSp. It follows from [Pri5] Lemma 2.26 that the latter is weakly equivalent to

A ↦ [MC(exp(DL ⊗ 𝔪(A)))/ exp(L^{0} ⊗ 𝔪(A_{0}))],

which is not fibrant in general. Now, [Pri4] Theorem 6.23 implies that this is isomorphic to

A ↦ [MC(Tot^{Π}(L ⊗ N𝔪(A)))/ exp(L^{0} ⊗ 𝔪(A_{0})],

which is just Spf N^{∗}Del(L). [Pri5] Lemma 1.62 then implies that RSpf N^{∗}F ≅ Spf N^{∗}F for all levelwise quasi-smooth functors F. □

Definition 5.14. Define DGdgC_{k} to be the category of Artinian local ℕ_{0} × ℕ_{0}-graded graded-commutative Λ-algebras A_{•}^{•} with differential of bidegree (1, −1) and residue field k. Let dgDGSp be the category of left-exact Set-valued functors on DGdgC_{k}.

Proposition 5.15. Under the equivalence of Proposition 5.6, an ℕ_{0}-graded DGLA L corresponds to the deformation functor Def(E(L)) ∈ scSp of Definition 4.4.

Proof. The equivalence of [Pri5] Corollary 4.57 is given by a functor RSpf D^{∗} : Ho(sDGSp) → Ho(dgDGSp), together with a functor

Spf Tot^{∗}MC : DG_{ℤ}LA → dgDGSp,

on ℤ-graded DGLAs, given by

Spf Tot^{∗}MC(L)(A) = MC(Tot L ⊗ 𝔪(A)).

By Corollary 5.13, it suffices to show that the objects RSpf D^{∗}Del(L) and Spf Tot^{∗}MC(L) are weakly equivalent in dgDGSp.

Taking A ∈ DGdgC_{k}, it follows from the definitions that Spf D^{∗}Del(L)(A) consists of maps Spf (DA) → MC(L)×^{exp(L^{0})} W (exp(L^{0})) in sDGSp, where DA ∈ (dgC_{k})^{Δ} is defined by cosimplicial denormalisation, and Spf (DA) ∈ sDGSp is the functor dgC_{k} → 𝕊 given in level n by Hom_{dgC_{k}}(D^{n}A, −).

Thus

Spf D^{∗}Del(L)(A) ⊂ MC(Tot L ⊗ A^{0}) × MC(D(exp(L^{0} ⊗ A_{0}^{•}))

consists of pairs (ω, g) with g ∗ ∂_{DA}^{0}ω = ∂_{DA}^{1}ω, corresponding in level n to the map

(Spf D^{n}A) → MC(L) × exp(L^{0})^{n}
(ω, g) ↦ (ω, (∂^{2})^{n−1}g, ∂^{0}(∂^{2})^{n−2}g, …, (∂^{0})^{n−1}g).

By [Pri4] Theorem 6.23, we know that MC(D(exp(L^{0} ⊗ A_{0}^{•})) ≅ MC(L^{0} ⊗ A_{0}^{•})), giving us γ ∈ MC(L^{0} ⊗ A_{0}^{•})). Note that g = exp(γ), so

g ∗ ∂_{DA}^{0}ω = ∂_{DA}^{0}ω + [γ, ω],

with all higher terms vanishing, since σ^{0}γ = 0, so ⟦γ, ⟦γ, v⟧⟧ = 0 for all v (and in particular when v = ∂_{DA}^{0}ω).

Thus we have

Spf D^{∗}Del(L)(A) = {(ω, γ) : ω ∈ MC(Tot L ⊗ A^{0}), γ ∈ MC(L^{0} ⊗ A_{0}^{•}), [γ, ω] + d_{c,A}ω = 0},

where d_{c,A} is the cochain differential on A

Now look at γ + ω ∈ (Tot (L ⊗ A))^{1}. The equations combine to show that α := γ + ω lies in MC(Tot (L ⊗ A)), since

dα + [α, α] = (d_{L}ω + d_{A}^{s}ω + [ω, ω]) + (d_{A}γ + [γ, γ]) + ([γ, ω] + d_{A,c}ω) = 0,

where d_{A}^{s} is the chain differential on A.

Thus we have defined a map

ψ : Spf D^{∗}Del(L) → Spf (Tot^{Π})^{∗}MC(L).

In fact, we have shown that

Spf D^{∗}Del(L)(A) ≅ MCTot ((L ⊗ A^{0}) ×_{(L^{0}⊗A^{0})} (L^{0} ⊗ A_{0})),

and [Pri5] Lemma 4.13 then implies that ψ is a weak equivalence (with similar reasoning to [Pri5] Lemma 2.26).

Finally, note that this gives cohomology groups (as defined in Definition 1.33) H^{n}(Spf D^{∗}Del(L)(A)) ≅ H^{n+1}(L), and that

H^{n}(RSpf D^{∗}Del(L)) ≅ H^{n}(Del(L)) ≅ H^{n+1}(L),

since the equivalence of [Pri5] Proposition 4.57 preserves cohomology groups. Therefore the morphism Spf D^{∗}Del(L) → RSpf D^{∗}Del(L) is also a weak equivalence by Corollary 1.29, and this completes the proof. □

5.4. An application.

Corollary 5.16. Let X be a smooth scheme, for which there exists a flabby DGLA resolution ℒ^{•} of the tangent sheaf 𝒯_{X}. Then for A ∈ sC_{k}, the simplicial nerve Σ(L)(NA) of the DGLA L^{•} = Γ(X, ℒ^{•}) is weakly equivalent to Def_{E}(A), where E is the SDC constructed in Example 3.8.1 (or Example 3.19 with Y = ∅).

Proof. E(ℒ^{•}) is a sheaf of SDCs on X, with Γ(X, E(ℒ^{•})) = E(L). We may then form the Godement sheaf resolution 𝒞 (E(ℒ)) of E(L), giving another sheaf of SDCs on X (as in [Pri2] §3.2) and quasi-isomorphisms

E(ℒ^{•}) → 𝒞 (E(ℒ^{•})) ← 𝒞 (E(𝒯_{X}))

of flabby sheaves of SDCs on X, and hence quasi-isomorphisms

E(L) → C(X, E(L)) ← C(X, E(𝒯_{X}))

of SDCs, where C(X, ℱ) = Γ(X, 𝒞 (ℱ)) is the Godement resolution.

Finally, note that E(𝒯_{X})(A) = exp(𝒯_{X} ⊗ 𝔪(A)) is the sheaf of infinitesimal algebra automorphisms of 𝒪_{X}, so the methods of [Pri2] §3.2.2 provide a quasi-isomorphism C(X, E(𝒯_{X})) → E of SDCs. Thus Def(E) ≃ Def(E(L)), and Proposition 5.15 shows that this is equivalent to L under the equivalences of Proposition 5.6. □

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