Deformations via Simplicial Deformation Complexes

J.P.Pridham^{∗}

September 11, 2008

Introduction

There has long been a philosophy that every deformation problem in characteristic 0 should give rise to a differential graded Lie algebra (DGLA). This DGLA should not only determine the deformation functor (and indeed the deformation groupoid up to equivalence), but also somehow encapsulate more information (corresponding to the higher cohomology) about the original problem. We have:

Deformation Problem ⇝ DGLA ⇝ Deformation Groupoid ⇝ Deformation Functor.

However, there are several respects in which DGLAs are not wholly satisfactory. Finding a DGLA to govern a given problem is not easy — Kontsevich describes it as an “art” ([Kon94]), so the arrow

Deformation Problem ⇝ DGLA

above is merely aspirational. Kontsevich also remarks that, where DGLAs are constructed for geometric problems, the methods used are predominantly analytic. This proves inadequate for the more algebraic problems, such as deformations of an algebraic variety. However, in [KS00], Kontsevich and Soibelman go some way towards constructing DGLAs for algebraic problems, by providing DGLAs to govern deformations over an operad. In non-zero characteristic, DGLAs do not, in general, even give rise to deformation functors.

The reason for these problems is that differential graded objects are usually natural things to consider only in characteristic zero. This statement is implied by omission in papers such as [Qui69], showing that DG objects suffice when considering homotopy theory over the rationals. In non-zero characteristic, it is necessary to use simplicial objects in their stead. This equivalence of DG and simplicial objects, together with the relative facility of DG methods, has resulted in a comparative neglect of simplicial methods. As I will demonstrate, there are may scenarios in which simplicial structures arise more naturally.

In this paper, I define a simplicial object to replace DGLAs, namely a Simplicial Deformation Complex (SDC). This gives rise to the picture:

Deformation Problem ⇝ SDC ⇝ Deformation Groupoid ⇝ Deformation Functor.

∗The author is supported by Trinity College, Cambridge and the Isle of Man Department of Education.

Here we have the advantage not only that this works in non-zero characteristic, but also that the first arrow is reasonably canonical.

Whereas Kontsevich and Soibelman used operads for their notion of algebraicity, I use a weaker notion — that of monadic adjunctions. Given the existence of a suitable monadic adjunction, an SDC can be constructed to govern the relevant deformation problem. In particular, all operads give rise to monadic adjunctions.

The frequent existence of such adjunctions, and their rôle in cohomology and homology, are well documented (e.g.[Mac71] Ch.VII §6, and Notes at the end of Ch. VI). It is thus unsurprising that monadic adjunctions should be useful in deformation theory, and they immediately enable the construction of SDCs for many deformation problems.

One major way in which this approach diverges from previous approaches is that the dual construction is equally straightforward, providing SDCs whenever there is a suitable comonadic adjunction. As an example, this immediately yields an SDC governing deformations of a local system on an arbitrary topological space.

The real power of this approach, however, lies in the ability to construct SDCs from a combination of monadic and comonadic adjunctions. For example, to deform a scheme X is equivalent to deforming its structure sheaf 𝒪_{X}. The algebra structure of 𝒪_{X} can be thought of as monadic, while the sheaf structure is comonadic. This provides an SDC for this example, under more general conditions than those for which a DGLA was constructed in [Hin05]. By contrast, most previous examples for which DGLAs were constructed were either purely comonadic (e.g. smooth schemes, for which the ring structure does not deform), or purely monadic (e.g. affine schemes, for which the sheaf structure does not deform). Another example of this type is deformation of a Hopf algebra.

Furthermore, I adapt the results of [Qui69] to prove that DGLAs and SDCs are equivalent in characteristic zero. However, while the arrow DGLA ⇝ SDC is very natural, the arrow SDC ⇝ DGLA is generally not, which helps to explain why constructing DGLAs is an “art”.

Contents

0 Pre-requisites 3 0.1 Functors on Artinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Differential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . 6 0.3 Monadic Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 Simplicial Deformation Complexes 13 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Deformation of algebras . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.2 Deformation of sheaves — Čech resolution . . . . . . . . . . . . . 21

2 A Procedure for Computing SDCs 22

2.1 The SDC associated to a monadic adjunction . . . . . . . . . . . . . . . 23 2.1.1 Deformation of sheaves — Godement resolution . . . . . . . . . . 24 2.1.2 Deformation of co-algebras . . . . . . . . . . . . . . . . . . . . . 25 2.2 The general approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Deformation of Hopf algebras . . . . . . . . . . . . . . . . . . . . 32

3 Deformation of Schemes 33 3.1 Separated Noetherian schemes . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Smooth schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Deformation of a subscheme . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Constrained deformation of schemes . . . . . . . . . . . . . . . . . . . . 38 3.5 Deformation of a pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Deformation of a group scheme . . . . . . . . . . . . . . . . . . . . . . . 40

3.6.1 Deformation of a sub-(group scheme) . . . . . . . . . . . . . . . . 41 3.7 G-invariant deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Representations of the Algebraic Fundamental Group 43 4.1 Representations to GL_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Representations to a smooth Noetherian group scheme . . . . . . . . . . 44

5 Equivalence between SDCs and DGLAs in Characteristic Zero 45 5.1 SDCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 DGLAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 SDC ⇝ DGLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 DGLA ⇝ SDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.4.1 An explicit description . . . . . . . . . . . . . . . . . . . . . . . . 58

Bibliography 59

0 Pre-requisites

This section contains a summary of pre-requisite material used throughout the rest of the paper, included for ease of reference.

0.1 Functors on Artinian Rings

This section consists merely of a list of well-known results and definitions, included for the convenience of the reader.

Fix Λ a complete Noetherian local ring, μ its maximal ideal, k its residue field. Define Ĉ_{Λ} to be the category of complete local Noetherian Λ-algebras with residue field k, and C_{Λ} to be the category of Artinian local Λ-algebras with residue field k

We require that all functors on C_{Λ} satisfy

(H0) F (k) = •, the one-point set.

We take the following definitions and results from [Sch68]:

Definition 0.1. For p : B → A in C_{Λ} surjective, p is a small extension if ker p = (t), a principal ideal, such that 𝔪_{B}(t) = (0).

Note that any surjection can be be factorised as a composition of small extensions. For F : C_{Λ} → Set, let F̂ : Ĉ_{Λ} → Set by F̂(R) = \underleftarrow{lim} F (R/𝔪_{R}^{n}). Note that

F̂(R) ⟶^{∼} Hom(h_{R}, F), where h_{R} : C_{Λ} → Set; A ↦ Hom(R, A). Definition 0.2. We will say a functor F : C_{Λ} → Set is pro-representable if it is isomorphic to h_{R}, for some R ∈ Ĉ_{Λ}. By the above remark, this isomorphism is determined by an element ξ ∈ F̂(R). We say the pro-couple (R, ξ) pro-represents F. Definition 0.3. A natural transformation ϕ : F → G in [C_{Λ}, Set] is called:

1. unramified if ϕ : t_{F} → t_{G} is injective, where t_{F} = F (k[ϵ]).

2. smooth if for every B ↠ A in C_{Λ}, we have F (B) ↠ G(B) ×_{G(A)} F (A)

3. étale if it is smooth and unramified. Definition 0.4. F : C_{Λ} → Set is smooth if and only if F → • is smooth. Definition 0.5. A pro-couple (R, ξ) is a hull for F if the induced map h_{R} → F is étale. Theorem 0.6. Let (R, ξ), (R′, ξ′) be hull of F. Then there exists an isomorphism u : R → R′ such that F (u)(ξ) = ξ′. Lemma 0.7. Suppose F is a functor such that

F (k[V] ×_{k} k[W]) ⟶^{∼} F (k[V]) × F (k[W])

for vector spaces V and W over k, where k[V] := k ⊕ V ∈ C_{Λ} in which V^{2} = 0. Then F (k[V]), and in particular t_{F}, has a canonical vector space structure, and F (k[V]) ≅ t_{F} ⊗ V. Theorem 0.8. Let F : C_{Λ} → Set. Let A′ → A and A′′ → A be morphisms in C_{Λ}, and consider the map: (†) F (A′ ×_{A} A′′) → F (A′) ×_{F (A)} F (A′′). Then

1. F has a hull if and only if F has properties (H1), (H2) and (H3) below: (H1) (†) is a surjection whenever A′′ → A is a small extension. (H2) (†) is a bijection when A = k, A′′ = k[ϵ]. (H3) dim_{k}(t_{F}) < ∞.

2. F is pro-representable if and only if F has the additional property (H4):

(H4) F (A′ ×_{A} A′′) ⟶^{∼} F (A′) ×_{F (A)} F (A′′).

for any small extension A′ → A. Proof. [Sch68], Theorem 2.11.

Definition 0.9. F : C_{Λ} → Set is homogeneous if η : F (B ×_{A} C) → F (B) ×_{F (A)} F (C) is an isomorphism for every B ↠ A.

Note that a homogeneous functor satisfies conditions (H1), (H2) and (H4).

Definition 0.10. F : C_{Λ} → Set is a deformation functor if:

1. η is surjective whenever B ↠ A.

2. η is an isomorphism whenever A = k.

Note that a deformation functor satisfies conditions (H1) and (H2).

The following results are proved by Manetti (in [Man99]):

Theorem 0.11. (Standard Smoothness Criterion) Given ϕ : F → G a morphism of

deformation functors, with (V, v_{e}) ⟶^{ϕ′} (W, w_{e}) a compatible morphism of obstruction the-

ories, if (V, v_{e}) is complete, V ⟶^{ϕ′} W injective, and t_{F} → t_{G} surjective, then ϕ is smooth.

Proof. [Man99], Proposition 2.17.

For functors F : C_{Λ} → Set and G : C_{Λ} → Grp, we say that G acts on F if we have a functorial group action G(A) × F (A) ⟶^{∗} F (A), for each A in C_{Λ}. The quotient functor F/G is defined by (F/G)(A) = F (A)/G(A).

Theorem 0.12. If F : C_{Λ} → Set, a deformation functor, and G : C_{Λ} → Grp a smooth deformation functor, with G acting on F, then D := F/G is a deformation functor, and if ν : t_{G} → t_{F} denotes h ↦ h ∗ 0, then t_{D} = coker ν, and the universal obstruction theories of D and F are isomorphic.

Proof. [Man99], Lemma 2.20.

Theorem 0.13. For F : C_{Λ} → Set is homogeneous, and G : C_{Λ} → Grp a deformation functor, given a, b ∈ F (R), define Iso(a, b) : C_{R} → Set by

Iso(a, b)(R ⟶^{f} A) = {g ∈ G(A)|g ∗ f(a) = f(b)}.

Then Iso(a, b) is a deformation functor, with tangent space ker ν and, if G is moreover smooth, complete obstruction space coker ν = t_{D}. Furthermore, if G is homogeneous, then so is Iso(a, b).

Proof. [Man99], Proposition 2.21.

Theorem 0.14. If G, G′ smooth deformation functors, acting on F, F′ respectively, with F, F′ homogeneous, ker ν → ker ν′ surjective, and coker ν → coker ν′ injective, then F/G → F′/G′ is injective.

Proof. [Man99], Corollary 2.22.

This final result does not appear in [Man99], but proves extremely useful:

Corollary 0.15. If F : C_{Λ} → Set and G : C_{Λ} → Grp are deformation functors, with G acting on F, let D := F/G, then:

1. If G is smooth, then η_{D} is surjective for every B ↠ A (i.e. D is a deformation functor).

2. If F is homogeneous and ker ν = 0, then η_{D} is injective for every B ↠ A.

Thus, in particular, F/G will be homogeneous if F is homogeneous, G is a smooth deformation functor and ker ν = 0.

Proof. The first part is just Theorem 0.12. For the second part, assume we are given

α, β ∈ D(B ×_{A} C) such that η(α) = η(β),

so α|_{B} = β|_{B}, α|_{C} = β|_{C}. Lift α, β to α̃, β̃ ∈ F (B ×_{A} C). We have g_{B} ∈ G(B), g_{C} ∈ G(C) such that

g_{B} ∗ α̃|_{B} = β̃|_{B}, g_{C} ∗ α̃|_{C} = β̃|_{C}.

Thus g_{B}|_{A}, g_{C}|_{A} ∈ Iso(α̃|_{A}, β̃|_{A})(A). By Theorem 0.13, Iso(α̃|_{A}, β̃|_{A}) has a hull, with tangent space ker ν = 0, so Iso(α̃|_{A}, β̃|_{A})(A) contains at most one element, and g_{B}|A = g_{C}|A. Hence

(g_{B}, g_{C}) ∈ G(B) ×_{G(A)} G(C),

and we may use surjectivity of η_{G} to lift this to g ∈ G(B ×_{A} C). Now η_{F} (g ∗α̃) = η_{F} (β̃), so homogeneity of F implies that g ∗ α̃ = β̃, so α = β.

To summarise the results concerning the pro-representability of the quotient D = F/G, we have:

1. If F is a deformation functor and G a smooth deformation functor, with coker ν finite dimensional, then D has a hull.

2. If F is homogeneous and G a smooth deformation functor, with ker ν = 0 and coker ν finite dimensional, then D is pro-representable.

0.2 Differential Graded Lie Algebras

The results of this section can all be found in [Man99].

Definition 0.16. A DGLA over a field k of characteristic 0 is a graded vector space L = ⨁_{i∈ℤ} 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.

Remark 0.17. Note that this definition of a DGLA does allow for negative degrees; however, it is natural to require that H^{<0}(L ⊗ k) = 0, and in Section 5 it will be shown that all the deformation problems we consider can be described by DGLAs with L^{<0} = 0.

Fix a DGLA L.

Definition 0.18. The Maurer-Cartan functor MC_{L} : C_{k} → Set is defined by

MC_{L}(A) = \{x ∈ L^{1} ⊗ 𝔪_{A}|dx + \frac{1}{2}[x, x] = 0\}.

Observe that for ω ∈ L^{1} ⊗ 𝔪_{A},

dω + \frac{1}{2}[ω, ω] = 0 ⇒ (d + ad_{ω}) ∘ (d + ad_{ω}) = 0,

so (L ⊗ A, [,], d + ad_{ω}) is a DGLA over A.

Definition 0.19. Define the gauge functor 𝔤𝔩 : C_{k} → Grp by 𝔤𝔩(A) = exp(L^{0} ⊗ 𝔪_{A}).

We may define another DGLA, L_{d}, by

L_{d}^{i} = \begin{cases}L^{1} ⊕ kd & i = 1 \\ L^{i} & i ≠ 1, \end{cases}

with

d_{d}(d) = 0, [d, d] = 0, [d, a]_{d} = da, ∀a ∈ L.

Lemma 0.20. exp(L^{0} ⊗ 𝔪_{A}) commutes with [,] when acting on L_{d} ⊗ A via the adjoint action.

Corollary 0.21. Since exp(L^{0} ⊗𝔪_{A}) preserves L^{1} ⊗𝔪_{A} +d ⊂ L_{d} ⊗A under the adjoint action, and

x ∈ MC_{L}(A) ⇔ [x + d, x + d] = 0,

the adjoint action of exp(L^{0}⊗𝔪_{A}) on L^{1}⊗A+d induces an action of 𝔤𝔩(A) on MC_{L}(A), which we will call the gauge action.

Definition 0.22. Def_{L} = MC_{L}/𝔤𝔩, the quotient being given by the gauge action α(x) = ad_{α}(x + d) − d. Observe that 𝔤𝔩 is homogeneous.

For a ∈ MC_{L}(A), define K_{a} : C_{A} → Grp by

K_{a}(B) = exp((d + ad_{a})L^{−1} ⊗ 𝔪_{B}).

Note that this makes sense, since (d + ad_{a})^{2} = 0, so

(d + ad_{a})L^{−1} ⊗ 𝔪_{B} ≤ L^{0} ⊗ 𝔪_{B}

is a Lie subalgebra. Note that K_{a} is then a subfunctor of Iso(a, a) ≤ 𝔤𝔩(A), so acts on Iso(a, b) by right multiplication.

Define the deformation groupoid 𝔇𝔢𝔣_{L} to have objects MC_{L}, and morphisms given by Iso(a, b)/K_{a}.

Now, t_{𝔤𝔩} = L^{0}ϵ, and t_{MC_{L}} = Z^{1}(Lϵ), with action

t_{𝔤𝔩} × t_{MC_{L}} → t_{MC_{L}};
(b, x) ↦ x + db, so
t_{Def_{L}} = H^{1}(L).

Lemma 0.23. H^{2}(L) is a complete obstruction space for MC_{L}. Proof. Given a small extension

e : 0 → J → A → B → 0,

and x ∈ MC_{L}(B), lift x to x̃ ∈ L^{1} ⊗ 𝔪_{A}, and let

h = dx̃ + \frac{1}{2}[x̃, x̃] ∈ L^{2} ⊗ 𝔪_{A}.

In fact, h ∈ L^{2} ⊗ J, as dx + \frac{1}{2}[x, x] = 0.

Now,

dh = d^{2}x̃ + [dx̃, x̃] = [h − \frac{1}{2}[x̃, x̃], x̃] = [h, x̃] = 0,

since [[x̃, x̃], x̃] = 0 and J.𝔪_{A} = 0. Let

v_{e}(x) = [h] ∈ H^{2}(L ⊗ J) = H^{2}(L) ⊗_{k} J.

This is well-defined: if y = x̃ + z, for z ∈ L^{1} ⊗ J, then

dy + \frac{1}{2}[y, y] = dx̃ + dz + \frac{1}{2}[x̃, x̃] + \frac{1}{2}[z, z] + [x̃, z] = h + dz,

as J.𝔪_{A} = 0.

This construction is clearly functorial, so it follows that (H^{2}(L), v_{e}) is a complete obstruction theory for MC_{L}.

Now Theorem 0.12 implies the following: Theorem 0.24. Def_{L} is a deformation functor, t_{Def_{L}} ≅ H^{1}(L), and H^{2}(L) is a complete obstruction theory for Def_{L}.

The other theorems can be used to prove: Theorem 0.25. If ϕ : L → M is a morphism of DGLAs, and

H^{i}(ϕ) : H^{i}(L) → H^{i}(M)

are the induced maps on cohomology, then:

1. If H^{1}(ϕ) is bijective, and H^{2}(ϕ) injective, then Def_{L} → Def_{M} is étale.

2. If also H^{0}(ϕ) is surjective, then Def_{L} → Def_{M} is an isomorphism.

3. Provided condition 1 holds, 𝔇𝔢𝔣_{L} → 𝔇𝔢𝔣_{M} is an equivalence of functors of groupoids if and only if H^{0}(ϕ) is an isomorphism.

Proof. [Man99], Theorem 3.1, mutatis mutandis.

Theorem 0.26. If H^{0}(L) = 0, then Def_{L} is homogeneous.

Proof. This is essentially Theorem 0.15, with some allowance made for K_{a}.

Thus, in particular, a quasi-isomorphism of DGLAs gives an isomorphism of deformation functors and deformation groupoids. Example 0.27. Deformation of algebras (due to [Kon94]).

We wish to describe flat deformations R_{A}/A of any algebra R_{0}/k:

X = Spec R_{0} ⟶ Spec R_{A} = X_{A}
flat↓  ↓_{flat}
Spec k ⟶  Spec A.

Explicitly, the deformation groupoid has objects flat R_{A}/A, such that R_{A} ⊗_{A} k = R_{0}, with morphisms given by infinitesimal isomorphisms of R_{A}, i.e. A-algebra isomorphisms θ such that θ ≡ id mod 𝔪_{A}.

Take any free resolution R_{•} → R of R, i.e. a free graded algebra R_{∗}, with a differential δ of degree 1, such that

H_{i}(R_{•}) = \begin{cases}R & i = 0 \\ 0 & i > 0. \end{cases}

Now, let

L^{∗} = Der_{k}(R_{∗}, R_{∗}),

with differential ad_{δ}.

Let 𝔇𝔢𝔣_{R_{0}/k}(A) denote the groupoid of deformations R_{A}/A of R_{0}, with morphisms given by infinitesimal isomorphisms. Given ω ∈ MC_{L}(A), it follows that (δ + ω)^{2} = 0. Define a map

𝔇𝔢𝔣_{L}(A) ⟶^{Φ} 𝔇𝔢𝔣_{R_{0}/k}(A),

sending ω ∈ MC_{L}(A) to the flat A-algebra

H_{0}(R_{∗} ⊗ A, δ + ω).

In fact, Φ gives an essential surjection.

Next, observe that 𝔤𝔩(A) consists precisely of infinitesimal A-algebra automorphisms of R^{∗} ⊗ A, and that the gauge action of θ sends δ + ω to θ ∘ (δ + ω) ∘ θ^{−1}.

Now, θ ∈ Hom(ω, ω′) maps under Φ to

H_{0}(θ) : H_{0}(R_{∗} ⊗ A, δ + ω) → H_{0}(R_{∗} ⊗ A, δ + ω′),

and it turns out that Φ is full.

To see that Φ is faithful, note that for θ ∈ Iso(ω, ω)(A),

H_{0}(θ) : H_{0}(R_{∗} ⊗ A, δ + ω) → H_{0}(R_{∗} ⊗ A, δ + ω)

is the identity if and only if θ ∈ K_{ω}(A).

0.3 Monadic Adjunctions

This section just recalls some standard definitions concerning adjunctions, and fixes notation which will be used throughout the paper.

Definition 0.28. For categories D, E, and a pair of functors

\xymatrix{D \ar[r]^{G} & E \ar[l]^{F}},

recall that an adjunction F ⊣ G is a natural isomorphism

Hom_{D}(F A, B) ≅ Hom_{E}(A, GB).

We say that F is left adjoint to G, or G is right adjoint to F. Let ⊥ = F G, and ⊤ = GF. To give an adjunction is equivalent to giving two natural transformations, the unit

η : id_{E} → ⊤,

and the co-unit

ε : ⊥ → id_{D},

satisfying the triangle identities:

\xymatrix{F \ar[r]^{F η} \ar[dr]_{_{id_{F}}} & F GF \ar[d]^{εF} & and & G \ar[r]^{ηG} \ar[dr]_{_{id_{G}}} & GF G \ar[d]^{Gε} \\  & F &  &  & G.}

In terms of the unit and co-unit, the adjunction is given as follows:

Hom_{D}(F A, B) ≅ Hom_{E}(A, GB)
θ ↦ Gθ ∘ η_{A}
ε_{B} ∘ F ϕ ↤ ϕ.

Note that the η_{A} and ε_{B} can be recovered from the adjunction, since they correspond under the adjunction to the identity maps on F A and GB, respectively.

Examples 0.29. 1. Given a morphism X ⟶^{f} Y of schemes, there is an adjunction

\xymatrix{Shf(X) \ar[r]^{f_{∗}} & Shf(Y) \ar[l]^{f^{−1}}|{⊥}},

between the category of sheaves on Y and the category of sheaves on X, i.e. a natural isomorphism

Hom_{X}(f^{−1}𝒢 , ℱ) ≅ Hom_{Y} (𝒢 , f_{∗}ℱ),

determined by the canonical maps

ε_{ℱ} : f^{−1}f_{∗}ℱ → ℱ , η_{𝒢} : 𝒢 → f_{∗}f^{−1}𝒢 .

2. There is also an adjunction

\xymatrix{𝒪_{X}−Mod \ar[r]^{f_{∗}} & 𝒪_{Y} −Mod \ar[l]^{f^{∗}}|{⊥}} ,

Hom_{𝒪_{X}} (f^{∗}𝒢 , ℱ) ≅ Hom_{𝒪_{Y}} (𝒢 , f_{∗}ℱ),
ε_{ℱ} : f^{∗}f_{∗}ℱ → ℱ , η_{𝒢} : 𝒢 → f_{∗}f^{∗}𝒢 .

If a functor has a left adjoint, then it preserves all (inverse) limits. Conversely, provided the categories involved satisfy various additional conditions, the Special Adjoint Functor Theorem ([Mac71] Ch. V.8) proves that any functor which preserves inverse limits has a left adjoint.

Given an adjunction

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

with unit η : id → UF and co-unit ε : F U → id, we let ⊤ = UF, and define the category of ⊤-algebras, E^{⊤}, to have objects

⊤E ⟶^{θ} E,

such that the following diagrams commute:

\xymatrix{E \ar[r]^{η_{E}} \ar[dr]_{id} & ⊤E \ar[d]^{θ} & and & ⊤^{2}E \ar[r]^{⊤θ} \ar[d]_{Uε_{F E}} & ⊤E \ar[d]^{θ} \\  & E &  & ⊤E \ar[r]^{θ} & E.}

A morphism

g : (\begin{matrix}⊤E_{1} \\ ↓ & θ \\ E_{1} \end{matrix}) → (\begin{matrix}⊤E_{2} \\ ↓ & ϕ \\ E_{2} \end{matrix})

of ⊤-algebras is a morphism g : E_{1} → E_{2} such that

⊤E_{1} ⟶^{⊤g} ⊤E_{2}
θ↓  ↓^{ϕ}
E_{1} ⟶_{g} E_{2}

commutes.

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

B ↦ (\begin{matrix}UF UB \\ ↓ & Uε_{B} \\ UB \end{matrix})

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

Definition 0.30. The adjunction

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

is said to be monadic if K : D → E^{⊤} is an equivalence.

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

\xymatrix{k−Alg \ar[r]^{U}_{⊤} & k−Vect \ar[l]^{_{Symm_{k}}}},

between k-algebras and vector spaces over k, U being the forgetful functor. Other examples are Lie algebras or non-commutative algebras, with their respective free functors, over vector spaces, or vector spaces over sets.

Example 0.31. Probably the clearest example of a monadic adjunction is that between sets and G-sets. Given a group G, The category G−Set consists of sets S, with a G-action G × S → S. We have the adjunction:

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

with η_{S} = (e, id) : S → G × S, and ε_{T} = ∗ : G × T → T.

A ⊤-algebra consists of a set S, together with a map ⊤S = G × S ⟶^{f} S, such that:

\xymatrix{S \ar[r]^{(e,id_{S})} \ar[dr]_{_{id_{S}}} & G × S \ar[d]^{f} & and & G × G × S \ar[r]^{(id_{G},f)} \ar[d]_{(∗,id_{S})} & G × S \ar[d]^{f} \\  & S &  & G × S \ar[r]^{f} & S}

commute, which is precisely the same as a G-set, hence the equivalence.

Dually, if we have an adjunction

\xymatrix{D \ar[r]^{V} & E \ar[l]^{G}|{⊥}},

with co-unit γ : V G → id and unit α : id → GV, we let ⊥ = V G, and can define the category of ⊥-coalgebras, E^{⊥}, with a functor K : D → E^{⊥}.

Definition 0.32. The adjunction

\xymatrix{D \ar[r]^{V} & E \ar[l]^{G}|{⊥}},

is said to be comonadic if K : D → E^{⊥} is an equivalence.

Finally, the equivalence version of Beck’s monadicity theorem gives a criterion for an adjunction to be monadic:

Definition 0.33. A split fork is a diagram

\xymatrix{A \ar[r]^{f} \ar[r]_{g} & B \ar[l]^{t} \ar[r]^{e} & C, \ar[l]^{s}}

such that ef = eg, es = 1, ft = 1, gt = se. A split coequaliser of f and g is the arrow e of some split fork.

Definition 0.34. A functor U : D → E is said to reflect coequalisers for a pair

f, g : A → B if for every diagram \xymatrix{A \ar[r]^{f} \ar[r]_{g} & B \ar[r]^{e} & C,} with ef = eg and Ue a coequaliser

of Uf and Ug, the morphism e is a coequaliser of f and g.

U is said to reflect isomorphisms if f is an isomorphism whenever Uf is. Observe that if U preserves a coequaliser of f, g in D, and U reflects isomorphisms, then U reflects all coequalisers of f, g.

Theorem 0.35. The following conditions are equivalent:

1. The adjunction

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

is monadic.

2. If f, g : D → D′ is a parallel pair in D for which Uf, Ug has a split coequaliser, then D has a coequaliser for f, g, and U preserves and reflects coequalisers for these pairs.

Proof. [Mac71] Ch. VI.7 Ex. 6.

Throughout this paper, the sign “=” will be used to denote canonical isomorphism.

1 Simplicial Deformation Complexes

1.1 Definitions

Definition 1.1. A simplicial deformation complex E^{•} consists of smooth homogeneous 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) = • 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}.

Remark 1.2. Observe that E^{•} is not quite a cosimplicial complex — we have omitted the maps ∂^{0} and ∂^{n+1}. In fact, if we define Ĕ_{•−1} = E^{•}, with σ_{i} = ∂^{i+1} and ∂_{i} = σ^{i}, then Ĕ_{•} is an augmented simplicial complex.

Definition 1.3. Let C^{•}(E) be the tangent space of E^{•}, i.e. C^{n}(E) = E^{n}(k[ϵ]).

Definition 1.4. Define the Maurer-Cartan functor MC_{E} by

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

Lemma 1.5. E^{0} is a group under multiplication.

Proof. We need only show existence of inverses. Given g ∈ E^{0}(A), we have a morphism ϕ_{g} : E^{0} → E^{0} of functors on C_{A}, given by h ↦ g ∗ h. Since E^{0} is smooth, its obstruction theory is trivial. Now, on tangent spaces, ϕ_{g}(a) = ḡ ∗ a, but the map A → k[ϵ] factors through k, so ḡ = 1, and ϕ_{g} : C^{0}(E) → C^{0}(E) is the identity. Thus the Standard Smoothness Criterion (Theorem 0.11) implies that ϕ_{g} is smooth, and in particular surjective, hence ϕ_{g} : E^{0}(A) → E^{0}(A) is surjective, and there exists an h with g ∗ h = 1. We construct a left inverse similarly, and then h_{l} = h_{l} ∗ g ∗ h_{r} = h_{r}. This also gives uniqueness of the inverse.

Now, if ω ∈ MC_{E}(A) and g ∈ E^{0}(A), then g ∗ ω ∗ g^{−1} ∈ MC_{E}(A). We may therefore make the following definition:

Definition 1.6.

Def_{E} = MC_{E}/E^{0},

the quotient being with respect to the adjoint action. The deformation groupoid

𝔇𝔢𝔣_{E}

has objects MC_{E}, and morphisms given by E^{0}.

Lemma 1.7. The action E^{0} × E^{n} → E^{n} is faithful (i.e. g ∗ e = e for some e only if g = 1).

Proof. From Theorem 0.13, it will suffice to show that ν : C(E)^{0} → C(E)^{n} is injective, since a deformation functor F with tangent space 0 has at most one element in F (A).

Let ω_{0} be the unique element of E^{1}(k); thus \overbrace{ω_{0} ∗ … ∗ ω_{0}}^{n} is the unique element of E^{n}(k). Note that σ^{0}(ω_{0}) = 1. Now, for g ∈ C^{0}(E),

(σ^{0})^{n}(ν(g)) = (σ^{0})^{n}(g ∗ ω_{0}^{n}) = (σ^{0})^{n−1}(g ∗ σ^{0}(ω_{0}) ∗ ω_{0}^{n−1}) = (σ^{0})^{n−1}(g ∗ ω_{0}^{n−1}) = g,

by induction. Therefore ν is injective.

Observe that, for ω ∈ MC_{E}(A),

σ^{0}(ω) ∗ ω = σ^{0}(ω ∗ ω) = σ^{0}∂^{1}(ω) = ω,

so the previous Lemma implies that σ^{0}(ω) = 1. As an immediate consequence, we have:

Lemma 1.8. For ω ∈ MC_{E}(A), if we let ∂_{ω}^{0}(e) = ω ∗ e, and ∂_{ω}^{n+1}(e) = e ∗ ω, for e ∈ E^{n}(A), then E^{∗}(A) becomes a cosimplicial complex, ∂_{ω}^{0} and ∂_{ω}^{n+1} also satisfying the following identities, for e ∈ E^{m}, f ∈ E^{n}:

∂_{ω}^{0}(e) ∗ f = ∂_{ω}^{0}(e ∗ f),
e ∗ ∂_{ω}^{n+1}(f) = ∂_{ω}^{m+n+1}(e ∗ f),
∂_{ω}^{m+1}(e) ∗ f = e ∗ ∂_{ω}^{0}(f).

Note, however, that not all the identities of Definition 1.1 hold for ∂_{ω}^{0} and ∂_{ω}^{n+1} — in general ∂_{ω}^{m+1}(e) ∗ f ≠ ∂^{m+1}(e ∗ f), and e ∗ ∂_{ω}^{0}(f) ≠ ∂^{m}(e ∗ f).

In particular, we obtain, on C_{k} ⊂ C_{Λ}, ∂^{0} := ∂_{ω_{0}}^{0} and ∂^{n+1} := ∂_{ω_{0}}^{n+1}. Thus C^{•}(E) has the natural structure of a cosimplicial complex.

Definition 1.9. Define the cohomology groups of E to be

H^{i}(E) := H^{i}(C^{•}(E)),

the cohomotopy groups of the cosimplicial complex C^{•}(E).

Lemma 1.10. The tangent space of MC_{E} is Z^{1}(C^{•}(E)), with the action of E^{0} giving ν(g) = ∂^{1}(g) − ∂^{0}(g).

Proof. On tangent spaces all maps are linear. Thus the map C^{1}(E) × C^{1}(E) → C^{2}(E) given by (α, β) ↦ α ∗ β can be written α ∗ β = f_{1}(α) + f_{2}(β). Evaluating this at β = ω_{0} we obtain f_{1}(α) = α ∗ ω_{0} = ∂^{2}(α), and similarly f_{2}(β) = ∂^{0}(β). Thus

t_{MC_{E}} = {ω ∈ C^{1}(E): ∂^{2}(ω) + ∂^{0}(ω) = ∂^{1}(ω)} = Z^{1}(C^{•}(E)).

Now, ν(g) = g ∗ ω_{0} ∗ g^{−1}, and, similarly to above,

f ∗ ω_{0} ∗ h = f ∗ ω_{0} ∗ 1 + 1 ∗ ω_{0} ∗ h = ∂^{1}(f) + ∂^{0}(h).

Hence ν(g) = ∂^{1}(g) − ∂^{0}(g).

Lemma 1.11. H^{2}(E) is a complete obstruction space for MC_{E}.

Proof. Take a small extension

e : 0 → k ⟶^{ϵ} A → B → 0.

Note that ϵ provides an isomorphism A ×_{k} k[ϵ] → A ×_{B} A, (α, aϵ) ↦ (α, α + aϵ), and thus, for any homogeneous functor F, F (A) × t_{F} ϵ → F (A ×_{B} A). Given f ∈ F (A) and v ∈ t_{F}, we will write the image under this isomorphism as (f, f + vϵ). Note that all the properties which this terminology suggests do hold.

Now, given ω ∈ MC_{E}(B), use the smoothness of E^{1} to take a lift ω̃ ∈ E^{1}(A). Since ω ∗ ω = ∂^{1}(ω),

(ω̃ ∗ ω̃, ∂^{1}(ω̃)) ∈ E^{2}(A) ×_{E^{2}(B)} E^{2}(A),

so we have ω̃ ∗ ω̃ = ∂^{1}(ω̃) + cϵ, for c ∈ C^{2}(E).

Now,

(ω̃ ∗ ω̃) ∗ ω̃ = ∂^{1}(ω̃) ∗ ω̃ + ∂^{3}(c)ϵ
ω̃ ∗ (ω̃ ∗ ω̃) = ω̃ ∗ ∂^{1}(ω̃) + ∂^{0}(c)ϵ,

so

∂^{1}(ω̃ ∗ ω̃) − ∂^{2}(ω̃ ∗ ω̃) = (∂^{0}(c) − ∂^{3}(c))ϵ.

Also,

∂^{1}(ω̃ ∗ ω̃) = ∂^{1}∂^{1}(ω̃) + ∂^{1}(c)ϵ
∂^{2}(ω̃ ∗ ω̃) = ∂^{2}∂^{1}(ω̃) + ∂^{2}(c)ϵ,

so

∂^{1}(ω̃ ∗ ω̃) − ∂^{2}(ω̃ ∗ ω̃) = (∂^{1}(c) − ∂^{2}(c))ϵ,

since ∂^{1}∂^{1} = ∂^{2}∂^{1}.

Thus dc = (∂^{0} − ∂^{1} + ∂^{2} − ∂^{3})(c) = 0, so c ∈ Z^{2}(C^{•}(E)).

If we take another lift ω̃_{1} = ω̃ + bϵ, then

ω̃_{1} ∗ ω̃_{1} = ω̃ ∗ ω̃ + ∂^{0}(b)ϵ + ∂^{2}(b)ϵ,

since ω̃ ∗ b = ω_{0} ∗ b + ∂^{0}(b), &c., and

∂^{1}(ω̃_{1}) = ∂^{1}(ω̃) + ∂^{1}(b)ϵ,

so c_{1} = c + db.

We therefore define v_{e}(ω) = [c]ϵ ∈ H^{2}(E), and it follows from the above working that this is a complete obstruction theory.

Theorems 0.11 to 0.15 now imply:

Theorem 1.12. Def_{E} is a deformation functor, with tangent space H^{1}(E) and complete obstruction space H^{2}(E). For ω, ω′ ∈ MC_{E}(A), Iso(ω, ω′) is homogeneous, with tangent space H^{0}(E) and complete obstruction space H^{1}(E). Moreover, if H^{0}(E) = 0, then Def_{E} is homogeneous.

Proof. Theorem 0.12 and Corollary 0.15. Theorem 1.13. If ϕ : E → F is a morphism of SDCs, and

H^{i}(ϕ) : H^{i}(E) → H^{i}(F)

are the induced maps on cohomology, then:

1. If H^{1}(ϕ) is bijective, and H^{2}(ϕ) injective, then Def_{E} → Def_{F} is étale.

2. If also H^{0}(ϕ) is surjective, then Def_{E} → Def_{F} is an isomorphism.

3. Provided condition 1 holds, 𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{F} is an equivalence of functors of

groupoids if and only if H^{0}(ϕ) is an isomorphism. Proof.

1. Theorems 0.11 and 0.12.

2. Theorem 0.14.

3. It remains only to show that ϕ^{0} : E^{0} → F^{0} is injective. E^{0} acts on F^{0} by multiplication, so ker ϕ^{0} = Iso(1, 1), which, by Theorem 0.13, is a deformation functor with tangent space ker H^{0}(ϕ) = 0. Hence ker ϕ^{0} = 0.

Call a morphism ϕ : E → F a quasi-isomorphism if the H^{i}(ϕ) : H^{i}(E) → H^{i}(F) are all isomorphisms. Definition 1.14. Given a morphism ϕ : E → F of SDCs, and a point x ∈ MC_{F} (Λ), define the groupoid

𝔇𝔢𝔣_{ϕ,x}

to be the fibre of the morphism

𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{F}

over x.

Explicitly, 𝔇𝔢𝔣_{ϕ,x}(A) has objects

{(ω, h) ∈ MC_{E}(A) × F^{0}(A) : hϕ(ω)h^{−1} = x},

and morphisms

E^{0}(A), where g(ω, h) = (gωg^{−1}, hϕ(g)^{−1}).

Theorem 1.15. Let E_{ϕ,x}^{n}(A) be the fibre of E^{n}(A) ⟶^{ϕ} F^{n}(A) over x^{n}. Then E_{x}^{•} is an SDC, and the canonical map

𝔇𝔢𝔣_{E_{ϕ,x}} → 𝔇𝔢𝔣_{ϕ,x}

is an equivalence of groupoids.

Proof. Def_{ϕ,x} is a deformation functor, with tangent space H^{1}(E, F) and complete obstruction space H^{2}(E, F). For ω, ω′ ∈ Ob 𝔇𝔢𝔣_{ϕ,x}(A), Iso(ω, ω′) is homogeneous, with tangent space H^{0}(E, F) and complete obstruction space H^{1}(E, F). Here, H^{i}(E, F) is the cohomology of the cone complex

[figure: ⋮; ⋮; C^{3}(E); d; −ϕ; ⨁; C^{2}(F); d; C^{2}(E); d; ϕ; ⨁; C^{1}(F); d; C^{1}(E); d; −ϕ; ⨁; C^{0}(F); d; C^{0}(E),; d; ϕ]

so we have the long exact sequence

[figure: 0; H^{0}(E, F); H^{0}(E); H^{0}(ϕ); H^{0}(F); H^{1}(E, F); H^{1}(E); H^{1}(ϕ); H^{1}(F); H^{2}(E, F); H^{2}(E); H^{2}(ϕ); H^{2}(F); ….]

Now, the definition of E_{x}^{•} provides a short exact sequence:

0 → C^{•}(E_{ϕ,x}) → C^{•}(E) ⟶^{ϕ} C^{•}(F) → 0,

giving the long exact sequence

[figure: 0; H^{0}(E_{ϕ,x}); H^{0}(E); H^{0}(ϕ); H^{0}(F); H^{1}(E_{ϕ,x}); H^{1}(E); H^{1}(ϕ); H^{1}(F); H^{2}(E_{ϕ,x}); H^{2}(E); H^{2}(ϕ); H^{2}(F); ….]

Thus the map 𝔇𝔢𝔣_{E_{ϕ,x}} → 𝔇𝔢𝔣_{ϕ,x} gives isomorphisms on all relevant tangent and obstruction spaces, so is an equivalence of groupoids.

1.2 Examples

1.2.1 Deformation of algebras

Let Λ_{n} := Λ/μ^{n+1}. Take a flat μ-adic system of algebras B_{n}/Λ_{n}, i.e flat algebras B_{n}/Λ_{n} such that B_{n+1} ⊗_{Λ_{n=1}} Λ_{n} = B_{n}, and let B := \underleftarrow{lim} B_{n}. The simplest (and most common) case to consider is when B_{n} = Λ_{n}. We wish to create an SDC describing flat deformations R_{A}/(B ⊗_{Λ} A) of any flat algebra R_{0}/B_{0}:

X_{0} = Spec R_{0} ⟶^{i} Spec R_{A} = X_{A}
flat↓  ↓_{flat}
Spec B_{0} ⟶ Spf B ×_{Spf Λ} Spec A.

Explicitly, the deformation groupoid has objects flat R_{A}/(B ⊗ A), such that R_{A} ⊗_{A} k = R_{0}, with morphisms given by infinitesimal isomorphisms of R_{A}, i.e. (B⊗A)- algebra isomorphisms θ such that θ ≡ id mod 𝔪_{A}.

There is, up to isomorphism, a unique flat μ-adic Λ-module M such that M ⊗k = R_{0} (since flat modules over nilpotent rings are free). There is an adjunction

\xymatrix{(B ⊗ A)FAlg \ar[r]^{U}_{⊤} & AFMod, \ar[l]^{F_{B⊗A}}}

between the categories of flat (B ⊗ A)-algebras and of flat A-modules, U being the forgetful functor, and

F_{B⊗A} = (B ⊗ A) ⊗_{A} Symm_{A} = B ⊗_{Λ} Symm_{A}.

Let ⊥ = F U and ⊤ = UF. We have the unit and co-unit:

η_{M} : M = S^{1}(M) ↪ Symm_{A}M → B ⊗_{Λ} Symm_{A}M = ⊤M,

and

ε_{R} : ⊥R → R
b[r_{1} ⊗ … ⊗ r_{n}] ↦ b · r_{1} … r_{n}.

As in [Wei94], we may form the canonical augmented cosimplicial complex

[figure: M; ⊤M; ⊤^{2}M; ⊤^{3}M; ·; ·; ·; ⊤^{4}M; . . . . . .]

with face operators

⊤^{i}η⊤^{n−i} : ⊤^{n}M → ⊤^{n+1}M

and degeneracy operators

⊤^{i}UεF ⊤^{n−i} : ⊤^{n+1}M → ⊤^{n}M.

Definition 1.16. Let

E^{n} = Hom_{Λ}(⊤^{n}M, M)_{U(ε_{R_{0}}∘...∘ε_{⊥^{n−1}R_{0}})},

where, for flat μ-adic Λ-modules M and N, and a k-linear map α : M ⊗ k → N ⊗ k, we define

Hom_{Λ}(M, N)_{α}(A)

to be the fibre of

Hom_{A}(M ⊗ A, N ⊗ A) → Hom_{k}(M ⊗ k, N ⊗ k)

over α.

We make E^{∗} into an SDC by giving it the product g ∗ h = g ∘ ⊤^{n}h, and Ĕ_{∗} is the dual complex to the canonical augmented complex M → ⊤^{∗}M.

Since flat modules over Artinian rings do not deform (being free), to give a deformation R/(B ⊗_{Λ} A) of R_{0}/B_{0} is the same as to give the module M ⊗_{Λ} A the structure of a (B ⊗_{Λ} A)-algebra, compatible with the B_{0}-algebra structure of R_{0}, and the A-module structure of M ⊗_{Λ} A.

A (B ⊗_{Λ} A)-algebra structure on module M_{A} := M ⊗_{Λ} A is the same as a map

f ∈ Hom_{AFMod}(⊤M_{A}, M_{A}); b[m_{1} ⊗ … ⊗ m_{n}] ↦ b · m_{1} … m_{n},

satisfying

f ∘ ⊤f = f ∘ Uε_{F M_{A}},
f ∘ η_{M_{A}} = id_{M_{A}}.

Hence elements of MC_{E}(A) correspond to (B ⊗ A)-algebra structures on M ⊗ A, compatible with R_{0} (recall that, for f ∈ MC_{E}, we automatically have σ^{0}(f) = 1). Elements f_{1}, f_{2} will give isomorphic ring structures precisely when we have g ∈ E^{0}(A) such that

⊤M ⊗ A ⟶^{f_{1}} M ⊗ A
⊤g↓  ↓^{g}
⊤M ⊗ A ⟶^{f_{2}} M ⊗ A

commutes.

Therefore 𝔇𝔢𝔣_{E} ≃ 𝔇𝔢𝔣_{R_{0}/B}.

Remark 1.17. Observe that, if ω ∈ MC_{E}(A) gives rise to a ring R, then we have an isomorphism of cosimplicial complexes

(E^{n}(A), ∂_{ω}) ≅ Hom_{B⊗A−Alg}(⊥_{∗}R, R)_{ε_{R}∘ε_{⊥R}∘...∘ε_{⊥^{n}R}},

where ⊥_{n}R = ⊥^{n+1}R. The natural product (f, g) ↦ f ∘ ⊥^{n+1}g, for f : ⊥_{n}R → R, corresponds to the product (α, β) ↦ α ∗ ω ∗ β in E^{∗}(A).

In particular, there is an isomorphism of cosimplicial complexes

C^{•}(E) ≅ Der_{B_{0}}(⊥_{•}R_{0}, R_{0}) = Hom_{R_{0}}(L_{•}^{R_{0}/B_{0}}, R_{0}),

where L_{•}^{R_{0}/B_{0}} is one presentation of the cotangent complex. Illusie ([Ill71] and [Ill72]) and André ([And67]) define the cotangent complex using the resolution arising from the adjunction between algebras and sets, whereas we have used the adjunction between flat algebras and flat modules. However, Quillen ([Qui70]) proves that all simplicial resolutions give the same object in the derived category.

Therefore H^{∗}(E) is André-Quillen cohomology.

If R_{0}/B_{0} is smooth, and lifts to some (μ-adic) smooth algebra R/B, then we have a morphism of SDCs

Hom_{BFAlg}(R, R)_{id} → E^{n},

where the complex on the left has the same object at every level, the obvious multiplication map, and all face and degeneracy operators given by the identity. Since R/B is smooth, this is a quasi-isomorphism, so the first complex also describes deformations of R_{0}.

1.2.2 Deformation of sheaves — Čech resolution

Take a sheaf of k-vector spaces ℳ_{0} on a topological space X. The deformation functor will associate to A, flat sheaves ℳ_{A} of A-modules such that ℳ_{A} ⊗ k = ℳ_{0}, modulo infinitesimal isomorphisms.

Take an open cover 𝔘 of X such that U_{i_{1}} ∩ U_{i_{2}} ∩ … ∩ U_{i_{n}} is contractible for all i_{1}, … i_{n}. Examples for which this is possible include X a manifold or a geometric simplicial complex. Set

X′ = ∐_{i} U_{i}.

We have inclusion maps u_{i} : U_{i} → X, so may combine them to give u : X′ → X.

Now, since flat modules do not deform, there are, up to isomorphism, unique μ-adic Λ-modules 𝒩_{i} lifting Γ(U_{i}, ℳ_{0}). Combining these, we obtain a sheaf 𝒩 on X′ with 𝒩 ⊗ k = u^{−1}ℳ_{0}, and there is an adjunction

\xymatrix{Shf(X′) \ar[r]^{u_{∗}}_{⊤} & Shf(X) \ar[l]^{u^{−1}}},

with unit

α : ℱ → u_{∗}u^{−1}ℱ ,
Γ(U, ℱ) → Γ(U, u_{∗}u^{−1}ℱ) = ∏_{i} Γ(U ∩ U_{i}, ℱ)
f ↦ ∏_{i} ρ_{U,U∩U_{i}}(f),

and co-unit

γ : u^{−1}u_{∗}𝒢 → 𝒢 ,
∏_{j}(𝒢_{j})|_{U_{ji}} = Γ(U_{i}, u^{−1}u_{∗}𝒢) → 𝒢_{i}
g ↦ g_{i} ∈ 𝒢_{ii} = 𝒢_{i}.

Definition 1.18. We define the SDC E^{∗} by:

E^{n} = Hom_{Λ}(𝒩 , (u^{−1}u_{∗})^{n}(𝒩 ))_{α^{n}},

where α^{n} = u^{−1}(α_{(u_{∗}u^{−1})^{n−1}ℳ_{0}} ∘ … ∘ α_{ℳ_{0}} ), with the SDC structure arising exactly as in the previous section (modulo contravariance).

Now, to give a sheaf ℳ of A-modules on X, such that u^{−1}ℳ = 𝒩 ⊗_{Λ} A, is to give transition functions

θ_{ji} : 𝒩_{i} ⊗ A → (𝒩_{j} ⊗ A)|_{U_{ij}},

satisfying

θ_{kj} ∘ θ_{ji} = θ_{ki}; θ_{ii} = id.

This is precisely the same as a map

Θ : 𝒩 ⊗ A → u^{−1}u_{∗}(𝒩 ⊗ A),

satisfying

(u^{−1}u_{∗}Θ) ∘ Θ = u^{−1}α_{u_{∗}𝒩_{A}} ∘ Θ; γ_{𝒩} ∘ Θ = id_{𝒩} .

If we include the additional condition that these transition functions must be compatible with those of ℳ_{0}, mod 𝔪_{A}, then we have precisely an element of MC_{E}(A). It is easy to see that equivalences of transition functions correspond to elements of E^{0}(A), acting via the adjoint action.

Note that we may regard E^{n} as

Γ(X′, ℋ om_{Λ}(𝒩 , (u^{−1}u_{∗})^{n}(𝒩 ))_{α^{n}}),

and that setting

ℰ^{n} = u_{∗}ℋ om_{Λ}(𝒩 , (u^{−1}u_{∗})^{n}(𝒩 ))_{α^{n}},

with tangent space C^{n}(ℰ), we have that the chain complex C^{•}(ℰ) is a flabby resolution of the sheaf ℋ om_{k}(ℳ_{0}, ℳ_{0}) (it is, in fact,a version of the Čech resolution), so that

H^{i}(E) = H^{i}(C^{•}(E)) = ℍ^{i}(X, C^{•}(ℰ)) = H^{i}(X, ℋ om_{k}(ℳ_{0}, ℳ_{0})),

as expected. Equivalently,

H^{i}(C^{•}(E)) = Ext_{k}^{i}(ℳ_{0}, ℳ_{0}) = H^{i}(X, ℋ om_{k}(ℳ_{0}, ℳ_{0})),

since we are working with sheaves of vector spaces over k, so ℋ om_{k} is exact.

2 A Procedure for Computing SDCs

In this section we make formal the approach which has been used so far to compute SDCs. The idea is that we throw away properties of the object which we wish to deform, until we obtain something whose deformations are trivial.

Throughout this section, we will encounter functors D : C_{Λ} → Cat. We will not require that these functors satisfy (H0) (the condition that F (k) = •).

Definition 2.1. Given a functor D : C_{Λ} → Cat, and an object D ∈ Ob D(k), define 𝔇𝔢𝔣_{D,D} : C_{Λ} → Grpd by setting 𝔇𝔢𝔣_{D,D}(A) to be the fibre of D(A) → D(k) over (D, id).

All the deformation problems we encounter are of this form.

Definition 2.2. We say a functor B : C_{Λ} → Cat has uniformly trivial deformation theory if 1. for all A ∈ C_{Λ} and all B_{1}, B_{2} ∈ Ob B(A), the functor Mor_{B}(B_{1}, B_{2}) : C_{A} → Set of

morphisms from B_{1} to B_{2} is smooth and homogeneous;

2. for A′ ↠ A in C_{Λ}, B(A′) → B(A) is essentially surjective.

A typical example of such a functor is that which sends A to the category of flat Amodules.

Observe that, if B is uniformly trivial, then given B ∈ Ob B(k), we may lift it to B̃ ∈ Ob B(Λ), and we have an equivalence of functors of groupoids

(B̃, Mor_{B}(B̃, B̃)) ⟶^{∼} 𝔇𝔢𝔣_{B,B}.

2.1 The SDC associated to a monadic adjunction

Assume we have a monadic adjunction

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

with unit η : id → UF = ⊤ and co-unit ε : F U → id, such that B has uniformly trivial deformation theory. Then, given D ∈ D(k), let B be any lift of UD to B(Λ), and define the SDC E^{∗} by

E^{n} = Hom_{B}(⊤^{n}B, B)_{U(ε_{D}∘...∘ε_{⊥^{n−1}D})},

the fibre of

Hom_{B}(⊤^{n}B, B) → Hom_{B(k)}(⊤^{n}B(k), B(k))

over U(ε_{D} ∘ … ∘ ε_{⊥^{n−1}D}).

We make E^{∗} into an SDC by giving it the product g ∗ h = g ∘ ⊤^{n}h, and Ĕ_{∗} is the dual complex to the canonical augmented complex B → ⊤^{∗}B. Explicitly, for g ∈ E^{n},

∂^{i}(g) = g ∘ ⊤^{i−1}Uε_{F ⊤^{n−i}B}
σ^{i}(g) = g ∘ ⊤^{i}η_{⊤^{n−i−1}B}.

Theorem 2.3. With the notation as above, we have an equivalence of functors of groupoids

𝔇𝔢𝔣_{D,D} ⟶^{∼} 𝔇𝔢𝔣_{E}.

Proof. Since the F ⊣ U is monadic, we have an equivalence K : D → B^{⊤}. Now observe that MC_{E} is precisely the fibre of

Ob B^{⊤} → Ob B^{⊤}(k)

over K(D) (note the comment after Lemma 1.7 gives σ^{0}(ω) = 1), with morphisms in the fibre

B^{⊤} → B^{⊤}(k)

over (K(D), id) corresponding to E^{0} acting on MC_{E} via the adjoint action. Hence

𝔇𝔢𝔣_{E} = 𝔇𝔢𝔣_{B^{⊤},K(D)}.

But K : D → B^{⊤} is an equivalence, giving an equivalence

K : 𝔇𝔢𝔣_{D,D} ⟶^{∼} 𝔇𝔢𝔣_{B^{⊤},K(D)} = 𝔇𝔢𝔣_{E}.

Observe that this result is precisely the same as the description above of deformations of algebras, and that the dual result (for comonadic adjunctions) describes deformations of sheaves.

2.1.1 Deformation of sheaves — Godement resolution

Take a sheaf of k-vector spaces ℳ_{0} on a site X with enough points. In examples, we will be considering the Zariski or étale sites of a scheme, or the site associated to a topological space. The deformation functor will associate to A, sheaves ℳ_{A} of flat A-modules such that ℳ_{A} ⊗ k = ℳ_{0}, modulo infinitesimal isomorphisms.

Let X′ be the set of points of X. Since X has enough points, the inverse image functor Shf(X) → ∏_{x∈X′} Shf(x) reflects isomorphisms. Explicitly, this says that a morphism θ : ℱ → 𝒢 of sheaves on X is an isomorphism whenever the morphisms θ_{x} : ℱ_{x} → 𝒢_{x} are for all x ∈ X′. In the reasoning which follows, it will suffice to replace X′ by any subset with this property.

If we are working on the Zariski site, we may take X′ = ∐_{x∈X} x. On the étale site, we may take

X′ = ∐_{x∈X} x̄,

where for each x ∈ X, a geometric point x̄ → X has been chosen. For Jacobson schemes, we may consider only closed points x ∈ X. We define the category Shf(X′) by Shf(X′) := ∏_{x∈X′} Shf(x).

There is an adjunction

\xymatrix{Shf(X′) \ar[r]^{u_{∗}}_{⊤} & Shf(X) \ar[l]^{u^{∗}}},

the maps u_{x} : x → X combining to form maps u^{∗} : Shf(X) → Shf(X′), and u_{∗} : Shf(X′) → Shf(X) given by u_{∗}ℱ = ∏_{x∈X′} u_{x∗}ℱ_{x}. Observe that the category of flat A-modules on X′ has uniformly trivial deformation theory.

It follows from Theorem 0.35 that this adjunction is comonadic, and from Theorem 2.5, the SDC governing this problem is

E^{n} = Hom_{Λ}(𝒩 , (u^{∗}u_{∗})^{n}(𝒩 ))_{α^{n}},

where 𝒩 is a flat μ-adic Λ-module on X′ with 𝒩 ⊗ k = u^{∗}ℳ_{0}, and α^{n} = u^{∗}(α_{(u_{∗}u^{∗})^{n−1}ℳ_{0}} ∘ … ∘ α_{ℳ_{0}} ).

To see this more clearly in the case of a topological space, observe, as in [God58], Remark II.4.3.2, that an element n ∈ Γ(U, u^{−1}u_{∗}𝒩 ) can be represented by a function of the form

n(x_{0}, x_{1}) ∈ 𝒩_{x_{1}},

defined on a set of the form

x_{0} ∈ U; x_{1} ∈ U(x_{0}),

where U(x_{0}) is some open étale neighbourhood of x_{0} in X. Two functions n_{1}, n_{2} define the same section if they agree on some set of the above form.

Thus an element Θ of MC_{E}(A) can be regarded as local transition functions; given n ∈ 𝒩_{x}, we obtain, for some étale neighbourhood U of x, Θ(n)_{xy} ∈ 𝒩_{y} for all y ∈ U, which can be regarded as lifting the germ and then projecting down to each stalk. The Maurer-Cartan equation is transitivity, and, as before, guarantees that Θ(n)_{xx} = n.

Again, we get

H^{i}(E) = H^{i}(C^{•}(E)) = ℍ^{i}(X, C^{•}(ℰ)) = H^{i}(X, ℋ om_{k}(ℳ_{0}, ℳ_{0})) = Ext_{k}^{i}(ℳ_{0}, ℳ_{0}),

as expected.

2.1.2 Deformation of co-algebras

Given a flat (co-associative) co-algebra (with co-unit) C_{0}/k, we wish to create an SDC describing flat deformations C_{A}/A such that C_{A} ⊗_{A} k = C_{0}, modulo infinitesimal isomorphisms.

There is, up to isomorphism, a unique flat μ-adic Λ-module M such that M ⊗_{Λ} k = C_{0}. There is an adjunction

\xymatrix{A−FCoAlg \ar[r]^{V} & AFMod \ar[l]^{G}|{⊥}},

between the category of flat co-algebras over A, and the category of flat modules over A, where V is the forgetful functor and the free functor G exists by the Special Adjoint Functor Theorem (since A−FCoAlg has all colimits, and V preserves these). Note that in this case the free functor is hard to write down explicitly, but this is unnecessary for our purposes. See [Swe69] for such a description.

By the Theorem 0.35, this adjunction is comonadic, so that deformations of C_{0} are given by the SDC

E^{n} = Hom_{Λ}(M, ⊥^{n}M)_{U(α_{⊤^{n−1}C_{0}}∘...∘α_{C_{0}})}.

2.2 The general approach

In general, we will not be able to pass from the category D to a category B with uniformly trivial deformation theory via a single monadic or comonadic adjunction. However, we should be able to pass from D to some B via a chain of monadic and comonadic adjunctions.

Not only should we have monadic and comonadic adjunctions, but, informally, the forgetful functors should commute with one another. More precisely, 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}^{⊣}}

where B has uniformly trivial deformation theory, with F ⊣ U monadic and G ⊢ V comonadic. Let

⊤_{h} = UF  ⊥_{h} = F U
⊥_{v} = V G  ⊤_{v} = GV,

with

η : 1 → ⊤_{h}, γ : ⊥_{v} → 1, ε : ⊥_{h} → 1 and α : 1 → ⊤_{v}.

The commutativity condition is that the following identities hold:

GU = UG or F V = V F  (1)
UV = V U  (2)
V ε = εV or Uα = αU  (3)
V η = ηV or Uγ = γU.  (4)

Observe that the adjoint properties ensure that identities on the same line are equivalent.

Lemma 2.4. For E ∈ E, A ∈ A, consider the diagram of isomorphisms given by the adjunctions:

[figure: Hom_{D}(F E, GA); Hom_{A}(V F E, A); Hom_{E}(E, UGA); Hom_{A}(F V E, A); Hom_{E}(E, GUA); Hom_{B}(V E, UA).]

The identities (1)–(4) ensure that all squares in this diagram commute. In particular, for any B ∈ B, the map

η_{B} ∘ γ_{B} : V GB → UF B

corresponds to a map

ρ_{B} = ε_{GF B} ∘ F Gη_{B} = GF γ_{B} ∘ α_{F GB} : F GB → GF B,

with

γ_{⊤_{h}B} ∘ UV (ρ_{B}) ∘ η_{⊥_{v}B} = η_{B} ∘ γ_{B},
U(ρ_{B}) ∘ η_{GB} = G(η_{B}),
γ_{F B} ∘ V (ρ_{B}) = F (γ_{B}).

By the naturality of ε : ⊥_{h} → id, we have ε ∘ (⊥_{h}ε) = ε ∘ (ε⊥_{h}), so we obtain a canonical map ε^{n} : ⊥_{h}^{n} → id, given by any such composition of ε’s. [For instance

ε^{n} = ε ∘ (⊥_{h}ε) ∘ … ∘ (⊥_{h}^{n−1}ε)

is one such composition.] We have similar maps for each of the units and co-units, giving:

ε^{n} : ⊥_{h}^{n} → id,  η^{n} : id → ⊤_{h}^{n}
α^{n} : id → ⊤_{v}^{n},  γ^{n} : ⊥_{v}^{n} → id

Let

δ = UV (ρ) : ⊤_{h}⊥_{v} → ⊥_{v}⊤_{h}.

Since δ is natural,

(δ⊥_{v}⊤_{h}) ∘ (⊤_{h}⊥_{v}δ) = (⊥_{v}⊤_{h}δ) ∘ (δ⊤_{h}⊥_{v}).

Therefore any composition of δ’s gives us the same canonical map

δ^{m,n} : ⊤_{h}^{m}⊥_{v}^{n} → ⊥_{v}^{n}⊤_{h}^{m}.

Theorem 2.5. Suppose we have a diagram as above. Then, for D ∈ Ob D(k), let B be any lift of UV D ∈ Ob B(k) to Ob B(Λ) (by the hypothesis on B, such a lift must exist and be unique up to isomorphism). Set

E^{n} = Hom_{B}(⊤_{h}^{n}B, ⊥_{v}^{n}B)_{UV (α_{D}^{n}∘ε_{D}^{n})}.

We give E^{∗} the multiplication

g ∗ h = ⊥_{v}^{n}(g) ∘ δ^{m,n} ∘ ⊤_{h}^{m}(h),

and Ĕ_{∗} gets its simplicial structure from the diagonal of the canonical double augmented complex

Hom_{B}(B → B^{∗}, B_{∗} → B).

Explicitly, for g ∈ E^{n},

∂^{i}(g) = ⊥_{v}^{n−i}V α_{G⊥_{v}^{i−1}B} ∘ g ∘ ⊤_{h}^{i−1}Uε_{F ⊤_{h}^{n−i}B}
σ^{i}(g) = ⊥_{v}^{n−i−1}γ_{⊥_{v}^{i}B} ∘ g ∘ ⊤_{h}^{i}η_{⊤_{h}^{n−i−1}B}.

Then

𝔇𝔢𝔣_{D,D} ≃ 𝔇𝔢𝔣_{E}.

Proof. First observe that this is, indeed, an SDC. We need to check that the product is associative. For f ∈ S^{l}, g ∈ S^{m}, h ∈ S^{n}, we have

f ∗ (g ∗ h) = ⊥_{v}^{m+n}f ∘ δ^{l,m+n} ∘ ⊤_{h}^{l}(⊥_{v}^{n}g ∘ δ^{m,n} ∘ ⊤_{h}^{m}h)
= ⊥_{v}^{m+n}f ∘ (⊥_{v}^{n}δ^{l,m} ∘ δ^{l,n}⊥_{v}^{m}) ∘ ⊤_{h}^{l}⊥_{v}^{n}g ∘ ⊤_{h}^{l}δ^{m,n} ∘ ⊤_{h}^{l+m}h
= ⊥_{v}^{m+n}f ∘ ⊥_{v}^{n}δ^{l,m} ∘ ⊥_{v}^{n}⊤_{h}^{l}g ∘ δ^{l,n}⊤_{h}^{m} ∘ ⊤_{h}^{l}δ^{m,n} ∘ ⊤_{h}^{l+m}h
= ⊥_{v}^{n}(⊥_{v}^{m}f ∘ δ^{l,m} ∘ ⊤_{h}^{l}g) ∘ δ^{l+m,n} ∘ ⊤_{h}^{l+m}h
= (f ∗ g) ∗ h.

We now have a lemma:

Lemma 2.6. To give a map f : ⊤_{h}B → ⊥_{v}B satisfying the Maurer-Cartan equation

⊥_{v}f ∘ δ ∘ ⊤_{h}f = V α_{GB} ∘ f ∘ Uε_{F B}

is the same as giving maps θ : ⊤_{h}B → B and ϕ : B → ⊥_{v}B satisfying the Maurer-Cartan equations:

θ ∘ ⊤_{h}θ = θ ∘ Uε_{F B}
⊥_{v}ϕ ∘ ϕ = V α_{GB} ∘ ϕ,

and the compatibility condition

⊥_{v}θ ∘ δ_{B} ∘ ⊤_{h}ϕ = ϕ ∘ θ.

Proof. (Of Lemma.)

Given such an f, we have

f = γ_{⊥_{v}B} ∘ V α_{GB} ∘ f ∘ Uε_{F B} ∘ η_{⊤_{h}B}
= γ_{⊥_{v}B} ∘ ⊥_{v}f ∘ δ_{B} ∘ ⊤_{h}f ∘ η_{⊤_{h}B}
= f ∘ γ_{⊤_{h}B} ∘ δ_{B} ∘ η_{⊥_{v}B} ∘ f
= f ∘ η_{B} ∘ γ_{B} ∘ f.

Let ϕ = f ∘ η_{B}, and θ = γ_{B} ∘ f. Now,

γ_{B} ∘ γ_{⊥_{v}B} ∘ ⊥_{v}f ∘ δ_{B} ∘ ⊤_{h}f = γ_{B} ∘ γ_{⊥_{v}B} ∘ V α_{GB} ∘ f ∘ Uε_{F B}
γ_{B} ∘ f ∘ γ_{⊤_{h}B} ∘ δ_{B} ∘ ⊤_{h}f = γ_{B} ∘ f ∘ Uε_{F B}
γ_{B} ∘ f ∘ U(γ_{F B} ∘ V (ρ_{B})) ∘ ⊤_{h}f = γ_{B} ∘ f ∘ Uε_{F B}
γ_{B} ∘ f ∘ ⊤_{h}γ_{B} ∘ ⊤_{h}f = γ_{B} ∘ f ∘ Uε_{F B}
θ ∘ ⊤_{h}θ = θ ∘ Uε_{F B},

using the Maurer-Cartan equations and Lemma 2.4. Similarly we obtain the Maurer-Cartan equation for ϕ, and finally

⊥_{v}θ ∘ δ_{B} ∘ ⊤_{h}ϕ = ⊥_{v}γ_{B} ∘ ⊥_{v}f ∘ δ_{B} ∘ ⊤_{h}f ∘ ⊤_{h}η_{B}
= ⊥_{v}γ_{B} ∘ V α_{GB} ∘ f ∘ Uε_{F B} ∘ ⊤_{h}η_{B}
= f
= ϕ ∘ θ.

Conversely, given such θ and ϕ, set f = ϕ ∘ θ. We obtain:

⊥_{v}f ∘ δ_{B} ∘ ⊤_{h}f = ⊥_{v}ϕ ∘ ⊥_{v}θ ∘ δ_{B} ∘ ⊤_{h}ϕ ∘ ⊤_{h}θ
= ⊥_{v}ϕ ∘ ϕ ∘ θ ∘ ⊤_{h}θ
= V α_{GB} ∘ ϕ ∘ θ ∘ Uε_{F B}
= V α_{GB} ∘ f ∘ Uε_{F B}.

Recall, from the comment after Lemma 1.7, that every element f of MC_{E}(A) must satisfy σ^{0}(f) = 1, which in this case is γ_{B} ∘ f ∘ η_{B} = id. This means that the pair (θ, ϕ) above satisfy θ ∘ η_{B} = id, and γ_{B} ∘ ϕ = id.

We obtain an adjunction

\xymatrix{E^{⊤_{h}} \ar[r]^{V} & B^{⊤_{h}} \ar[l]^{G}|{⊥}} ,

[figure: (; ⊤_{h}E; f; E; ); ↦; (; ⊤_{h}V E; V f; V E; )]

( \xymatrix{⊤_{h}GB \ar[d]_{G(g)∘U(ρ_{B})} \\ GB} ) ↤ ( \xymatrix{⊤_{h}B \ar[d]_{g} \\ B} ) ,

and such a pair (θ, ϕ) is precisely the same as an element of (B^{⊤_{h}})^{⊥_{v}}. We also have a correspondence between the action of E^{0} on MC_{E} and morphisms in (B^{⊤_{h}})^{⊥_{v}}, so that 𝔇𝔢𝔣_{E} corresponds to the fibre of

(B^{⊤_{h}})^{⊥_{v}} → (B^{⊤_{h}})^{⊥_{v}}(k)

over the image of (D, id).

Now, the equivalence

K : A → B^{⊤_{h}}
A ↦ ( \xymatrix{⊤_{h}UA \ar[d]_{Uε_{A}} \\ UA} )

arising from the monadic adjunction F ⊣ U satisfies

⊥_{v}KA = V G ( \xymatrix{⊤_{h}UA \ar[d]_{Uε_{A}} \\ UA} )

= ( \xymatrix{⊤_{h}⊥_{v}UA \ar[d]_{⊥_{v}Uε_{A}∘V U(ρ_{UA})} \\ ⊥_{v}UA} ) .

Moreover,

⊥_{v}Uε_{A} ∘ V U(ρ_{UA}) = UV (Gε_{A} ∘ (ε_{GF UA} ∘ F Gη_{UA}))
= UV ((ε_{GA} ∘ F UGε_{A}) ∘ F Gη_{UA})
= UV (ε_{GA} ∘ F G(Uε_{A} ∘ η_{UA}))
= UV (ε_{GA}),

so

⊥_{v}KA = ( \xymatrix{⊤_{h}⊥_{v}UA \ar[d]_{UV (ε_{GA})} \\ ⊥_{v}UA} ) = K⊥_{v}A.

This induces an equivalence of categories

A^{⊥_{v}} ⟶^{K_{h}} (B^{⊤_{h}})^{⊥_{v}}.

But G ⊢ V is comonadic, so

D ⟶^{K_{v}} A^{⊥_{v}}

is an equivalence. Thus we have an equivalence of groupoids

𝔇𝔢𝔣_{D,D} ⟶^{∼} 𝔇𝔢𝔣_{A^{⊥v} ,K_{v}D} ⟶^{∼} 𝔇𝔢𝔣_{(B^{⊤_{h}})^{⊥v} ,K_{h}K_{v}D} = 𝔇𝔢𝔣_{E}.

Given a category D, we now need a procedure for finding a category B with uniformly trivial deformation theory. We do this by thinking of the functors U and V as being forgetful functors, and looking for structure in D to discard.

Definition 2.7. Given a category B : C_{Λ} → Cat, define a set of structures Σ over B to consist of the following data:

1. A finite set Σ = Σ^{+} ⊔ Σ^{−},

2. For each subset S ⊂ Σ, a category B^{S}, with B^{∅} = B,

3. (a) for each s ∈ Σ^{+} and each S ⊂ Σ not containing s, a monadic adjunction

\xymatrix{B^{S∪\{s\}} \ar[r]^{U_{s}}_{⊤} & B^{S} \ar[l]^{F_{s}}}.

(b) for each s ∈ Σ^{−} and each S ⊂ Σ not containing s, a comonadic adjunction

\xymatrix{B^{S∪\{s\}} \ar[r]^{V_{s}} & B^{S} \ar[l]^{G_{s}}|{⊥}},

satisfying the commutativity conditions given below.

For s ∈ Σ^{+} we write

⊤_{s} = U_{s}F_{s} and ⊥_{s} = F_{s}U_{s},

with

η_{s} : 1 → ⊤_{s} and ε_{s} : ⊥_{s} → 1.

For s ∈ Σ^{−} we write

⊥_{s} = V_{s}G_{s} and ⊤_{s} = G_{s}V_{s},

with

γ_{s} : ⊥_{s} → 1, and α_{s} : 1 → ⊤_{s}.

The commutativity conditions are that the following hold:

1. For each distinct pair s_{1}, s_{2} ∈ Σ^{+}, and each S ⊂ Σ not containing s_{1}, s_{2}, the morphisms in the diagram

\xymatrix{B^{S∪\{s_{1},s_{2}\}} \ar[r]^{U_{1}}_{⊤} \ar[d]_{U_{2}}^{⊢} & B^{S∪\{s_{2}\}} \ar[l]^{F_{1}} \ar[d]_{U_{2}}^{⊢} \\ B^{S∪\{s_{1}\}} \ar[u]_{F_{2}} \ar[r]^{U_{1}}_{⊤} & B^{S} \ar[l]^{F_{1}} \ar[u]_{F_{2}}}

satisfy

F_{1}F_{2} = F_{2}F_{1} or U_{1}U_{2} = U_{2}U_{1}
U_{1}ε_{2} = ε_{2}U_{1}  U_{2}ε_{1} = ε_{1}U_{2}
U_{1}η_{2} = η_{2}U_{1}  U_{2}η_{1} = η_{1}U_{2}.

2. For each distinct pair s_{1}, s_{2} ∈ Σ^{−}, and each S ⊂ Σ not containing s_{1}, s_{2}, the morphisms in the diagram

\xymatrix{B^{S∪\{s_{1},s_{2}\}} \ar[r]^{V_{1}} \ar[d]_{V_{2}} & B^{S∪\{s_{2}\}} \ar[l]^{G_{1}}|{⊥} \ar[d]_{V_{2}} \\ B^{S∪\{s_{1}\}} \ar[u]_{G_{2}}^{⊣} \ar[r]^{V_{1}} & B^{S} \ar[l]^{G_{1}}|{⊥} \ar[u]_{G_{2}}^{⊣}}

satisfy

G_{1}G_{2} = G_{2}G_{1} or V_{1}V_{2} = V_{2}V_{1}
V_{1}α_{2} = α_{2}V_{1}  V_{2}α_{1} = α_{1}V_{2}
V_{1}γ_{2} = γ_{2}V_{1}  V_{2}γ_{1} = γ_{1}V_{2}.

3. For each s_{1} ∈ Σ^{+} and s_{2} ∈ Σ^{−}, and each S ⊂ Σ not containing s_{1}, s_{2}, the morphisms in the diagram

\xymatrix{B^{S∪\{s_{1},s_{2}\}} \ar[r]^{U}_{⊤} \ar[d]_{V} & B^{S∪\{s_{2}\}} \ar[l]^{F} \ar[d]_{V} \\ B^{S∪\{s_{1}\}} \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & B^{S}, \ar[l]^{F} \ar[u]_{G}^{⊣}}

satisfy

GU = UG or F V = V F

UV = V U

V ε = εV or Uα = αU
V η = ηV or Uγ = γU.

Lemma 2.8. Given a set of structures Σ over a category B, we have a diagram

\xymatrix{B^{Σ} \ar[r]^{U}_{⊤} \ar[d]_{V} & B^{Σ^{−}} \ar[l]^{F} \ar[d]_{V} \\ B^{Σ^{+}} \ar[u]_{G}^{⊣} \ar[r]^{U}_{⊤} & B, \ar[l]^{F} \ar[u]_{G}^{⊣}}

satisfying the equations (1)–(4) on p. 26, with F ⊣ U monadic and G ⊢ V comonadic.

Proof. We define F (resp. U) to be the composition of the F_{s} (resp U_{s}) for all s ∈ Σ^{+}, noting that the order of composition does not matter, since these functors commute with one another. We define G and V analogously. It is immediate that F ⊣ U and G ⊢ V are adjoint pairs, and that the commutativity conditions (1)–(4) are satisfied.

It thus remains only to show that F ⊣ U is monadic, and G ⊢ V comonadic. This can be done by using a similar approach to the proof of Theorem 2.5. The statement is that B^{Σ^{+}} ≃ B^{⊤_{+}}, where ⊤_{+} = F U, which is proved by induction on the cardinality of Σ^{+}.

Remarks 2.9. 1. This lemma shows that every set of structures can be replaced by a set of at most two elements, so it might seem that introducing the notion of a set of structures was not helpful. However, it is frequently easier to find the individual adjunctions than the composites, as will be seen in later examples (notably the deformation of a group scheme in Section 3.6).

2. When considering those deformation problems for which DGLAs have been successfully constructed, it seems that there is a set of structures which is either wholly monadic or wholly comonadic.

2.2.1 Deformation of Hopf algebras

Given a flat (associative, commutative, co-associative) Hopf algebra (with unit and counit) R_{0}/k, we wish to create an SDC describing flat deformations R_{A}/A of R_{0} such that R_{A} ⊗_{A} k = R_{0}, modulo infinitesimal isomorphisms.

The structures are:

Σ^{+} = {Algebra}, Σ^{−} = {Co-Algebra},

over the category of flat A-modules.

This gives the following commutative diagram of monadic and comonadic adjunctions:

\xymatrix{A−FHopfAlg \ar[r]_{⊤} \ar[d] & A−FCoAlg \ar[l]^{_{Symm_{A}}} \ar[d] \\ AFAlg \ar[u]_{G}^{⊣} \ar[r]_{⊤} & AFMod, \ar[u]_{G}^{⊣} \ar[l]^{_{Symm_{A}}}}

where A−FHopfAlg is the category of flat Hopf algebras over A, A−FCoAlg is the category of flat co-associative co-algebras with co-unit and co-inverse over A, and G is the free co-algebra functor of Section 2.1.2. Since we cannot describe G explicitly, we use the alternative form of the axioms involving Symm_{A} instead, when verifying the conditions of Theorem 2.5.

We thus obtain the SDC

E^{n} = Hom_{Λ}((Symm_{Λ})^{n}M, G^{n}M)_{α^{n}∘ε^{n}},

where M is the flat μ-adic Λ-module (unique up to isomorphism) lifting the k-vector space R_{0}. with α^{n}, ε^{n} the canonical maps

ε^{n} : (Symm_{k})^{n}R_{0} → R_{0}, α^{n} : R_{0} → G^{n}R_{0}

associated to R_{0}.

3 Deformation of Schemes

Given a flat μ-adic system of schemes S_{n}/Spec Λ_{n}, and a flat scheme X_{0}/S_{0}, our deformation functor consists of flat schemes X_{A}/S_{A} such that X_{A} ×_{Spec A} Spec k = X_{0}, modulo infinitesimal isomorphisms (isomorphisms which pull back to the identity on X_{0}), where 𝔖 = \underrightarrow{lim} S_{n} and S_{A} = 𝔖 ×_{Spf Λ} Spec A:

X_{0} ⟶ X_{A}
f flat↓  ↓_{flat}
S_{0} ⟶ S_{A}.

Since the topological space |X_{0}| underlying X_{0} does not deform, this is just a question of deforming the sheaf 𝒪_{X_{0}} of algebras. Observe that, since X_{0} is flat over S_{0}, any scheme X_{A}/S_{A} deforming X, and flat over A, will necessarily be flat over S_{A}. It therefore suffices to deform 𝒪_{X_{0}} as an 𝒪_{S_{A}}-algebra, flat over A.

In the notation of Section 2.2, we have the set of structures

Σ^{+} = {Algebra, 𝒪_{S_{A}}-Module}, Σ^{−} = {X_{0}-Sheaf},

over the category of sheaves of flat A-modules on X′_{0}, where X′_{0} is defined as in Section 2.1.1. To these structures correspond the monadic and comonadic adjunctions

{Symm_{A} ⊣ U, 𝒪_{𝔖} ⊗_{Λ} ⊣ U}, {u_{∗} ⊢ u^{−1}},

where U denotes the relevant forgetful functor.

This yields the following diagram of Cat-valued functors:

\xymatrix{𝒪_{S_{A}}\(AFAlg(X_{0})) \ar[r]_{⊤} \ar[d]_{u^{−1}} & AFMod(X_{0}) \ar[l]^{𝒪_{𝔖}⊗_{Λ}Symm_{A}} \ar[d]_{u^{−1}} \\ 𝒪_{S_{A}}\(AFAlg(X′_{0})) \ar[u]_{u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(X′_{0}). \ar[l]^{𝒪_{𝔖}⊗_{Λ}Symm_{A}} \ar[u]_{u_{∗}}^{⊣}}

where we write 𝒪_{𝔖} for the sheaf f^{−1}𝒪_{𝔖} (resp. u^{−1}f^{−1}𝒪_{𝔖}) on X_{0} (resp. X′_{0}), and 𝒪_{S_{A}}\(AFAlg) consists of those 𝒪_{Y_{A}}-algebras which are flat over A. The only nontrivial commutativity condition is the observation that pull-backs commute with tensor operations.

Hence, by Theorem 2.5, deformations are described by the SDC

E^{n} = Hom_{Λ}((𝒪_{𝔖} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (u^{−1}u_{∗})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})},

where ℳ is a lift of the sheaf u^{−1}𝒪_{X_{0}} of vector spaces on X′_{0} to a sheaf of flat μ-adic Λ-modules, and α^{n}, ε^{n} are the canonical maps α_{𝒪_{X_{0}}}^{n} , ε_{𝒪_{X_{0}}}^{n} in Theorem 2.5, given by the adjunctions. Remark 3.1. Observe that the description above allows us to construct an SDC governing deformations of a sheaf of algebras on any site with enough points. This is more general than the problem in [Hin05] for which a DGLA was constructed.

Define a sheaf of SDCs on X by

ℰ^{n} = u_{∗}ℋ om_{Λ}((𝒪_{𝔖} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (u^{−1}u_{∗})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{m})}.

Combining the observations in Sections 1.2.1, 2.1.1, we see that this has tangent space

C^{n}(ℰ) = u_{∗}𝒟er_{𝒪_{S_{0}}} ((𝒪_{S_{0}} ⊗_{k} Symm_{k})^{n+1}u^{−1}𝒪_{X_{0}}, (u^{−1}u_{∗})^{n}u^{−1}𝒪_{X_{0}})
≅ 𝒟er_{𝒪_{S_{0}}} ((𝒪_{S_{0}} ⊗_{k} Symm_{k})^{n+1}𝒪_{X_{0}}, (u_{∗}u^{−1})^{n+1}𝒪_{X_{0}})
≅ ℋ om_{𝒪_{X_{0}}} (L_{n}^{X_{0}/S_{0}}, 𝒞^{n}(𝒪_{X_{0}})),

where L_{•}^{X_{0}/S_{0}} is the standard form of the cotangent complex, as described in [Ill71], and 𝒞^{n} denotes the Godement resolution.

Since C^{n}(ℰ) has the structure of a diagonal complex of a bicosimplicial complex, it follows from the Eilenberg-Zilber Theorem that it is quasi-isomorphic to the total complex of the double complex

ℋ om_{𝒪_{X_{0}}} (L_{•}^{X_{0}/S_{0}}, 𝒞^{•}(𝒪_{X_{0}}))

Therefore the cohomology of our SDC is

H^{i}(C^{•}(E)) = H^{i}(Γ(X_{0}, C^{•}(ℰ))) ≅ 𝔼xt_{𝒪_{X_{0}}}^{i} (L_{•}^{X_{0}/S_{0}}, 𝒪_{X_{0}}),

which is André-Quillen hypercohomology, the second isomorphism following because L_{•}^{X_{0}/S_{0}} is locally projective as an 𝒪_{X_{0}}-module, and 𝒞^{•}(𝒪_{X_{0}}) is flabby.

3.1 Separated Noetherian schemes

If X_{0} is separated and Noetherian, then we may replace Godement resolutions by Čech resolutions. Take an open affine cover (X_{α})_{α∈I} of X_{0}, and set X̌ := ∐_{α∈I} X_{α}. We then have a diagram

\xymatrix{X̌′ \ar[r]^{ǔ} \ar[dr]^{w} \ar[d]_{v′} & X̌ \ar[d]^{v} \\ X′_{0} \ar[r]^{u} & X_{0}.}

Since v^{−1}𝒪_{X_{0}} = 𝒪X̌ is a quasi-coherent sheaf on X̌, Ext^{i}(𝒪X̌, 𝒪X̌) = 0 for all i > 0, so deformations of the sheaf 𝒪X̌ of k-vector spaces on X̌ are unobstructed and we may lift v^{−1}𝒪_{X_{0}} to some sheaf 𝒩 of flat μ-adic Λ-modules on X̌, unique up to non-unique isomorphism. There must also be (non-canonical) isomorphisms ǔ^{−1}𝒩 ≅ v′^{−1}ℳ. If we define Ě^{•} by

Ě^{n} = Hom_{Λ}((𝒪_{𝔖} ⊗_{Λ} Symm_{Λ})^{n}𝒩 , (v^{−1}v_{∗})^{n}𝒩 )_{v^{−1}(α^{n}∘ε^{n})},

and F^{•} similarly, replacing v by w, then we have morphisms of SDCs

E^{•} ⟶^{γ_{v′}^{•}} F^{•} ⟵^{γ• ; ǔ} Ě^{•}.

To see that these are quasi-isomorphisms, let 𝒯 ^{•} := ℋ om_{𝒪_{X_{0}}} (L_{•}^{X_{0}/S_{0}}, 𝒪_{X_{0}}), which is a complex of quasi-coherent sheaves on X_{0}, and observe that the induced maps on cohomology are

ℍ^{i}(X_{0}, 𝒯 ^{•}) → ℍ^{i}(X_{0}, 𝒯 ^{•}) ← ℍ̌^{i}(X_{0}, 𝒯 ^{•}).

The final map is an isomorphism since X_{0} is separated and Noetherian, so Čech cohomology agrees with sheaf cohomology.

3.2 Smooth schemes

If X_{0}/S_{0} is smooth, then the cohomology groups are just H^{i}(X_{0}, 𝒯_{X_{0}/S_{0}}), the cohomology of the tangent sheaf. If X_{0} is also separated and Noetherian (for instance if S_{0} is so), then we may lift X̌ to some smooth μ-adic formal scheme 𝔛̌ over 𝔖.

Consider the diagram

\xymatrix{𝒪_{S_{A}}\(AFAlg(X̌)) \ar[r]^{v_{∗}}_{⊤} & 𝒪_{S_{A}}\(AFAlg(X_{0})) \ar[l]^{v^{∗}}} ;

although the former category does not have uniformly trivial deformation theory, all the morphisms in it which we encounter do, which gives us an SDC and canonical maps

Hom_{𝒪_{𝔖}−Alg} (𝒪𝔛̌, (v^{−1}v_{∗})^{n}𝒪𝔛̌)_{v^{−1}(α^{n})} ⟶^{(ε^{n})^{∗}} Ě^{n},

which give a quasi-isomorphism of SDCs.

Explicitly, we may write our SDC as

E^{n} = ∏_{α_{0},...,α_{n}∈I} Γ(X_{α_{0},...,α_{n}}, ℋ om_{𝒪_{𝔖}−Alg} (𝒪_{𝔛_{αn}}, 𝒪_{𝔛_{α_{0}}})_{id}),

where X_{α_{0},...,α_{n}} = ⋂_{i=1}^{n} X_{α_{i}}, and 𝔛̌ = ∐ 𝔛_{α}. The product is given by

(ϕ ∗ ψ)_{α_{0},...,α_{m+n}} = ϕ_{α_{0},...,α_{m}} ∘ ψ_{α_{m},...,α_{m+n}},

and operations are

(∂^{i}ϕ)_{α_{0},...,α_{n+1}} = ϕ_{α_{0},...,α_{i−1},α_{i+1},...,α_{n+1}}
(σ^{i}ϕ)_{α_{0},...,α_{n−1}} = ϕ_{α_{0},...,α_{i},α_{i},...,α_{n−1}}

3.3 Deformation of a subscheme

The standard application of the previous section would be to set S_{n} = Spec Λ_{n}. However, exactly the same proof applies to a far wider class of problems. Rather than requiring both X_{A}/S_{A} and 𝔖/Spf Λ to be flat, we only used the fact that X_{A}/Spec A was flat (flatness of X_{A}/S_{A} then being automatic, since X_{0}/S_{0} flat). This gives us the following immediate generalisation:

Given

\xymatrix{X_{0} \ar[dr]_{flat} \ar[r] & Y_{0} \ar[d]^{flat} \\  & S_{0} \ar[d] \\  & Spec k,}

for flat μ-adic families

𝔜 ⟶^{flat} 𝔖 ⟶^{flat} Spf Λ,

let Y_{A} := 𝔜 ⊗_{Λ} A, S_{A} := 𝔖 ⊗_{Λ} A, and consider deformations

[figure: X_{0}; Y_{0}; S_{0}; X_{A}; flat; Y_{A}; flat; S_{A},]

such that X_{A} ⊗_{A} k = X_{0}.

Thus we need to consider 𝒪_{Y_{A}}-algebras deforming 𝒪_{X_{0}}, flat over 𝒪_{S_{A}}. These are the same as 𝒪_{Y_{A}}-algebras deforming 𝒪_{X_{0}}, flat over A.

We have the set of structures

Σ^{+} = {Algebra, 𝒪_{Y_{A}}-Module}, Σ^{−} = {X_{0}-Sheaf},

over the category AFMod(X′_{0}), giving the diagram:

\xymatrix{𝒪_{Y_{A}}\(AFAlg(X_{0})) \ar[r]_{⊤} \ar[d]_{u^{−1}} & AFMod(X_{0}) \ar[l]^{𝒪_{𝔜}⊗_{Λ}Symm_{A}} \ar[d]_{u^{−1}} \\ 𝒪_{Y_{A}}\(AFAlg(X′_{0})) \ar[u]_{u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(X′_{0}) \ar[l]^{𝒪_{𝔜}⊗_{Λ}Symm_{A}} \ar[u]_{u_{∗}}^{⊣}}

and we obtain the same SDC as before, with 𝔖 replaced by 𝔜. Hence deformations are described by the SDC

E^{n} = Hom_{Λ}((𝒪_{𝔜} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (u^{−1}u_{∗})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})},

for a flat μ-adic Λ-module ℳ lifting u^{−1}𝒪_{X_{0}}.

In particular this applies to deformation of a subscheme X_{0} ↪ Y_{0}, since a flat deformation of a subscheme as a Y_{A}-scheme will be a subscheme. If X_{0} ↪ Y_{0} is a regular embedding (e.g. if X_{0} and Y_{0} are both smooth over S_{0}), then the cohomology groups will be

H^{i}(C^{•}(E)) = H^{i+1}(X_{0}, 𝒩_{X_{0}/Y_{0}}),

where 𝒩_{X_{0}/Y_{0}} is the normal sheaf.

If X_{0} and Y_{0} are both smooth over S_{0}, we can simplify the SDC of Section 3.1 still further. Consider the diagram

\xymatrix{𝒪_{Y_{A}}\(AFAlg(X_{0})) \ar[r]_{⊤} \ar[d]_{v^{−1}} & 𝒪_{S_{A}}\(AFAlg(X_{0})) \ar[l]^{𝒪_{𝔜}⊗_{𝒪_{𝔖}}} \ar[d]_{v^{−1}} \\ 𝒪_{Y_{A}}\(AFAlg(X̌)) \ar[u]_{v_{∗}}^{⊣} \ar[r]_{⊤} & 𝒪_{S_{A}}\(AFAlg(X̌)). \ar[l]^{𝒪_{𝔜}⊗_{𝒪_{𝔖}}} \ar[u]_{v_{∗}}^{⊣}}

Although the last category does not have uniformly trivial deformation theory, all the morphisms in it which we encounter do. If we let (f^{♯})^{⊗n} denote the canonical ring homomorphism 𝒪_{𝔜}^{⊗_{𝒪_{𝔖}} n} ⊗_{𝒪_{𝔖}} 𝒪𝔛̂ → 𝒪𝔛̂, then there is an SDC and a quasi-isomorphism

Hom_{𝒪_{𝔖}−Alg} (𝒪_{𝔜}^{⊗_{𝒪_{𝔖}} n} ⊗_{𝒪_{𝔖}} 𝒪𝔛̂, (v^{−1}v_{∗})^{n}𝒪𝔛̂)_{v^{−1}α^{n}∘(f^{♯})^{⊗n}} → Ě^{n}.

The map is defined by composing the maps Symm_{Λ}(𝒪_{𝔜}^{⊗n} ⊗ 𝒪𝔛̂) → 𝒪_{𝔜}^{⊗n} ⊗ 𝒪𝔛̂, arising from the natural ring structure on the tensor product.

Explicitly, we may write our SDC as

E^{n} = ∏_{α_{0},...,α_{n}∈I} Γ(X_{α_{0},...,α_{n}}, ℋ om_{𝒪_{𝔖}−Alg} (𝒪_{𝔜}^{⊗_{𝒪_{𝔖}} n} ⊗_{𝒪_{𝔖}} 𝒪_{𝔛_{αn}}, 𝒪_{𝔛_{α_{0}}})_{(f^{♯})^{⊗n}}),

The product is given by

(ϕ ∗ ψ)_{α_{0},...,α_{m+n}}(r_{1} ⊗ … ⊗ r_{m+n} ⊗ a) =
ϕ_{α_{0},...,α_{m}}(r_{1} ⊗ … ⊗ r_{m} ⊗ ψ_{α_{m},...,α_{m+n}}(r_{m+1} ⊗ … ⊗ r_{m+n} ⊗ a)) ,

and operations are

(∂^{i}ϕ)_{α_{0},...,α_{n+1}}(r_{1} ⊗ … ⊗ r_{n+1} ⊗ a) =
ϕ_{α_{0},...,α_{i−1},α_{i+1},...,α_{n+1}}(r_{1} ⊗ … ⊗ r_{i−1} ⊗ (r_{i}r_{i+1}) ⊗ r_{i+2} … ⊗ r_{n+1} ⊗ a)
(σ^{i}ϕ)_{α_{0},...,α_{n−1}}(r_{1} ⊗ … ⊗ r_{n−1} ⊗ a) =
ϕ_{α_{0},...,α_{i},α_{i},...,α_{n−1}}(r_{1} ⊗ … ⊗ r_{i} ⊗ 1 ⊗ r_{i+1} … ⊗ r_{n−1} ⊗ a).

To see that the map is a quasi-isomorphism, observe that the Eilenberg-Zilber Theorem allows us to regard the cohomology of E as hypercohomology of the complex 𝒯_{X_{0}/S_{0}} → f^{∗}𝒯_{Y_{0}/S_{0}}[−1], which is quasi-isomorphic to the tangent complex of the morphism X_{0} → Y_{0}. Note that this SDC works for any morphism of smooth schemes, not only for embeddings.

3.4 Constrained deformation of schemes

Given a morphism of flat μ-adic schemes ℨ ⟶^{h} 𝔜 over 𝔖, and a diagram

Z_{0} ⟶^{g} X_{0} ⟶^{f} Y_{0}

over S_{0}, with fg = h, our deformation functor consists of diagrams

Z_{A} ⟶^{g_{A}} X_{A} ⟶^{f_{A}} Y_{A},

over S_{A}, with X_{A} → S_{A} flat and f_{A} ∘ g_{A} = h, which pull back along S_{0} → S_{A} to our original diagram. Here, Z_{A} is the scheme ℨ ×_{Spf Λ} Spec A, and Z_{0} = Z_{k}. An example of such a deformation problem would be deformations of a subscheme, constrained to pass through a fixed set of points.

Since the topological space |X_{0}| underlying X_{0} does not deform, it suffices to deform 𝒪_{X_{0}} as an 𝒪_{Y_{A}}-algebra with an 𝒪_{Z_{A}}-augmentation, flat over A.

In the notation of Section 2.2, we have the set of structures

Σ^{+} = {algebra, 𝒪_{Y_{A}}-module}, Σ^{−} = {X_{0}-sheaf, 𝒪_{Z_{A}}-augmented},

over the category of sheaves of flat A-modules on X′_{0}, where X′_{0} is defined as in Section 2.1.1.

To these structures correspond the monadic and comonadic adjunctions

{Symm_{A} ⊣ U, 𝒪_{𝔜} ⊗_{Λ} ⊣ U}, {u_{∗} ⊢ u^{−1}, 𝒪_{ℨ} × ⊢ V },

where U, V denote the relevant forgetful functors.

This yields the following diagram of Cat-valued functors:

\xymatrix{𝒪_{Y_{A}}\(AFAlg(X_{0}))/𝒪_{Z_{A}} \ar[r]_{⊤} \ar[d]_{u^{−1}} & AFMod(X_{0})/𝒪_{Z_{A}} \ar[l]^{𝒪_{𝔜}⊗_{Λ}Symm_{A}} \ar[d]_{u^{−1}} \\ 𝒪_{Y_{A}}\(AFAlg(X′_{0})) \ar[u]_{𝒪_{Z_{A}}×u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(X′_{0}). \ar[l]^{𝒪_{𝔜}⊗_{Λ}Symm_{A}} \ar[u]_{𝒪_{Z_{A}}×u_{∗}}^{⊣}}

where we write 𝒪_{𝔜} for the sheaf f^{−1}𝒪_{𝔜} (resp. u^{−1}f^{−1}𝒪_{𝔜}), and 𝒪_{ℨ} for the sheaf g_{∗}𝒪_{ℨ} (resp. u^{−1}g_{∗}𝒪_{ℨ}) on X_{0} (resp. X′_{0}). The category 𝒪_{Y_{A}}\(AFAlg)/𝒪_{Z_{A}} consists of those 𝒪_{Y_{A}}-algebras which are flat over A, and equipped with an augmentation to 𝒪_{Z_{A}}, compatible with h^{♯}. The only non-trivial commutativity condition is the observation that pull-backs commute with tensor operations.

By Theorem 2.5, this deformation problem is governed by the SDC

E^{n} = Hom_{Λ}((𝒪_{𝔜} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (𝒪_{ℨ} × u^{−1}u_{∗})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})},

where ℳ is a lift of the sheaf u^{−1}𝒪_{X_{0}} of vector spaces on X′_{0} to a sheaf of flat μ-adic Λ-modules, and α^{n}, ε^{n} are the canonical maps α_{𝒪_{X_{0}}}^{n} , ε_{𝒪_{X_{0}}}^{n} in Theorem 2.5, given by the adjunctions. The cohomology is

𝔼xt_{𝒪_{X_{0}}}^{i} (L_{•}^{X_{0}/Y_{0}}, 𝒪_{X_{0}} ⟶^{g^{♯}} g_{∗}𝒪_{Z_{0}}[−1]),

If X_{0} and Y_{0} are both smooth over S_{0}, we may consider the diagram

\xymatrix{𝒪_{Y_{A}}\(AFAlg(X_{0}))/𝒪_{Z_{A}} \ar[r]_{⊤} \ar[d]_{v^{−1}} & 𝒪_{S_{A}}\(AFAlg(X_{0}))/𝒪_{Z_{A}} \ar[l]^{𝒪_{𝔜}⊗_{𝒪_{𝔖}}} \ar[d]_{v^{−1}} \\ 𝒪_{Y_{A}}\(AFAlg(X̌))/𝒪_{Z_{A}} \ar[u]_{𝒪_{Z_{A}}×v_{∗}}|{⊣} \ar[r]_{⊤} & 𝒪_{S_{A}}\(AFAlg(X̌)). \ar[l]^{𝒪_{𝔜}⊗_{𝒪_{S_{A}}}} \ar[u]_{𝒪_{Z_{A}}×v_{∗}}^{⊣}}

All the morphisms which we encounter in the last category have uniformly trivial deformation theory, allowing us to replace this SDC by

E^{n} = ∏_{α_{0},...,α_{n}∈I} Γ(X_{α_{0},...,α_{n}}, ℋ om_{𝒪_{𝔖}−Alg} (𝒪_{𝔜}^{⊗_{𝒪_{𝔖}} n} ⊗ 𝒪_{𝔖}𝒪_{𝔛_{αn}}, 𝒪_{𝔛_{α_{0}}} × 𝒪_{ℨ}^{n})_{(g^{♯})^{n}∘(f^{♯})^{⊗n}}),

since cohomology in this case is hypercohomology of the complex

𝒯_{X_{0}/S_{0}} → (f^{∗}𝒯_{Y_{0}/S_{0}} ⊕ 𝒯_{X_{0}/S_{0}} ⊗_{𝒪_{X_{0}}} g_{∗}𝒪_{Z_{0}})[−1] → f^{∗}𝒯_{Y_{0}/S_{0}} ⊗_{𝒪_{X_{0}}} g_{∗}𝒪_{Z_{0}}[−2].

3.5 Deformation of a pair

Given a flat μ-adic formal scheme 𝔖, and flat schemes

\xymatrix{X_{0} \ar[rr]^{f_{0}} \ar[dr] &  & Y_{0} \ar[dl] \\  & S_{0},}

let S_{A} := 𝔖 ×_{Spf Λ} Spec A. Our deformation functor consists of (infinitesimal) isomorphism classes of:

[figure: X_{0}; f_{0}; Y_{0}; S_{0}; X_{A}; f_{A}; flat; Y_{A}; flat; S_{A},]

such that X_{A} ×_{Spec A} Spec k = X_{0}, Y_{A} ×_{Spec A} Spec k = Y_{0}, and f_{A} ⊗_{A} k = f_{0}.

In this case, we obtain the SDC from the square of adjunctions

(\begin{matrix}𝒪_{S_{A}}\(AFAlg(Y_{0})) \\ ↓ & f_{0}^{♯} \\ 𝒪_{S_{A}}\(AFAlg(X_{0})) \end{matrix} ) ⟶ ; ⊤ ; ⟵_{𝒪_{𝔖}⊗_{Λ}Symm_{A}}  (\begin{matrix}AFMod(Y_{0}) \\ ↓ & f_{0}^{♯} \\ AFMod(X_{0}) \end{matrix})

u_{∗} ↑⊢↓ u^{−1}  u_{∗} ↑⊢↓ u^{−1}

(\begin{matrix}𝒪_{S_{A}}\(AFAlg(Y′_{0})) \\ ↓ & (f′_{0})^{♯} \\ 𝒪_{S_{A}}\(AFAlg(X′_{0})) \end{matrix} ) ⟶ ; ⊤ ; ⟵_{𝒪_{𝔖}⊗_{Λ}Symm_{A}} (\begin{matrix}AFMod(Y′_{0}) \\ ↓ & (f′_{0})^{♯} \\ AFMod(X′_{0}) \end{matrix}) ,

where for categories A, B, with a functor F : A → B, (A ⟶^{F} B) is the category whose objects are triples

(A, B, F A ⟶^{h} B),

and whose morphisms are pairs f : A → A′, g : B → B′ such that the following square commutes:

F A ⟶^{h} B
F f↓  ↓^{g}
F A′ ⟶^{h′} B′.

3.6 Deformation of a group scheme

Given a group scheme G_{0}/k, we consider deformations

G_{0} ⟶ G_{A}
↓  ↓_{flat}
Spec k ⟶ Spec A,

where G_{A} is a group scheme over A with G_{A} ⊗_{A} k = G_{0}.

Since the topological space |G_{0}| does not deform, we need only consider deformations of 𝒪_{G_{0}} as a sheaf of Hopf algebras. The structures are:

Σ^{+} = {Algebra}, Σ^{−} = {G_{0}-Sheaf, Co-Algebra},

over the category of sheaves of flat A-modules on G′_{0}.

This gives the following commutative diagram of monadic and comonadic adjunctions:

\xymatrix{A−FHopfAlg(G_{0}) \ar[r]_{⊤} \ar[d]_{u^{−1}} & A−FCoAlg(G_{0}) \ar[l]^{_{Symm_{A}}} \ar[d]_{u^{−1}} \\ AFAlg(G′_{0}) \ar[u]_{Q∘u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(G′_{0}), \ar[u]_{Q∘u_{∗}}^{⊣} \ar[l]^{_{Symm_{A}}}}

where A−FHopfAlg is the category of flat Hopf algebras over A, A−FCoAlg is the category of flat co-associative co-algebras with co-unit and co-inverse over A, and Q is the free co-algebra functor of Section 2.1.2. We thus obtain the SDC

E^{n} = Hom_{Λ}((Symm_{Λ})^{n}ℳ , (u_{∗} ∘ Q ∘ u^{−1})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})},

where ℳ is the lift (unique up to isomorphism) of the sheaf u^{−1}𝒪_{G_{0}} of k-vector spaces to a flat μ-adic sheaf on G′, with α^{n}, ε^{n} the canonical maps

ε^{n} : (Symm_{k})^{n}𝒪_{G_{0}} → 𝒪_{G_{0}}, α^{n} : 𝒪_{G_{0}} → (u_{∗} ∘ Q ∘ u^{−1})^{n}𝒪_{G_{0}}

associated to 𝒪_{G_{0}}.

3.6.1 Deformation of a sub-(group scheme)

The problem is now, given a flat μ-adic formal group scheme ℌ over Λ, to deform a sub-(group scheme) G_{0} → H_{0} as a sub-(group scheme) of H_{A} := ℌ ⊗_{Λ} A, flat over A. In fact, we will not need to assume that G_{0} → H_{0} is a subscheme — any morphism will do. The picture is:

[figure: G_{0}; H_{0}; Spec k; G_{A}; flat; H_{A}; flat; Spec A,]

all morphisms as group schemes over Λ.

The structures are:

Σ^{+} = {Algebra, 𝒪_{H_{A}}-Module}, Σ^{−} = {G_{0}-Sheaf, Coalgebra},

so the diagram

\xymatrix{𝒪_{H_{A}}\(A−FHopfAlg(G_{0})) \ar[r]_{⊤} \ar[d]_{u^{−1}} & A−FCoAlg(G_{0}) \ar[l]^{𝒪_{ℌ}⊗_{Λ}Symm_{A}} \ar[d]_{u^{−1}} \\ 𝒪_{H_{A}}\(AFAlg(G′_{0})) \ar[u]_{Q∘u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(G′_{0}) \ar[l]^{𝒪_{ℌ}⊗_{Λ}Symm_{A}} \ar[u]_{Q∘u_{∗}}^{⊣}}

gives us our SDC,

E^{n} = Hom_{Λ}((𝒪_{ℌ} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (u_{∗} ∘ Q ∘ u^{−1})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})}.

3.7 G-invariant deformations

Given a flat μ-adic formal group scheme 𝔊 over Spf Λ, a flat μ-adic formal scheme 𝔗/Λ, and a flat scheme X_{0}/k, with a morphism X_{0} → T_{0}, and compatible group actions

𝔗 ˆ×_{Λ}𝔊 → 𝔗, r : X_{0} ×_{k} G_{0} → X_{0},

where G_{0} = 𝔊 ⊗_{Λ} k, we seek deformations

[figure: X_{0}; T_{0}; Spec k; X_{A}; flat; T_{A}; flat; Spec A,]

where T_{A} = 𝔗 ⊗_{Λ} A and G_{A} = 𝔊 ⊗_{Λ} A, together with a compatible group action X_{A} ×_{A} G_{A} → X_{A}. Since the underlying topological space does not deform, we need only consider G_{A}-linearised sheaves ℱ of algebras, i.e sheaves with co-associative maps ℱ → r_{∗}(ℱ ⊗_{A} 𝒪_{G_{A}}), deforming 𝒪_{X_{0}}.

The structures are:

Σ^{+} = {Algebra, 𝒪_{T_{A}}-Module}, Σ^{−} = {X_{0}-Sheaf, G_{A}-linear},

where a G_{A}-linearised module is a module M equipped with a co-associative morphism

M → M ⊗ 𝒪_{G_{A}}.

The relevant diagram is then:

\xymatrix{𝒪_{T_{A}}\((AFAlg(X_{0}))^{G_{A}}) \ar[r]_{⊤} \ar[d]_{u^{−1}} & (AFMod(X_{0}))^{G_{A}} \ar[l]^{𝒪_{𝔗}⊗_{Λ}Symm_{A}} \ar[d]_{u^{−1}} \\ 𝒪_{T_{A}}\(AFAlg(X′_{0})) \ar[u]_{R∘u_{∗}}^{⊣} \ar[r]_{⊤} & AFMod(X′_{0}), \ar[u]_{R∘u_{∗}}^{⊣} \ar[l]^{𝒪_{𝔗}⊗_{Λ}Symm_{A}}}

where (AFMod)^{G_{A}} denotes G_{A}-linearised module, and R is the co-free functor

R(ℱ) = r_{∗}(ℱ ⊗_{Λ} 𝒪_{𝔊}),

whose 𝔊-action is given by:

r_{∗}(ℱ ⊗_{Λ} 𝒪_{𝔊}) ⟶^{r_{∗}(id⊗m^{♯})} r_{∗}(ℱ ⊗ m_{∗}(𝒪_{𝔊} ⊗ 𝒪_{𝔊})) = r_{∗}(r_{∗}(ℱ ⊗_{Λ} 𝒪_{𝔊}) ⊗ 𝒪_{𝔊}),

as required. The unit of the adjunction V ⊣ R, for forgetful functor V, is

ρ : ℱ → RV (ℱ) = r_{∗}(ℱ ⊗_{Λ} 𝒪_{𝔊}),

where ρ is the structure map of ℱ as a 𝔊-linearised sheaf. The co-unit is

r_{∗}(id ⊗ e^{♯}) : V R(ℱ) = r_{∗}(ℱ ⊗_{Λ} 𝒪_{𝔊}) → ℱ .

This gives the SDC

E^{n} = Hom_{Λ}((𝒪_{𝔗} ⊗_{Λ} Symm_{Λ})^{n}ℳ , (u_{∗} ∘ R ∘ u^{−1})^{n}ℳ)_{u^{−1}(α^{n}∘ε^{n})},

where ℳ is the lift (unique up to isomorphism) of the sheaf u^{−1}𝒪_{X_{0}} of k-vector spaces to a flat μ-adic sheaf on X′.

4 Representations of the Algebraic Fundamental Group

Fix a connected scheme X, a geometric point x̄ → X, and a ring Λ as in Section 0.1, with finite residue field k. Throughout, A will denote a ring in C_{Λ}. Denote π_{1}(X, x̄) by Γ.

4.1 Representations to GL_{n}

It is well known (e.g. [Mil80] Theorem 5.3), that there is an equivalence of categories between finite π_{1}(X, x̄)-sets, and locally constant étale sheaves on X, with finite stalks. There is thus an equivalence of categories between representations

ρ : π_{1}(X, x̄) → GL_{n}(A),

and locally constant étale sheaves 𝕍 on X with stalks A^{n}. Given ρ, the corresponding sheaf 𝕍_{ρ} is given by

𝕍_{ρ} = A^{n} ×_{Γ,ρ} ℱ_{Y} = (π_{∗}A^{n})^{Γ,ρ},

where ℱ_{Y} is the sheaf on X whose espace étalé is Y, and ρ|_{π_{1}(Y,ȳ)} = 1, which must happen for some Galois cover Y ⟶^{π} X, A being finite.

If we now fix a representation ρ_{0} : Γ → GL_{n}(k), then we have an equivalence of categories (or, rather, groupoids)

ℜ_{A}(ρ_{0}) ↔ 𝔙_{A}(𝕍_{0}),

where ℜ_{A}(ρ_{0}) has objects

ρ ≡ ρ_{0} mod 𝔪_{A},

while 𝔙_{A}(𝕍_{0}) has objects

𝕍 ≡ 𝕍_{0} mod 𝔪_{A}.

In both cases morphisms must be the identity mod 𝔪_{A}. Explicitly, a morphism in ℜ_{A}(ρ_{0}) is an element g ∈ ker(GL_{n}(A) → GL_{n}(k)), acting by conjugation.

Hence this deformation problem is just the same as deforming sheaves of modules, as in Section 2.1.1, giving the SDC

E^{n} = Hom_{Λ}(𝒱 , (u^{∗}u_{∗})^{n}(𝒱 ))_{u^{∗}α^{n}},

where 𝒱 is the lift (unique up to isomorphism) of the sheaf u^{−1}𝕍_{0} of k-vector spaces to a flat μ-adic sheaf on X′. The cohomotopy groups are, of course, isomorphic to

H^{∗}(X, ℋ om_{k}(𝕍_{0}, 𝕍_{0})).

Note that

ℋ om_{k}(𝕍_{0}, 𝕍_{0}) = (π_{∗}𝔤𝔩_{n}(k))^{Γ,ρ_{0}},

so the cohomotopy groups are

H^{i}(X, (π_{∗}𝔤𝔩_{n}(k))^{Γ,ρ_{0}}),

where Γ acts via the adjoint action.

4.2 Representations to a smooth Noetherian group scheme

Fix a system {G_{n}/Spec Λ_{n}}_{n∈ℕ} of smooth Noetherian group schemes, and let G be the μ-adic formal group scheme \underrightarrow{lim} G_{n}. Fix a representation ρ_{0} : Γ → G(k).

Definition 4.1. A principal G(A)-sheaf on X is an étale sheaf 𝔹 on X, together with an action 𝒢 × 𝔹 → 𝔹, where 𝒢 denotes the constant sheaf G(A), such that

𝒢 × 𝔹 ≅ 𝔹 × 𝔹, (g, b) ↦ (g · b, b),

and the stalks 𝔹x̄ are non-empty.

Given a map A → B in C_{Λ}, there is a functorial map from G(A)-sheaves to G(B)- sheaves, given by

𝔹 ↦ G(B) ×^{G(A)} 𝔹,

i.e. (gh, b) ∼ (h, gb), for g ∈ G(A), h ∈ G(B) and b ∈ B.

Definition 4.2. Given a representation ρ : Γ → G(A), let

𝔹_{ρ} := ℱ_{Y} ×^{Γ,ρ} G(A) = (π_{∗}G(A))^{Γ,ρ},

where ℱ_{Y} is the sheaf on X whose espace étalé is Y, and ρ|_{π_{1}(Y,ȳ)} = 1, which must happen for some Galois cover Y ⟶^{π} X, G(A) being finite (since G is Noetherian and A finite).

More precisely,

ℱ_{Y} (U ⟶^{u} X) = \{ \xymatrix{U \ar[r]^{ũ} \ar[dr]_{u} & Y \ar[d] \\ & X} a lift\} ,

and the equivalence relation forming 𝔹_{ρ} is (γ ∘ u, g) ∼ (u, gρ(γ)), or equivalently η ∼ γ^{∗}ηρ(γ). 𝔹_{0} := {𝔹_{ρ_{0}|_{Λn}} }_{n∈ℕ}.

Lemma 4.3. There is an equivalence of categories

ℜ_{A}(ρ_{0}) ↔ 𝔅_{A}(𝔹_{0}),

where ℜ_{A}(ρ_{0}) has as objects representations

ρ ≡ ρ_{0} mod 𝔪_{A},

while 𝔅_{A}(𝔹_{0}) has as objects principal G(A)-sheaves

𝔹 ≡ 𝔹_{0} mod 𝔪_{A}

on X. In both cases morphisms must be the identity mod 𝔪_{A}.

Proof. We need to show that the functor Φ : ρ ↦ 𝔹_{ρ} is full, faithful and essentially surjective.

1. Φ is full and faithful:

Given θ : 𝔹_{σ} ⟶^{∼} 𝔹_{τ}, take some finite Galois cover Y ⟶^{π} X such that π^{−1}𝔹_{σ} and π^{−1}𝔹_{τ} are both isomorphic to the trivial sheaf G(A). Thus π^{−1}(θ) acts as a unique element α of G(A) with α ≡ 1 mod 𝔪_{A}, so θ = Φ(α).

2. Φ is essentially surjective:

Given 𝔹 ∈ 𝔅_{A}(𝔹_{0}), 𝔹 is locally constant with finite stalks (since G(A) is finite), so there is a Galois cover Y ⟶^{π} X such that π^{−1}𝔹 is constant. Choose a geometric point ȳ → Y, above x̄. Γ acts on (π^{−1}𝔹)ȳ via:

(π^{−1}𝔹)ȳ ≅ (π^{−1}𝔹)_{γ∘ȳ} ≅ 𝔹x̄ ≅ (π^{−1}𝔹)ȳ,

the first isomorphism being given by the constancy of π^{−1}𝔹, and the other two canonical. This is an isomorphism of G(A)-sets , so gives an element ρ ∈ ℜ_{A}(ρ_{0}). It follows that Φ(ρ) ≅ 𝔹.

Observe that deformations of 𝔹_{0} as a principal G-sheaf are the same as its deformations as a faithful G-sheaf. We have the functorial comonadic adjunction:

\xymatrix{Faithful G(A)-Sheaves(X) \ar[d]_{u^{−1}} \\ Faithful G(A)-Sheaves(X′) \ar[u]_{u_{∗}}^{⊣}}

using the canonical injection G(A) → u_{∗}G(A) to make u_{∗}ℬ a faithful G(A)-sheaf. Observe that the second category has uniformly trivial deformation theory, G being smooth, so that deformations are described by the SDC

E^{n} = Hom(ℬ, (u^{∗}u_{∗})^{n}ℬ)_{u^{∗}α^{n}}^{G(Λ)},

where ℬ is the lift (unique up to isomorphism) of the sheaf u^{−1}𝔹_{0} of G(k)-sheaves to a sheaf of continuous (in the μ-adic topology) G(Λ)-sheaves on X′. The cohomotopy groups are isomorphic to

H^{∗}(X, ad𝔹_{0}),

where ad𝔹_{0} is the tangent space of 𝔞𝔲𝔱(𝔹_{0}) at the identity. Equivalently,

ad𝔹_{0} = 𝔤 ×^{G(k)} 𝔹_{0} = (π_{∗}𝔤)^{Γ,ρ_{0}},

for 𝔤 the tangent space of G(k) at the identity, and G(k) acting on 𝔤 via the adjoint action, so the cohomotopy groups are

H^{∗}(X, (π_{∗}𝔤)^{Γ,ρ_{0}}).

5 Equivalence between SDCs and DGLAs in Characteristic Zero

Throughout this section, we will assume that Λ = k, a field of characteristic zero. We will also assume that all DGLAs are in non-negative degrees, i.e. L^{<0} = 0.

Remark 5.1. Throughout this section, we will encounter both chain complexes and simplicial complexes. In order to avoid confusion, chain complexes will be denoted by a bullet (C_{•}), while simplicial complexes will be denoted by an asterisk (C_{∗}). Asterisks will also be used to denote complexes without differentials, so that, for a chain complex C_{•}, we may write C_{•} = (C_{∗}, d). Homology groups of a chain complex C_{•} will be denoted H_{i}(C), while homotopy groups of a simplicial complex C_{∗} will be denoted π_{i}(C). The dual conventions will be used for cochain and cosimplicial complexes.

1. Consider, for simplicity, an SDC E^{∗} for which each tangent space 𝔢^{n} := C^{n}(E) is finite dimensional. Thus each E^{n} is pro-representable, by a smooth complete local Noetherian k-algebra Q_{n}, say. Since ∂_{ω_{0}}^{0} and ∂_{ω_{0}}^{n+1} are defined on C_{k}, E^{∗} is a cosimplicial complex, so Q_{∗} has the structure of a simplicial complex of algebras. The product on E^{∗} gives us a comultiplication

∗_{m,n}^{♯} : Q_{m+n} → Q_{m} ˆ⊗Q_{n},

satisfying various axioms. 𝔢^{∗} = Der_{k}(Q_{∗}, k), so π^{i}(𝔢^{∗}) = π^{i}(Der_{k}(Q_{∗}, k)).

2. To give a graded Lie algebra L^{∗} (in non-negative degrees) over k is the same as giving the smooth homogeneous functor

exp(L) : C_{k}^{ℕ_{0}} → Grp
A_{∗} ↦ exp(⨁_{n} L^{n} ⊗ (𝔪_{A})_{n}),

where C_{k}^{ℕ_{0}} is the category of nilpotent local Noetherian ℕ_{0}-graded (supercommutative) k-algebras with residue field k (concentrated in degree 0). Observe that ⨁_{n} L^{n} ⊗ (𝔪_{A})_{n} is a Lie algebra, so the formula makes sense.

The category C_{k}^{ℕ_{0}} behaves similarly to C_{k}, in that the analogue of Theorem 0.8 holds; in particular, if L^{∗} is finite dimensional, then exp(L) can be pro-represented by a smooth complete local Noetherian graded k-algebra R_{∗}, the group structure making R_{∗} into a graded Hopf algebra:

∗_{m,n}^{♯} : R_{m+n} → R_{m} ˆ⊗R_{n}.

If we take a finite dimensional DGLA L^{•}, the the differential gives rise to a differential on R_{∗}, giving it the structure of a DG Hopf algebra R_{•}. Since L^{•} = Der_{k}(R_{•}, k), we have H^{i}(L^{•}) = H^{i}(Der_{k}(R_{•}, k).

Hence, given the equivalence, described in [Qui69], between DG and simplicial algebras in characteristic zero, it should be possible to obtain DGLAs from SDCs and vice versa. Moreover, given the similarity between our cohomology groups and André-Quillen cohomology, and the result in [Qui70] that, in characteristic 0, André-Quillen cohomology can be computed by either simplicial or DG resolutions, it seems that an SDC should have the same cohomology as the corresponding DGLA. It is then plausible that they should have the same deformation groupoid. The purpose of this section is to make this reasoning rigorous.

5.1 SDCs

Fix an SDC E^{∗}. Although each E^{n} will only be pro-representable if dim 𝔢^{n} < ∞, in general we may make use of the following lemma:

Lemma 5.2. Given a smooth homogeneous functor F : C_{Λ} → Set, let {t_{i} : i ∈ I} be a basis for t_{F}. Then F is isomorphic to Hom_{cts.}(R, −), where R is the pro-Artinian completion of Λ[T_{i} : i ∈ I], localised at (T_{i} : i ∈ I), for formal symbols T_{i}. Note that, if t_{F} is finite dimensional, this is just the expected ring Λ[[T_{1}, …, T_{m}]].

Proof. This is very similar to the proof of Theorem 0.8. A full proof in the analogous case of nilpotent Lie algebras (rather than Artinian rings) is given in [Pri07] Theorem ??.

Thus an SDC E^{∗} is equivalent to a system of smooth local complete pro-Artinian rings Q_{∗}, with the dual structures to those described in Section 1.1. In particular, since we are only considering the case Λ = k, we have canonical ∂_{ω_{0}}^{0}, ∂_{ω_{0}}^{n+1}. Explicitly:

Lemma 5.3. Q_{∗} is a simplicial complex of smooth local complete pro-Artinian kalgebras (with residue field k), together with a co-associative comultiplication

ρ_{m,n} := ∗_{m,n}^{♯} : Q_{m+n} → Q_{m} ˆ⊗Q_{n},

such that Q_{0} → k is the co-unit, satisfying:

1. (∂_{i} ⊗ id) ∘ ρ_{m+1,n} = ρ_{m,n} ∘ ∂_{i} for i ≤ m.

2. (id ⊗ ∂_{i}) ∘ ρ_{m,n+1} = ρ_{m,n} ∘ ∂_{m+i} for i ≥ 1.

3. (∂_{m+1} ⊗ id) ∘ ρ_{m+1,n} = (id ⊗ ∂_{0}) ∘ ρ_{m,n+1}.

4. (σ_{i} ⊗ id) ∘ ρ_{m−1,n} = ρ_{m,n} ∘ σ_{i}.

5. (id ⊗ σ_{i}) ∘ ρ_{m,n−1} = ρ_{m,n} ∘ σ_{m+i}.

Of course, given a system Q_{∗} satisfying these conditions, we can clearly recover an SDC as Hom_{cts.}(Q_{∗}, −), so that we may regard this as an equivalent definition of an SDC.

Remark 5.4. We say that a simplicial complex A_{∗} of local Λ-algebras is Artinian if it satisfies DCC on simplicial ideals. This is equivalent to saying that the algebras A_{n} are Artinian, and the normalised complex N(A_{∗}) is bounded. It is easy to see that Q_{∗} = \underleftarrow{lim} Q_{∗}, the limit being over quotient complexes Q_{∗} → Q_{∗} Artinian. Moreover, Q_{∗} is really just a shorthand for this inverse system. Thus, for any functor F, by F (Q_{∗}) we mean \underleftarrow{lim} F (Q_{∗}). In particular, π_{∗}(Q_{∗}) := \underleftarrow{lim} π_{∗}(Q_{∗}). We will apply the same convention to the limits of DG complexes in the next section.

5.2 DGLAs

Define C_{k}^{ℕ_{0}} to be the category of nilpotent local Noetherian graded (super-commutative) k-algebras (in non-negative degrees) with residue field k (concentrated in degree 0).

A GLA L^{∗} over k (in non-negative degrees) gives rise to a smooth homogeneous functor

exp(L) : C_{k}^{ℕ_{0}} → Grp
A_{∗} ↦ exp(⨁_{n} L^{n} ⊗ (𝔪_{A})_{n}).

Given a functor F : C_{k}^{ℕ_{0}} → Set, define t_{F}^{i} := F (k[ϵ_{i}]), where ϵ_{i} is a variable in degree i, with ϵ_{i}^{2} = 0.

Lemma 5.5. Given a smooth homogeneous functor F : C_{k}^{ℕ_{0}} → Set, let {t_{i} : i ∈ I} be a homogeneous basis for t_{F}. Then F is isomorphic to Hom_{cts.}(R, −), where R is the pro-Artinian completion of the ring Λ[T_{i} : i ∈ I]_{gr}, localised at (T_{i} : i ∈ I), for formal symbols T_{i} ∈ deg(t_{i}). Note that, if t_{F} is finite dimensional, this is just the expected ring Λ[[T_{1}, …, T_{m}]]_{gr}.

Proof. The analogue of Theorem 0.8 holds for graded rings, and this is just a generalisation of Lemma 5.2.

Let exp(L) be “represented” by the DG algebra T_{∗}. The group structure of exp(L) corresponds to a graded Hopf algebra map

ρ : T_{∗} → Tot (T_{∗} ˆ⊗T_{∗}),

where

Tot (T_{∗} ˆ⊗T_{∗})_{n} = ⨁_{r+s=n} T_{r} ˆ⊗T_{s},

and the multiplication follows the usual (graded) convention:

(u ⊗ v)(x ⊗ y) = (−1)^{v̄x̄}ux ⊗ vy,

for homogeneous elements. We can, of course, recover L^{∗} from T_{∗} by

L^{∗} = Der_{cts.}(T_{∗}, k).

Lemma 5.6. Conversely, given a “representable” functor G : C_{k}^{ℕ_{0}} → Grp, with tangent space L^{∗}, there is a canonical isomorphism

exp(L) ≅ G.

In particular, this implies that G must be smooth.

Proof. Let G be “represented” by T_{∗}. Then

G(A_{∗}) = Hom_{k−Alg}(T_{∗}, A_{∗}),  ⨁_{n} L^{n} ⊗ (𝔪_{A})_{n} = Der_{k}(T_{∗}, A_{∗}).

We can embed both of these into Hom_{k}(T_{∗}, A_{∗}), on which we define the associative product f ∗ g = (f ⊗ g) ∘ ρ. Now the maps

G(A_{∗}) ≅ ⨁_{n} L^{n} ⊗ (𝔪_{A})_{n}
g ↦ ∑_{n≥1}(−1)^{n−1} \frac{(g − e)^{∗n}}{n}

∑_{n≥0} \frac{l^{∗n}}{n!} ↤ l,

where e is the identity map T_{∗} → k, give us our isomorphism.

Given a DGLA L^{•}, it remains only to understand what the differential d on L^{•} gives rise to on T_{∗}.

To give d is the same as giving the map

\xymatrix{L^{∗} \ar[r]^{id+dϵ_{1}} \ar[dr]_{id} & L^{∗}[ϵ_{1}] \ar[d] \\  & L^{∗},}

and exp(L^{∗}[ϵ_{1}]) is “represented” by T_{∗}[ϵ_{1}]. Thus d corresponds to a derivation d^{♯} : T_{n+1} → T_{n}. Since d^{2} = 0, we get (d^{♯})^{2} = 0. The condition that d be a Lie algebra derivation corresponds to the statement

ρ ∘ d^{♯} = (id ⊗ d^{♯} ± d^{♯} ⊗ id) ∘ ρ, where
(±d^{♯} ⊗ id)(u ⊗ v) = (−1)v̄d^{♯}(u) · v.

In summary, a DGLA L^{•} is equivalent to a smooth local complete pro-Artinian DG k-algebra T_{•}, with a co-associative chain map

ρ : T_{•} → Tot (T_{•} ˆ⊗T_{•}),

for which T_{•} → k is the co-unit.

5.3 SDC ⇝ DGLA

Given Q_{∗}, as in Section 5.1, we form a DG Hopf algebra C(Q)_{•}, with C(Q)_{n} = Q_{n} as a graded module. We let

d = ∑_{i=0}^{n+1}(−1)^{i}∂_{i} : C(Q)_{n+1} → C(Q)_{n},

and give it an algebra structure via the Eilenberg-Zilber shuffle product: for x ∈ C(Q)_{m} and y ∈ C(Q)_{n}, let

x∇y = ∑_{(μ,ν)∈Sh(m,n)}(−1)^{(μ,ν)}(σ_{ν_{n}} … σ_{ν_{1}}x) · (σ_{μ_{m}} … σ_{μ_{1}}y),

where Sh(m, n) is the set of (m, n) shuffle permutations, i.e. permutations

(μ_{1}, …, μ_{m}, ν_{1}, …, ν_{n}) of {0, …, m + n − 1}

such that

μ_{1} < … < μ_{m} and ν_{1} < … < ν_{n},

and (−1)^{(μ,ν)} denotes the sign of the permutation (μ, ν). As explained in [Wei94] or [Qui69], the resulting product is supercommutative

x∇y = (−1)^{mn}y∇x

and associative, and d is a graded derivation with respect to ∇, i.e.

d(x∇y) = (dx)∇y + (−1)^{m}x∇(dy).

Lemma 5.7. C(Q)_{∗} is a Hopf algebra under the map ρ.

Proof. Given x ∈ C(Q)_{m}, y ∈ C(Q)_{n}, and any p + q = m + n, we have:

ρ_{p,q}(x∇y) = ρ_{p,q}( ∑_{(μ,ν)}(−1)^{(μ,ν)}σ_{ν_{n}} … σ_{ν_{1}}x · σ_{μ_{m}} … σ_{μ_{1}}y)
= ∑_{(μ,ν)}(−1)^{(μ,ν)}ρ_{p,q}(σ_{ν_{n}} … σ_{ν_{1}}x) · ρ_{p,q}(σ_{μ_{m}} … σ_{μ_{1}}y)
= ∑ ±(σ_{ν′_{n′}} … σ_{ν′_{1}} ⊗ σ_{ν′′_{n′′}} … σ_{ν′′_{1}}) ∘ ρ_{m′,m′′}(x) · (σ_{μ′_{m′}} … σ_{μ′_{1}} ⊗ σ_{μ′′_{m′′}} … σ_{μ′′_{1}}) ∘ ρ_{n′,n′′}(y),

the final sum being over

m′ + m′′ = m, n′ + n′′ = n, m′ + n′ = p, m′′ + n′′ = q,
(μ′, ν′) ∈ Sh(m′, n′), (μ′′, ν′′) ∈ Sh(m′′, n′′),

with

± = (−1)^{(μ′,μ′′+p,ν′,ν′′+p)}.

For this map to be a Hopf algebra map, we would need this expression to equal

∑_{\begin{matrix}m′+m′′=m,n′+n′′=n, \\ m′+n′=p,m′′+n′′=q \end{matrix}} ρ_{m′,m′′}(x)∇ρ_{n′,n′′}(y),

The (p, q) component of ρ(x)∇ρ(y). Now,

ρ_{m′,m′′}(x)∇ρ_{n′,n′′}(y) =

∑_{\begin{matrix}(μ′,ν′) \\ (μ′′,ν′′) \end{matrix}} ±(σ_{ν′_{n′}} … σ_{ν′_{1}} ⊗ σ_{ν′′_{n′′}} … σ_{ν′′_{1}}) ∘ ρ_{m′,m′′}(x) · (σ_{μ′_{m′}} … σ_{μ′_{1}} ⊗ σ_{μ′′_{m′′}} … σ_{μ′′_{1}}) ∘ ρ_{n′,n′′}(y),

where

± = (−1)^{m′′n′}(−1)^{(μ′,ν′)}(−1)^{(μ′′,ν′′)},

the (−1)^{m′′n′} arising from the (required) supercommutativity of C(Q)_{∗} ⊗ C(Q)_{∗}. The result now follows from the observation that

(−1)^{(μ′,μ′′+p,ν′,ν′′+p)} = (−1)^{m′′n′}(−1)^{(μ′,ν′)}(−1)^{(μ′′,ν′′)}.

Lemma 5.8. ρ is a chain map.

Proof. For x ∈ C(Q)_{m+1},

ρ(dx) = ∑_{p+q=m} ρ_{p,q}(∑_{i=0}^{m+1}(−1)^{i}∂_{i}x)

= ∑_{p+q=n}(∑_{i=0}^{p}(−1)^{i}(∂_{i} ⊗ id) ∘ ρ_{p+1,q}(x) + (−1)^{p} ∑_{j=1}^{q+1}(−1)^{j}(id ⊗ ∂_{j}) ∘ ρ_{p,q+1}(x))
= ∑_{p+q=m}(((d − (−1)^{p+1}∂_{p+1}) ⊗ id) ∘ ρ_{p+1,q}(x) + (−1)^{p}(id ⊗ (d − ∂_{0})) ∘ ρ_{p,q+1}(x))
= ∑_{p+q=m}(d ⊗ id) ∘ ρ_{p+1,q}(x) − (−1)^{p+1}(∂_{p+1} ⊗ id) ∘ ρ_{p+1,q}(x)

+ ∑_{p+q=m}(−1)^{p}(id ⊗ d) ∘ ρ_{p,q+1}(x) − (−1)^{p}(id ⊗ ∂_{0}) ∘ ρ_{p,q+1}(x)

= ∑_{p+q=m}(d ⊗ id) ∘ ρ_{p+1,q}(x) + ∑_{p+q=m}(−1)^{p}(id ⊗ d) ∘ ρ_{p,q+1}(x)
= dρ(x).

There is now a problem with C(Q_{∗}) as a candidate to define a DGLA L — it is not local. In [Qui69], rather than taking the unnormalised chain complex C(Q_{∗}), Quillen takes the normalised chain complex N(Q_{∗}). Recall that

C(Q)_{n} = N(Q)_{n} ⊕ D(Q)_{n}, where

N(Q)_{n} = ⋂_{i=1}^{n} ker(∂_{i} : Q_{n} → Q_{n−1}), D(Q)_{n} = ∑_{i=1}^{n−1} σ_{i}(Q_{n−1}).

Observe that N(Q)_{•} is local, as ker(∂_{1}) ⊂ 𝔪_{Q}. It is pro-Artinian following Remark 5.4, since the normalisation of an ideal is a ∇-ideal, and N(I)∇N(J) ⊂ N(IJ). However, we cannot give N(Q) the Hopf algebra structure ρ: in general, for x ∈ N(Q)_{m+n}, (∂_{m} ⊗ id) ∘ ρ_{m,n}(x) ≠ 0. Instead, we take the complex

N̄(Q) := C(Q)/D(Q).

It follows immediately from the identities in Lemma 5.3 that

ρ_{m,n} : N̄(Q)_{m+n} → N̄(Q)_{m} ˆ⊗N̄(Q)_{n}

is well defined. We need only check that ∇ is well defined on N̄(Q), since then we will have N̄(Q) isomorphic to N(Q) as an algebra.

Lemma 5.9. For x ∈ D(Q)_{m} and y ∈ C(Q)_{n}, x∇y ∈ D(Q)_{m+n}.

Proof. Without loss of generality, x = σ_{r}u, for some 0 ≤ r < m. We will show that each term in the sum of the shuffle product is in D(Q)_{m+n}. Consider

σ_{ν_{n}} … σ_{ν_{1}}x · σ_{μ_{m}} … σ_{μ_{1}}y,

and let a := max{j : ν_{j} − j < r}, which makes sense, since ν_{j} − j is a non-decreasing function. With the obvious convention if this set is empty, we have 0 ≤ a ≤ n. Now, the simplicial identities give

σ_{ν_{n}} … σ_{ν_{1}}σ_{r}u = σ_{ν_{n}} … σ_{ν_{a+1}}σ_{r+a}σ_{ν_{a}} … σ_{ν_{1}}u,

with

ν_{1} < … < ν_{a} < r + a < ν_{a+1} < … < ν_{n}.

Now r + a ≤ p − 1 + q, and r + a does not equal any of the ν_{j}, so r + a = μ_{i}, for some i. This gives

(σ_{ν_{n}} … σ_{ν_{1}}x) · (σ_{μ_{m}} … σ_{μ_{1}}y) = (σ_{ν_{n}} … σ_{ν_{a+1}}σ_{μ_{i}}σ_{ν_{a}} … σ_{ν_{1}}u) · (σ_{μ_{m}} … σ_{μ_{1}}y)
= (σ_{μ_{i}}σ_{ν_{n}−1} … σ_{ν_{a+1}−1}σ_{ν_{a}} … σ_{ν_{1}}u) · (σ_{μ_{i}}σ_{μ_{m}−1} … σ_{μ_{i+1}−1}σ_{μ_{i−1}} … σ_{μ_{1}}y)
= σ_{μ_{i}}((σ_{ν_{n}−1} … σ_{ν_{a+1}−1}σ_{ν_{a}} … σ_{ν_{1}}u) · (σ_{μ_{m}−1} … σ_{μ_{i+1}−1}σ_{μ_{i−1}} … σ_{μ_{1}}y)),

which is in D(Q)_{m+n}, as required.

Definition 5.10. Given an SDC E^{∗}, we define the DGLA L(E)^{•} by

L(E)^{•} = Der_{k,cts.}(N̄(Q)_{•}, k),

where Q_{∗} “represents” E^{∗}.

Lemma 5.11. There are natural maps π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•}), where 𝔢^{∗} is the tangent space of E^{∗}.

Proof. We may compute π^{i} using the conormalised cocomplex

N^{n}(K^{∗}) = ⋂_{i=0}^{n−1} ker(σ^{i} : K^{n} → K^{n−1}), d = ∑_{i=0}^{n}(−1)^{i}∂^{i}.

Thus an element α ∈ N^{i}(𝔢^{∗}) is a derivation (with respect to the product on Q) from N̄_{i}(Q) to k (by the definition of N̄). A derivation to k is just something which annihilates both k and 𝔪_{Q}^{2}, but 𝔪_{Q}∇𝔪_{Q} ⊂ 𝔪_{Q}^{2}, so α gives a derivation (with respect to ∇) from N̄_{•}(Q) to k, of degree i, i.e.

α ∈ L(E)^{i}.

This correspondence preserves closed and exact forms, so we get

π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•}).

Lemma 5.12. There is a natural map 𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{L(E)}, with the maps on cohomology compatible with the corresponding maps on tangent and obstruction spaces.

Proof. 1. Given ω ∈ MC_{E}, we may regard ω as a continuous ring homomorphism ω : Q_{1} → A, satisfying ω ∗ ω = ∂^{1}(ω). We also know that σ^{0}(ω) = e. Set α = ω − ω_{0}. This gives σ^{0}(α) = 0, therefore α : N̄_{1}(Q) → A. In fact, α is a ∇-derivation: for x ∈ Q_{0}, y ∈ Q_{1},

α(x∇y) = α(σ_{0}(x)y)
= (ω − ω_{0})(σ_{0}(x)y)
= σ^{0}(ω)(x)ȳ − σ^{0}(ω_{0})(x)ȳ + \overline{σ_{0}(x)}ω(y) − \overline{σ_{0}(x)}ω_{0}(y)
= x̄α(y)
= x̄α(y) + α(x)ȳ.

Moreover, the Maurer-Cartan equation for ω becomes

(ω_{0} + α) ∗ (ω_{0} + α) = ∂^{1}(ω_{0} + α)
ω_{0}^{2} + ∂^{0}(α) + ∂^{2}(α) + α ∗ α = ω_{0}^{2} + ∂^{1}(α)
∂^{0}(α) − ∂^{1}(α) + ∂^{2}(α) + α ∗ α = 0
dα + \frac{1}{2}[α, α] = 0.

Therefore α ∈ MC_{L(E)}.

2. Given g ∈ E^{0}(A), g is an algebra homomorphism g : Q_{0} → A. Since ∇ agrees with the usual product on Q_{0}, and N̄_{0}(Q) = Q_{0}, g is a ∇-algebra homomorphism g : N̄_{0}(Q) → A. Therefore

g ∈ exp(L(E) ⊗ 𝔪_{A}).

It is easy to see that the adjoint action on E^{∗} corresponds to the gauge action on L(E)^{•}.

We therefore have the map 𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{L(E)}. The results on tangent and obstruction spaces follow immediately.

We now need only prove that the maps π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•}) are isomorphisms, from which it will follow that the morphism of groupoids is an equivalence. This will be proved in Theorem 5.22.

5.4 DGLA ⇝ SDC

Definition 5.13. Given a chain complex V_{•} of vector spaces, the left adjoint N^{−1} of the normalisation functor N is defined by

N^{−1}(V)_{n} = ⨁_{\begin{matrix}m+s=n \\ 0≤ν_{1}<ν_{2}<...<ν_{s}<n \end{matrix}} σ_{ν_{s}} … σ_{ν_{2}}σ_{ν_{1}}V_{m},

where σ_{ν_{s}} … σ_{ν_{2}}σ_{ν_{1}}V_{m} is just a copy of V_{m}, and we define σ and ∂ on N^{−1}(V) using the simplicial identities, subject to

∂_{i}|_{V} = \begin{cases}d & i = 0 \\ 0 & i > 0. \end{cases}

Note that N(N^{−1}(V)) ≅ V, and, given v ∈ V_{n}, we write N^{−1}v for

v ∈ (id)V_{n} ⊂ N^{−1}(V)_{n}.

Definition 5.14. Given a local complete pro-Artinian DG k-algebra T_{•}, we can follow Quillen ([Qui69] Proposition 4.4) in defining the left adjoint to N, on complexes of algebras (adapted slightly for local algebras), by

N^{∗}(T )_{n} = F (N^{−1}(𝔪_{T}))_{n}/I_{n},

recalling that F is our free functor for local pro-Artinian algebras, where I_{n} is the ideal generated by

{N^{−1}(x)∇N^{−1}(y) − N^{−1}(x ∧ y): x ∈ (𝔪_{T})_{r}, y ∈ (𝔪_{T})_{s}, r + s = n},

where ∧ is the product on T_{•}, and ∇ is the shuffle product on F (N^{−1}(𝔪_{T})), defined in terms of the (free) product on F (N^{−1}(𝔪_{T})). We regard (k ⊂ N^{∗}(T )_{n}) as σ_{0}^{n}(k ⊂ T_{0}) so that there is a copy of N^{−1}(T ) (and not just N^{−1}(𝔪_{T})) in N^{∗}(T ).

Theorem 5.15. If T_{•} is smooth, then so is N^{∗}(T )_{∗}.

Proof. This is essentially [Qui69] Proposition 4.4.

Theorem 5.16. If T_{•} is smooth, then the unit η : T_{•} → N(N^{∗}(T ))_{•} is such that the H_{i}(η) are isomorphisms for all i. Thus H_{i}(T_{•}) ≅ π_{i}(N^{∗}(T )_{∗}).

Proof. Essentially [Qui69] (Theorem 4.6 and the remark following). Note that the assumption of characteristic zero is vital here.

It therefore seems that, if T_{•} is the smooth DG Hopf algebra corresponding to the DGLA L^{•}, then a good candidate for the corresponding SDC E(L)^{∗} should be that represented by N^{∗}(T ). However, we need first to define Hopf maps on N^{∗}(T ). Since the Hopf maps did not behave under N, it is unsurprising that they do not behave under N^{∗}. We get round this by making use of the isomorphism N̄ ≅ N.

Lemma 5.17. If Q_{∗} “represents” an SDC E^{∗}, then, under the isomorphism

N(Q)_{•} → N̄(Q)_{•},

the map ρ on N̄ corresponds to Δ on N, where

Δ_{m,n} = ((id + ∑_{i=0}^{m−1}(−1)^{m−i}σ_{i}∂_{m}) ⊗ id) ∘ ρ_{m,n} :

N(Q)_{•} ⟶^{Δ} N(Q)_{•} ˆ⊗N(Q)_{•}
≅↓  ↓≅
N̄(Q)_{•} ⟶^{ρ} N̄(Q)_{•} ˆ⊗N̄(Q)_{•}.

On N, we may also recover ρ from Δ.

Proof. Observe that

Δ(x) − ρ(x) ∈ D(Q) ˆ⊗Q,

and verify that Δ(x) ∈ N(Q) ˆ⊗N(Q).

Finally, note that we can apply the identities from Lemma 5.3 to get

Δ_{m,n} = ρ_{m,n} + ∑_{i=0}^{m−1}(−1)^{m−i}(σ_{i} ⊗ ∂_{0}) ∘ ρ_{m−1,n+1}.

Hence

Δ_{m,n} − ∑_{i=0}^{m−1}(−1)^{m−i}(σ_{i} ⊗ ∂_{0}) ∘ Δ_{m−1,n+1}

= ρ_{m,n} + ∑_{i=0}^{m−1}(−1)^{m−i}(σ_{i} ⊗ ∂_{0}) ∘ ρ_{m−1,n+1} − ∑_{i=0}^{m−1}(−1)^{m−i}(σ_{i} ⊗ ∂_{0}) ∘ ρ_{m−1,n+1}

− ∑_{i=0}^{m−1} ∑_{j=0}^{m−2}(−1)^{m−i}(−1)^{m−1−j}(σ_{i} ⊗ ∂_{0}) ∘ (σ_{j} ⊗ ∂_{0}) ∘ ρ_{m−2,n+2}

= ρ_{m,n} − ∑_{i=0}^{m−1} ∑_{j=0}^{m−2}(−1)^{m−i}(−1)^{m−1−j}(σ_{i}σ_{j} ⊗ ∂_{0}∂_{0}) ∘ ρ_{m−2,n+2}
= ρ_{m,n}

on N, since ∂_{0}∂_{0} = ∂_{0}∂_{1}, and id ⊗ ∂_{1} is zero on ρ(N).

Definition 5.18. Therefore, given a DGLA L, “represented” by the smooth DG Hopf algebra T_{•} with comultiplication Δ, we define N̄^{∗}(T ) to be the algebra N^{∗}(T ) (noting that this is smooth by Theorem 5.15), with Hopf maps defined on

{N^{−1}(t) : t ∈ 𝔪_{T} } ⊂ N^{−1}(𝔪_{T})

by

ρ_{m,n} := Δ_{m,n} − ∑_{i=0}^{m−1}(−1)^{m−i}(σ_{i} ⊗ ∂_{0}) ∘ Δ_{m−1,n+1},

extending to the rest of N^{−1}(𝔪_{T}) by the rules in Lemma 5.3, and extending to N^{∗}(T ) by multiplicativity. Note that Lemma 5.7 ensures that this is well-defined. Call the resulting SDC E(L)^{∗}.

Theorem 5.19. There are canonical isomorphisms H^{i}(L) → π^{i}(e(L)^{∗}), where e(L)^{∗} is the tangent space of E(L)^{∗}, and an equivalence 𝔇𝔢𝔣_{L} → 𝔇𝔢𝔣_{E(L)}.

Proof.

1. e(L)^{∗} = Der(N̄^{∗}(T )_{∗}, k). From the definition of N̄^{∗}(T ), it follows that

Der(N̄^{∗}(T )_{∗}, k) ≅ {α ∈ Hom(N^{−1}(T )^{∗}, k): α(1) = 0, α(N^{−1}(x∧y)) = 0 ∀x, y ∈ 𝔪_{T} }.

Now, one possible definition of cohomotopy groups is as the cohomology of the complex

N̄^{n}(e) = e^{n}/(∑_{i=1}^{n} ∂^{i}e^{n−1}).

Therefore

π^{i}(e(L)^{∗}) = H^{i}(N̄(e(L))^{•})
= H^{i}(N̄(Der(N̄^{∗}(T )_{∗}, k)))
= H^{i}(N̄({α ∈ Hom(N^{−1}(𝔪_{T})_{∗}, k): α(N^{−1}(x ∧ y)) = 0 ∀x, y ∈ 𝔪_{T} }))
= H^{i}({α ∈ Hom(NN^{−1}(𝔪_{T})_{•}, k): α(N^{−1}(x ∧ y)) = 0 ∀x, y ∈ 𝔪_{T} })
= H^{i}(Der(T_{•}, k))
= H^{i}(L),

since NN^{−1}(𝔪_{T})_{•} = (𝔪_{T})_{•}.

2. First observe that N̄^{∗}(T )_{0} = T_{0}, with the same product. Thus 𝔤𝔩 = E(L)^{0}. Call the identity of both these groups e. Given α ∈ MC_{L}(A), let

ω : N̄^{∗}(T )_{1} → A

be the multiplicative map defined on the generators

N^{−1}(T )_{1} = T_{1} ⊕ σ_{0}T_{0} by ω = (α, 0) + ω_{0},

where ω_{0} = e∘∂_{0} [effectively this gives us ω = (α, e∘(σ_{0})^{−1})]. The proof of Lemma 5.12, and the discussion comparing N and N̄, show that this is well-defined and satisfies the Maurer-Cartan equation. The gauge action of 𝔤𝔩 corresponds to the adjoint action of E^{0}, combining to give us a morphism

𝔇𝔢𝔣_{L} → 𝔇𝔢𝔣_{E(L)}.

That this is an equivalence follows from the first part, and Theorems 0.11 to 0.14.

Lemma 5.20. There is a quasi-isomorphism E^{∗} → E(L(E))^{∗} of SDCs, for any SDC E^{∗}.

Proof. To see that there is a morphism E^{∗} → E(L(E))^{∗} is immediate. It is the map

N̄^{∗}N̄(Q)_{∗} → Q_{∗}

corresponding to the canonical map N^{∗}N(Q_{∗}) → Q_{∗}. Moreover

π_{∗}(N̄^{∗}N̄(Q)_{∗}) ≅ H_{∗}(N̄(Q)_{•}) ≅ π_{∗}(Q_{∗}),

the first isomorphism being a consequence of Theorem 5.16. Thus the map N̄^{∗}W N̄(Q)_{∗} → Q_{∗} induces an isomorphism on homotopy groups. Now, we may use the completeness of these simplicial algebras to show that they are free in the sense of [Qui69] Proposition I 4.4. Since free simplicial complexes are the analogue of projective resolutions, and we have a pair of homotopic free complexes, it follows that

Der(Q_{∗}, k) → Der(N̄^{∗}N̄(Q)_{∗}, k)

gives an isomorphism on cohomotopy groups, i.e.

π^{i}(𝔢^{∗}) ≅ π^{i}(e(L(E))^{∗}),

the proof being the same as the result in [Qui70] that the quasi-isomorphism class of the tangent complex is independent of the choice of cofibrant resolution. Hence E^{∗} → E(L(E))^{∗} is a quasi-isomorphism of SDCs.

Corollary 5.21. For any DGLA L^{•}, there is a quasi-isomorphism L^{•} → L(E(L))^{•}.

Proof. By Lemma 5.20, there is a quasi-isomorphism of SDCs

E(L)^{∗} → E(L(E(L)))^{∗}.

Now, Theorem 5.19 gives a canonical isomorphism N̄(e(L))^{•} ≅ L^{•}, so we may regard N̄(e(L))^{•} as a DGLA, with bracket

[α, β] = (α ⊗ β) ∘ Δ ∓ (β ⊗ α) ∘ Δ.

(Note that N̄(𝔢^{∗}), for the tangent space 𝔢^{∗} of an arbitrary SDC E^{∗} will not be closed under this bracket.)

The quasi-isomorphism of SDCs then gives us a quasi-isomorphism of DGLAs

L^{•} ≅ N̄(e(L))^{•} ⟶^{∼} N̄(e(L(E(L))))^{•} ≅ L(E(L))^{•}.

Theorem 5.22. The maps π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•}) in Lemma 5.11 are isomorphisms, and hence the morphism 𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{L(E)} in Lemma 5.12 is an equivalence.

Proof. We can factor the isomorphism

π^{i}(𝔢^{∗}) ⟶^{≅} π^{i}(e(L(E))^{∗})

from Lemma 5.20 as

π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•}) ⟶^{≅} π^{i}(e(L(E))^{∗}),

the second isomorphism coming from Theorem 5.19. Thus

π^{i}(𝔢^{∗}) → H^{i}(L(E)^{•})

must be an isomorphism, and by Theorems 0.11 to 0.14 and Lemma 5.12,

𝔇𝔢𝔣_{E} → 𝔇𝔢𝔣_{L(E)}

must be an equivalence.

5.4.1 An explicit description

Given a DGLA L, first form the denormalised cosimplicial complex

D^{n}L := ( ⨁_{\begin{matrix}m+s=n \\ 0≤j_{1}<...<j_{s}≤n \end{matrix}} ∂^{j_{s}} … ∂^{j_{1}}L^{m})/(dv ∼ ∑_{i=0}^{n+1}(−1)^{i}∂^{i}v ∀v ∈ L^{n}),

where we define the ∂^{j} and σ^{i} using the simplicial identities, subject to the condition that σ^{i}L = 0.

We now have to define the Lie bracket ⟦, ⟧ from D^{n}L ⊗ D^{n}L to D^{n}L. On the conormalised complex N^{n}(DL ⊗ DL) ≤ D^{n}L ⊗ D^{n}L, we define this as the composition

N^{n}(DL ⊗ DL) \underrightarrow{∑ ∇^{pq}} ⨁_{p+q=n} L^{p} ⊗ L^{q} ⟶^{[,]} L^{n},

where

∇^{pq}(v ⊗ w) = ∑_{(μ,ν)∈Sh(p,q)}(−1)^{(μ,ν)}(σ^{ν_{1}} … σ^{ν_{q}}v) ⊗ (σ^{μ_{1}} … σ^{μ_{p}}w).

We then extend this bracket to the whole of D^{n}L ⊗ D^{n}L by setting

⟦∂^{i}a, ∂^{i}b⟧ := ∂^{i}⟦a, b⟧.

Now set

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}.

References

[And67] Michel André. Méthode simpliciale en algèbre homologique et algèbre commutative. Lecture Notes in Mathematics, Vol. 32. Springer-Verlag, Berlin, 1967.

[God58] Roger Godement. Topologie algébrique et théorie des faisceaux. Hermann, Paris, 1958.

[Hin05] Vladimir Hinich. Deformations of sheaves of algebras. Adv. Math., 195(1):102– 164, 2005.

[Ill71] Luc Illusie. Complexe cotangent et déformations. I. Springer-Verlag, Berlin, 1971. Lecture Notes in Mathematics, Vol. 239.

[Ill72] Luc Illusie. Complexe cotangent et déformations. II. Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 283.

[Kon94] Maxim Kontsevich. Topics in algebra — deformation theory. Lecture Notes, 1994.

[KS00] Maxim Kontsevich and Yan Soibelman. Deformations of algebras over operads and the Deligne conjecture. In Conférence Moshé Flato 1999, Vol. I (Dijon), volume 21 of Math. Phys. Stud., pages 255–307. Kluwer Acad. Publ., Dordrecht, 2000.

[Mac71] Saunders MacLane. Categories for the working mathematician. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.

[Man99] Marco Manetti. Deformation theory via differential graded Lie algebras. In Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), pages 21–48. Scuola Norm. Sup., Pisa, 1999. arXiv math.AG/0507284.

[Mil80] James S. Milne. Étale cohomology. Princeton University Press, Princeton, N.J., 1980.

[Pri07] J. P. Pridham. The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold. Ann. Fac. Sci. Toulouse Math. (6), 16(1):147–178, 2007. arXiv math.CV/0502451 v5.

[Qui69] Daniel Quillen. Rational homotopy theory. Ann. of Math. (2), 90:205–295, 1969.

[Qui70] Daniel Quillen. On the (co-) homology of commutative rings. In Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pages 65–87. Amer. Math. Soc., Providence, R.I., 1970.

[Sch68] Michael Schlessinger. Functors of Artin rings. Trans. Amer. Math. Soc., 130:208–222, 1968.

[Swe69] Moss E. Sweedler. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York, 1969.

[Wei94] Charles A. Weibel. An introduction to homological algebra. Cambridge University Press, Cambridge, 1994.