Homology, Homotopy and Applications, vol.6(1), 2004, pp.299–340

HOCHSCHILD COHOMOLOGY OF COMPLEX SPACES AND NOETHERIAN SCHEMES

FRANK SCHUHMACHER

(communicated by Jean-Louis Loday)

Abstract The classical HKR-theorem gives an isomorphism between the n-th Hochschild cohomology of smooth algebras and the nth exterior power of their module of Kähler differentials. Here, we generalize it for simplicial graded commutative objects in “good pairs of categories”. We apply this generalization to complex spaces and Noetherian schemes and deduce several theorems on the decomposition of their respective (relative) Hochschild (co)homologies.

Introduction

One motivation for considering Hochschild cohomology in geometry is its possible application in noncommutative deformation theory. Even if the geometrical objects (complex spaces, schemes) considered in this paper belong to classical commutative geometry, it is interesting to find out how they can be deformed into noncommutative spaces, and in which way classical deformation theory is contained in noncommutative deformation theory^{1}. Classical deformations of schemes and complex spaces are governed by tangent cohomology (see for example [16]), and since deformations of (noncommutative) associative algebras are governed by Hochschild cohomology (see [8]), we may assume that Hochschild cohomology of schemes and complex spaces governs their noncommutative deformation theories. This leads to the question of how their Hochschild and tangent cohomologies are related, which is the subject of the present paper.

Hochschild complexes and Hochschild homology for schemes were first defined by Weibel/ Geller [21]. They defined the Hochschild complex as sheafification of the cyclic bar complex. An alternative definition is due to Yekutieli [23]. Hochschild cohomology for schemes was defined by Gerstenhaber and Schack [8]. Kontsevich [14] suggested to define Hochschild cohomology of a complex manifold X as Ext_{X^{2}}(O_{X}, O_{X}). Swan [19] showed that for schemes, the Gerstenhaber/Schack-

Supported by: Doktorandenstipendium des Deutschen Akademischen Austauschdienstes im Rahmen des gemeinsamen Hochschulsonderprogramms III des Bundes und der Länder. Received May 5, 2004, revised July 19, 2004; published on August 6, 2004. 2000 Mathematics Subject Classification: 13D03, 14F43. Key words and phrases: admissible pair of categories, complex space, Hochschild cohomology, regular sequence, scheme. \begin{matrix}c \\ ◯ \end{matrix} 2004, Frank Schuhmacher. Permission to copy for private use granted.

1So far, deformation theory for noncommutative schemes has not been developed and noncommutative complex spaces with singularities are not jet defined.

definition is equivalent to Kontsevich’s definition. The definition of Hochschild complexes and Hochschild (co)homology for complex spaces is due to Buchweitz/Flenner [6]. Weibels definition of Hochschild complexes is not recommendable for complex spaces. This is due to the fact that the (cyclic) bar complex of an analytic algebra A (in which tensor products are replaced by analytic tensor products) is not a complex of projective A-modules. Thus, if we defined the Hochschild cohomology as cohomology of the dual of the cyclic bar complex of A, we would not arrive at the desired Ext-interpretation. Instead, we must define the Hochschild complex piecewise (on Stein compacts) via free algebra resolutions, and globally via a Čech construction (see [6] or Section 4).

In order to avoid having to prove each statement for each situation, we unify the algebraic and analytic theories. To do so, we follow an approach due to Bingener/Kosarew [1] who extracted the common features of both situations and listed them as axioms for “admissible pairs of categories”. These are pairs (C, M) where C is a suitable category of algebras and M a category of modules over C. Admissible pairs of categories enable the description of “affine” spaces. For example C may be the category of sections on Stein compacts or the category of sections on affine schemes (i.e. the category of algebras). In the present paper we will mainly talk about admissible pairs of categories. These are interesting by themselves, as they are useful in many more situations (see Examples 1.1) than affine schemes and Stein compacts. We will show how to apply results for admissible pairs to schemes and complex spaces only in Section 4.

The main results for admissible pairs (C, M) of categories are the following: (1) In Section 1.7, we characterize regular sequences in graded commutative algebras in C. (2) We define the Hochschild complex for simplicial algebras in C (i.e. functors from a small category N to C) and show that the Hochschild complex is homotopy invariant (Proposition 2.2). (3) We prove a HKR theorem for free commutative graded DG algebras in C, if the ground ring k contains ℚ. Loday’s textbook [15] contains the special case of this theorem where C is just the category of k-algebras. Loday’s sketch of proof does not apply in the general case, as for an analytic algebra A, the bar construction is not a projective resolution of A over A ˆ⊗A. A corollary of this HKR theorem is the following Quillen-type theorem for a k-algebra in C: There is a quasi-isomorphism

ℍ(a/k) ≈ ∧𝕃(a/k)

from the Hochschild complex to the exterior algebra of the tangent complex. (Quillen [18] stated this result in the case where C is the category of k-algebras.) We generalize this theorem for objects in C^{N} and in Section 4, we deduce the main result of this paper by means of a Čech construction. (4) If X ⟶ Y is a morphism of complex spaces (paracompact and separated) or a separated morphism of finite type of Noetherian schemes in characteristic zero, then there exists a quasi-isomorphism

ℍ(X/Y) ≈ ∧𝕃(X/Y)  (0.1)

over O_{X}, where ℍ(X/Y) is the relative Hochschild complex of X over Y (see Section 4) and 𝕃(X/Y) is the relative cotangent complex. From the main result we deduce several decomposition theorems for Hochschild (co)homology. (5) Hochschild cohomology contains tangent cohomology:

HH^{n}(X/Y, M) ≅ ∐_{i+j=n} Ext^{i}(∧^{j}𝕃(X/Y), M).  (0.2)

The left side is the n-th Hochschild cohomology of X over Y with values in M. The right side contains the (i − 1)-th relative tangent cohomology Ext^{i−1}(𝕃(X/Y), M) as a direct factor. For complex spaces, this decomposition, as well as equation (0.1), has already been proved in a completely different way by Buchweitz/Flenner [6].

(6) The second corollary is a decomposition theorem for the Hochschild cohomology of complex analytic manifolds and smooth schemes in characteristic zero:

HH^{n}(X) ≅ ∐_{i+j=n} H^{i}(X, ∧^{j}T_{X}).  (0.3)

On the right, we have the sheaf cohomology of the exterior powers of the tangent complex. A proof of this result for complex analytic manifolds has been announced (but not yet published) by Kontsevich. For smooth schemes, decomposition (0.3) was proved in a different way by Yekutieli [23]. A similar statement for quasiprojective smooth schemes is due to Gerstenhaber/Schack [8] and Swan [19]. (7) If X is a smooth scheme in characteristic zero, or a manifold, then we can deduce the “Hodge decomposition” of the Hochschild homology

HH_{n}(X) ≅ ∏_{i−j=n} H^{j}(X, ∧^{i}Ω_{X})  (0.4)

Remark that the difference with the Hodge composition of the De Rham cohomology is that we sum over the columns of the “Hodge diamond” instead of over the lines. For schemes, this result was shown in a different way (using the λ-decomposition of the Hochschild complex) by Weibel [22].

Acknowledgments: I want to express my gratitude to Siegmund Kosarew for his encouragement and the many discussions and suggestions that were indispensable for the progress of my work on the present paper. Furthermore, I am grateful to Ragnaz-Olaf Buchweitz and Hubert Flenner for sending me a not-yet-published version of their paper on Hochschild cohomology from which I have learned many techniques applied in this work.

Conventions: For a ring k, we denote the category of k-modules by k-𝔐𝔬𝔡. For a morphism f : A ⟶ B in any category, we denote the kernel of f in the categorical sense (see [20]) by kern f, i.e. kern f is a morphism K ⟶ A, where K is an object, determined up to a canonical isomorphism. By Kern f, we mean the object K. We use the notions cokern, Cokern, im and Im in the same way. For example, we have Im f ≅ Kern(cokern f). We write ≈ for quasi-isomorphisms and ≃ for homotopy equivalences. We use the letter D to denote derived categories and K to denote homotopy categories, i.e. the localization of categories by homotopy equivalences. The differential of a DG object is always of degree +1. If the degree g(a) of a homogeneous element a of a graded ring or module arises in an exponent, we just write a instead of g(a). By convention, (−1)^{ab} means (−1)^{g(a)·g(b)} and not (−1)^{g(ab)}.

1. Admissible pairs of categories

In order to describe geometric objects locally by means of algebraic objects, one has to handle pairs of categories (C, M), where C is a category of algebras and M a category of modules over algebras in C. If the algebraic calculus should include the local description of commutative schemes, complex spaces and even infinite dimensional spaces, like Banach analytic spaces, then the frame of admissible pairs of categories is a good choice. Before listing the axioms defining an admissible pair of categories (figured out by Bingener and Kosarew in [1]), we give several examples:

Examples 1.1. The following pairs (C, M) are admissible pairs of categories:

(1) Let C^{(0)} be the category of all commutative 𝕂-algebras and M^{(0)} the category of modules over algebras in C^{(0)}.

(2) Recall that a Stein compact is a compact subset X of a complex space, admitting a base of open neighborhoods 𝔘, such that each U ∈ 𝔘 is a Stein space. Let C^{(1)} be the category of all analytic ℂ-algebras, i.e. the category of all sections of the structure sheaf of a Stein compact. Then each algebra in C^{(1)} is a DFN-algebra, i.e. a topological algebra with respect to the dual Frechet nuclear topology (see [17], for instance) and each homomorphism of such algebras is continuous. Let M^{(1)} be the category of all DFN-modules over algebras in C^{(1)}.

(3) In the first example, we can replace C^{(0)} by the category of all Noetherian, commutative 𝕂-algebras.

(4) In the second example, we can replace C^{(1)} by the category of local analytic algebras.

(5) For ϵ ∈ (0, 1], let C^{(ϵ)} be the category of commutative complete P O-algebras in the sense of [16] and M^{(ϵ)} the category of all complete PO-modules and PO_{ϵ}-homomorphisms.

The reader, only interested in schemes or algebraic varieties, doesn’t have to care about the following definition and may always take (C, M) as in Example (1) instead.

We fix a ground ring 𝕂 (in our main reference [1], 𝕂 is the field ℚ, so here we start with a more general setting). Denote by C a category of commutative 𝕂algebras and by C-𝔐𝔬𝔡 the category of all modules over algebras in C. For objects A, B in C and M in A-𝔐𝔬𝔡, N in B-𝔐𝔬𝔡, a homomorphism M ⟶ N in C-𝔐𝔬𝔡 is a pair (ϕ, f), where ϕ : A ⟶ B is a homomorphism in C and f : M ⟶ N_{[ϕ]} is a homomorphism in A-𝔐𝔬𝔡. Let M be a subcategory of C-𝔐𝔬𝔡. Then the pair (C, M) is called an admissible pair of categories if the following conditions hold:

(1) In C there exist finite fibered sums that we denote as usual by A ⊗_{K}^{C} B.

(2) If ϕ : A ⟶ B is a homomorphism in C and N a module in M(B), then N is via ϕ an object of M(A), and for each module M in M(A), Hom_{M(A)}(M, N_{[ϕ]}) is the set of all homomorphisms f : M ⟶ N in M, such that (ϕ, f) is a homomorphism in C-𝔐𝔬𝔡.

(3) Let A be an algebra in C. Then M(A) is an additive category, in which kernels and cokernels exist. Further, C_{A} is a subcategory of M(A) and the functor of M(A) in A-𝔐𝔬𝔡 commutes with kernels and finite direct sums.

(4) Let ϕ : A ⟶ B a homomorphism in C and u : M ⟶ N a homomorphism in M(B). Let L (resp. L′) be the kernel of u (resp. u_{[ϕ]}) in M(B) resp. M(A). Then the canonical map L′ ⟶ L_{[ϕ]} is an isomorphism in M(A).

(5) Let A be an algebra in C and N a module in M(A). For each finite family M_{i}; i ∈ I of modules in M(A), there is a given 𝕂-submodule

Mult_{M(A)}(M_{i}, i ∈ I; N)

of the module Mult_{A}(M_{i}, i ∈ I; N) of A-multilinear forms ∏_{i∈I} M_{i} ⟶ N, which is functorial in each M_{i} and N and has the following properties:

(5.1) Let i_{0} be an element of I and u : M′_{i_{0}} ⟶ M_{i_{0}} a homomorphism in M(A). Set M′′_{i_{0}} := Cokern(u) and M′_{i} := M′′_{i} := M_{i}, for i ∈ I \{i_{0}}. The sequence

0 → Mult_{M(A)}(M′′_{i}, i ∈ I; N) → Mult_{M(A)}(M_{i}, i ∈ I; N) →
Mult_{M(A)}(M′_{i}, i ∈ I; N)

induced by u is exact. (5.2) For modules M, N ∈ M(A), there is a canonical isomorphism

Mult_{M(A)}(M; N) ⟶ Hom_{M(A)}(M; N). (5.3) For M in M(A), the multiplication map μ_{M} : A × M ⟶ M is in

Mult_{M(A)}(A × M; M). (5.4) If σ : I ⟶ J is a bijective map, then the restriction of the isomorphism

Mult_{A}(M_{i}, i ∈ I; N) ⟶ Mult_{A}(M_{σ^{−1}(j)}, j ∈ J; N)

defined by σ, defines an isomorphism

Mult_{M(A)}(M_{i}, i ∈ I; N) ⟶ Mult_{M(A)}(M_{σ^{−1}(j)}, j ∈ J; N).

(5.5) Each homomorphism ϕ : A ⟶ B in C induces a cartesian diagram

\xymatrix{Mult_{M(B)}(M_{i}, i ∈ I; N) \ar[d] \ar[r] & Mult_{M(A)}((M_{i})_{[ϕ]}, i ∈ I; N_{[ϕ]}) \ar[d] \\ Mult_{B}(M_{i}, i ∈ I; N) \ar[r] & Mult_{A}((M_{i})_{[ϕ]}, i ∈ I; N_{[ϕ]})}

(5.6) For each i ∈ I, let L_{j}, j ∈ J_{i} a nonempty finite family of modules in M(A). Set J := ⨿_{i∈I}J_{i}. The canonical map

(∏_{i∈I} Mult_{M(A)}(L_{j}, j ∈ J_{i}; M_{i})) × Mult_{M(A)}(M_{i}, i ∈ I; N) ⟶

Mult_{A}(L_{j}, j ∈ J; N)

factorises through Mult_{M(A)}(L_{j}, j ∈ J; N). (5.7) The functor N ↦ Mult_{M(A)}(M_{i}, i ∈ I; N) on M(A) is represented by a

module _{⨂i∈I} _{A}^{M}M_{i} in M(A).

(5.8) If I is a disjoint union ⋃_{j∈J} I_{j} with I_{j} ≠ ∅ for all j, then the canonical homomorphism

⨂_{i∈I}_{A}^{M}M_{i} ⟶ ⨂_{j∈J}_{A}^{M}(⨂_{i∈I_{j}}_{A}^{M}M_{i})

is an isomorphism in M(A). (5.9) The canonical map A ⊗_{A}^{M} M ⟶ M is an isomorphism in M(A).

(6) Let ϕ : A ⟶ B be a homomorphism in C and M a module in M(A) and N a module in M(B). The module N_{[ϕ]} ⊗_{A}^{M} M is via the canonical A-bilinear map^{2}

B × N_{[ϕ]} ⊗_{A}^{M} M ⟶ N_{[ϕ]} ⊗_{A}^{M} M

a module in M(B). The analogue statement holds for M ⊗_{A}^{M} N_{[ϕ]}.

(7) Let k ⟶ A and k ⟶ B be two homomorphisms in C and ϕ (resp. ψ) the canonical maps of A (resp. B) in C := A ⊗_{k}^{M} B. Let M be a module in M(k) and ρ : C × M ⟶ M an element of Mult_{M(k)}(C × M; M) such that

(a) ρ extends the multiplication of k on M.

(b) M is via ρ a C-module.

(c) M_{[ϕ]} is in M(A) and M_{[ψ]} in M(B). Then M is in M(C).

(8) For algebras A and B in C_{k}, the canonical map A ⊗_{k}^{M} B ⟶ A ⊗_{k}^{C} B is an isomorphism in M(k).

Axioms (1) - (8) hold in the algebraic as well as in the analytic context. The difference between both contexts is manifest in the difference between Axiom (S1) and (S1’):

Axioms. Let A be an algebra in C.

(S1) If u : M ⟶ N is a homomorphism of finite A-modules in M(A), then the cokernel of u in M(A) coincides with the cokernel of u in A-𝔐𝔬𝔡 and for N = A the cokernel of u is an algebra in C_{A} with respect to the canonical projection A ⟶ Cokern(u). (S1’) For any homomorphism u : M ⟶ N of A-modules, the cokernel of u in M(A) coincides with the cokernel of u in A-𝔐𝔬𝔡 and for N = A the cokernel of u is an algebra in C_{A} with respect to the canonical projection A ⟶ Cokern(u). (S2) Bijective homomorphisms in M(A) are isomorphisms.

Examples 1.2. Again, consider the Examples 1.1.

(1) (C^{(0)}, M^{(0)}) is an admissible pair of categories that satisfies Axioms (S1’) and (S2).

(2) Let Mult_{M^{(1)}}( ) be the group of all continuous multilinear forms. Then (C^{(1)}, M^{(1)}) is an admissible pair of categories that satisfies Axioms (S1) and (S2).

2The existence of this map is a consequence of (2), (5.7) and (5.8).

Let (C, M) be an admissible pair of categories. Let k be an algebra in C and A, B, M and Nmodules in M(k).

Remark 1.3. (1) For elements a, a′ ∈ A and b, b′ ∈ B, we have (a ⊗ b)(a′ ⊗ b′) =

aa′ ⊗ bb′.

(2) Each homomorphism h ∈ Hom_{M(k)}(A ⊗_{k}^{M} B, N) is uniquely determined by its values on the elements of the form a ⊗ b.

Remark that in the graded commutative context (see below) we will have (a ⊗ b)(a′ ⊗ b′) = (−1)^{ba′}aa′ ⊗ bb′ for homogeneous a, b, a′, b′.

1.1. Free modules and algebras

To motivate the definition of markings on categories of algebras and modules: Let X be a compact n-dimensional polydisk. This is a Stein compact. We want to consider the algebra A = colim_{U⊃X} Γ(U, O_{ℂ^{n}}) of global sections on X as free algebra in C^{(1)}, generated by n algebra generators x_{1}, ..., x_{n}. If B is the algebra of sections on a second polydisk Q, we can’t just choose n elements of B as images of x_{1}, ..., x_{n} to define a homomorphism in C^{(1)}, since the corresponding map Q ⟶ ℂ^{n} would not, in general, land in P. There are restrictions on the values each x_{i} may take. A marking on the category C is such a restriction. A non-restrictive marking is called canonical. Usually, when the modules in M are non-graded, the marking on M will be the canonical one. Let (C, M) be an admissible pair of categories. graphFree algebras: A marking on C is a family (F_{τ})_{τ∈T} of subfunctors F_{τ} : C ⟶ (sets) of the identity functor such that F_{τ} (A) contains 0, for all τ, and all objects A in C. For a given object k of C and a family (τ_{i})_{i∈I}, we consider the functor F_{I,k} : A ↦ ∏_{i∈I} F_{τ_{i}}(A) on the category C_{k}. If F_{I,k} is representable, i.e. there is a k-algebra A and a canonical bijection

b : Hom_{k}^{C}(A, B) ⟶ ∏_{i∈I} F_{τ_{i}}(B)

for each algebra B in C_{k}, then A together with the family (e_{i})_{i∈I} = b(Id_{A}) is called the free algebra over k with free algebra generators e_{i}, i ∈ I. We will write A = k⟨e_{i}⟩_{i∈I}. The marking F is called representable, if F_{I,k} is representable for each k in C and each finite family (τ_{i})_{i∈I}.

1.1.0.1. Free modules: A marking on M is a family (G_{u})_{u∈U} of subfunctors G_{u} : M ⟶ (sets) of the identity functor such that, for each u ∈ U, the following condition holds: For each homomorphism ϕ : A ⟶ B in C and each module N in M(B), we have G_{u}(N_{[ϕ]}) = G_{u}(N). For a given algebra A in C and a family (u_{i})_{i∈I}, we consider the functor G_{I,A} : M ↦ ∏_{i∈I} G_{u_{i}}(M) on the category M(A). If G_{I,A} is representable, i.e. there is an A-module M and a canonical bijection

b : Hom_{M(A)}(M, N) ⟶ ∏_{i∈I} G_{u_{i}}(N)

for each A-module N, then M together with the family (e_{i})_{i∈I} = b(Id_{M}) is called the free module over A with free module generators e_{i}, i ∈ I. We will write M = ∐_{i∈I} Ae_{i}. The marking G is called representable, if G_{I,A} is representable, for each A in C and each finite family (u_{i})_{i∈I}. A marking on (C, M) is a pair (F, G) of a marking F = (F_{τ})_{τ∈T} on C and a marking G = (G_{u})_{u∈U} on M together with a map η : T ⟶ U such that F_{τ} (A) ⊆ G_{η(τ)}(A), for each A in C and each τ in T.

Axioms. Let (F, G) be a marking on (C, M).

(F1) F is representable.

(F2) Let k be an algebra in C and A = k⟨e_{i}⟩_{i∈I} be a free k-algebra in C. The canonical homomorphism k[e_{i}]_{i∈I} ⟶ k⟨e_{i}⟩_{i∈I} in k-𝔐𝔬𝔡 is flat and the functor M ↦ A ⊗_{k}^{M} M is exact on the category of finite modules in M(k).

(F3) Let A be like in (F2) and A′ = k⟨e′_{i}⟩_{i∈I′} be another free k-algebra in C with I ⊆ I′. Then A′ is flat over A via the homomorphism A ⟶ A′ with e_{i} ↦ e′_{i}.

(F4) G is representable.

(F5) For each u ∈ U and each A in C, G_{u} is a right exact functor on M(A).

(F6) Let A be an algebra in C and E = ∐_{i∈I} Ae_{i} be a free A-module with respect to G with finite basis (e_{i})_{i∈I} and let M be a module in M(A). The canonical homomorphism M^{I} ⟶ M ⊗_{A} E in A-𝔐𝔬𝔡 is bijective.

(F7) Let k be an algebra in C and A = k⟨e_{i}⟩_{i∈I} be a free k-algebra in C with finite I. Then Ω_{A/k} is a free A-module with base de_{i} ∈ G_{η(τ_{i})}(Ω_{A/k}); i ∈ I.

Remark that Axiom (F2) implies that free algebra generators (of degree 0) are no zero divisors.

Definition 1.4. The marking (F, G) is called good, if Axioms (F1), (F4), (F5), (F6) and (F7) hold. An admissible pair of categories (C, M) equipped with a good marking (F, G) is called a good pair of categories if it satisfies Axioms (S1) and (S2).

Examples 1.5. (1) On the admissible pair (C^{(0)}, M^{(0)}) of Example 1.1, we work with the canonical marking, i.e. F (A) = A, for each algebra A in C, and G(M) = M, for each module M in M(A). With this marking, (C^{(0)}, M^{(0)}) is a good pair of categories, that satisfies additionally Axioms (F2) and (F3).

(2) Consider the admissible pair (C^{(1)}, M^{(1)}). For A in C^{(1)} and t ∈ T := (0, ∞), let F_{t}(A) be the set of all elements of A such that the Gelfand-transformation (see [4] for the definition) χ(A) ⟶ ℂ factorises through {z ∈ ℂ : |z| ⩽ t}. Further, let G be the canonical marking on M^{(1)}. The pair (C^{(1)}, M^{(1)}), together with the marking (F, G) is a good pair of categories, that satisfies Axioms (F2) and (F3).

(3) If C is the category of local analytic algebras and M the category of DFNmodules over C, then G is set to be the canonical marking and for objects A, we set F (A) to be the maximal ideal 𝔪_{A} of A. The pair (C, M) is a good pair of categories that satisfies Axioms (F2) and (F3).

1.2. Graded objects

Let (C, M) be an admissible pair of categories. As in [1], we can construct a new admissible pair (gr(C), gr(M)) as follows: Let gr(C) be the category of graded commutative^{3} rings A = ∐_{i⩽0} A^{i} with A^{0} in C, all A^{i} in M(A^{0}) such that the multiplication maps A^{i} × A^{j} ⟶ A^{i+j} belong to Mult_{M(A^{0})}(A^{i} × A^{j}, A^{i+j}). A homomorphism ϕ : A ⟶ B in gr(C) is a homomorphism of graded commutative rings such that ϕ^{0} is a homomorphism in C and all ϕ^{i} : A^{i} ⟶ B^{i} are ϕ^{0}- linear homomorphisms in M. Let gr(M) be the category over gr(C) whose objects over an algebra A in gr(C) are the graded A-modules M = ∐_{i∈ℤ} M^{i}, with M^{i} = 0, for almost all i > 0, such that each M^{i} is in M(A^{0}) and the maps A^{i} × M^{j} ⟶ M^{i+j} belong to Mult_{M(A^{0})}(A^{i} × M^{j}, M^{i+j}). If B is another algebra in gr(C) and N is a module in gr(M)(B), then Hom_{gr(M)}(M, N) is the set of all pairs (ϕ, f), where ϕ : A ⟶ B is a homomorphism in gr(M) and f : M ⟶ N is a ϕ-linear homomorphism of degree zero, such that all f^{i} : M^{i} ⟶ N^{i} are homomorphisms in M over ϕ^{0}.

For modules M_{1}, …, M_{n} and N in gr(M)(A), let Mult_{gr(M)(A)}(M_{1} ×...×M_{n}, N) be the 𝕂-module of all maps f : M_{1} ×…×M_{n} ⟶ N with the following properties:

(1) For k_{1}, …, k_{n} in ℤ, the restriction f|_{M_{1}^{k_{1}} ×...×M_{n}^{kn}} factorises through a map in

Mult_{M(A^{0})}(M_{1}^{k_{1}} × … × M_{n}^{k_{n}}, N^{k_{1}+...+k_{n}}).

(2) For elements a ∈ A and m_{i} ∈ M_{i}, we have f(m_{1}, …, m_{r}a, m_{r+1} … m_{n}) = f(m_{1}, …, m_{r}, am_{r+1}, …, m_{n}), for 1 ⩽ r < n, and f(m_{1}, …, m_{n}a) = f(m_{1}, …, m_{n})a

We just have made use of the fact that we can make each M in gr(M)(A) a graded symmetrical A-bimodule by setting m · a := (−1)^{g(m)g(a)}a · m, for homogeneous elements a ∈ A and m ∈ M. To define free algebras in gr(C), we modify the definition in Section 1.1 as follows: There is a map g : T ⟶ ℤ_{⩽0} and each functor F_{τ} is a subfunctor of the functor A ↦ A^{g(τ)}. In this context, if F is representable, then each functor F_{I,A} is representable, if for n ⩽ 0, the set of all τ_{i} with g(τ_{i}) = n is finite. In this case the free algebra A⟨e_{i}⟩_{i∈I} is called g-finite free algebra. Of course, the degree g(e_{i}) of a free generator e_{i} is just g(τ_{i}).

To define free modules in gr(M), we modify the Definition in Section 1.1 as follows: There is a map g : U ⟶ ℤ and each functor G_{u} is a subfunctor of the functor M ↦ M^{g(u)}. In this graded context, when G is representable, then each functor G_{I,A} is representable, if for each n, the set of all u_{i} with g(u_{i}) = n is finite. In this case the free module ∐_{i∈I} Ae_{i} is called g-finite free A-module. We have g(e_{i}) = g(u_{i}) for i ∈ I.

To define a marking on (gr(C), gr(M)), we have to add in Section 1.1 the condition, that the map η : T ⟶ U is compatible with g.

Example 1.6. If G is a marking on M, then we get a marking gr(G) = (gr(G))_{u′∈U′} on gr(M) in the following way: Set U′ := U × ℤ. For A in gr(C), M in gr(M) and

3i.e. ab = (−1)^{g(a)g(b)}ba, for homogeneous a, b ∈ A

u′ = (u, n) ∈ U′ set gr_{u′}(G)(M) := G_{u}(M^{n}). Here we have g(u′) = n.

If (F, G) is a marking on (C, M), then we get a marking gr_{G}(F) = (gr_{G}(F)_{τ′})_{τ′∈T′} in the following way: Let T′ be the disjoint union of T × {0} and U × ℤ_{<0}. For A in gr(C) and τ′ = (τ, n), we set gr_{G}(F)_{τ′}(A) = F_{τ} (A), if n = 0, and gr_{G}(F)_{τ′}(A) = G_{τ} (A^{n}), if n < 0. If (F, G) is a marking on (C, M), then (gr_{G}(F), gr(G)) is a marking on (gr(C), gr(M)) with the map η′ : T′ ⟶ U′ given by (τ, 0) ↦ (η(τ), 0) and (u, n) ↦ (u, n), for n < 0.

Remark that by Lemma I.7.6 of [1], free algebra generators of negative degree behave much like polynomial variables^{4} and if (C, M) is a good pair of categories, then (gr(C), gr(M)) is a good pair of categories as well. If (C, M) is an admissible pair of categories that satisfies Axiom (S2), then by Proposition I.6.9 of [1], the admissible pair (gr(C), gr(M)) also satisfies Axiom (S2). In general, this is not true for Axiom (S1). But we have:

Remark 1.7. Let (C, M) satisfy Axiom (S1) and let A be an object of gr(C) such that all A^{i} are finite A^{0}-modules. For g-finite modules M, N in gr(M)(A) each homomorphism f : M ⟶ N in gr(M)(A) is strict, i.e. the cokernel of f in gr(M) coincides with the set-theoretical cokernel.

Remark 1.8. Let (C, M) be an admissible pair of categories with a marking (F, G), where G is canonical. Suppose that Axiom (S1) holds. Let k be an algebra in C and let M_{1}, M_{2} and N be modules in M(k) such that M_{1} and M_{2} are finite k-modules with M_{i} ⊆ N and M_{1} ∩ M_{2} = {0}. Then we have

(1) The inclusions M_{i} ↪ N are homomorphisms in M(k).

(2) M_{1} + M_{2} is in M(k).

(3) The inclusions M_{i} ⟶ M_{1} + M_{2} are homomorphisms in M(k).

(4) The projections p_{i} : M_{1} + M_{2} ⟶ M_{i} are homomorphisms in M(k).

(5) M_{1} + M_{2} = M_{1} ⊕ M_{2}. In (gr(C), gr(M)) the same statement is true if we suppose that all k^{i} are finite k^{0}-modules and M_{1}, M_{2} and N are g-finite.

The proof is left to the reader as exercise. In the sequel, we will denote the full subcategory of gr(C) generated by all algebras A such that each A^{i} is a finite A^{0}-module by gr′(C).

Remark 1.9. (1) Suppose that (C, M) is a good pair of categories and that k is an algebra in gr′(C). Let R = k⟨T ⟩ be a free g-finite algebra over k in gr(C). We have the following decomposition

R = k ⊕ ∑_{t∈T} Rt

in the category gr(M)(k).

4For a more precise statement, see Proposition 1.17.

(2) Suppose that additionally the marking G on M is trivial and that Axiom (F2) holds. Then, for each n ⩾ 0, R decomposes as

R = k ⊕ ∑_{t_{1}∈T} t_{1}k ⊕ ... ⊕ ∑_{t_{1},...,t_{n}∈T} t_{1} · . . . · t_{n}k ⊕ ∑_{t_{1},...,t_{n+1}∈T} t_{1} · . . . · t_{n+1}R.

Proof. We only prove (i) and leave the proof of (ii) as exercise. We can form the free g-finite R-module M = ∐_{t∈T} Re(t), where to each free algebra generator t ∈ gr_{G}(F)_{τ′} we have associated a free module generator e(t) ∈ gr(G)_{η(τ′)}(M). Consider the homomorphism M ⟶ R in gr(M)(R) with e(t) ↦ t. By Remark 1.7, the cokernel map of this homomorphism coincides with the cokernel map in R-𝔐𝔬𝔡, which is just the projection p : R ⟶ R/(T) and R/(T) is an algebra in gr(C). Now there is a diagram

\xymatrix{R \ar[r]^{π} \ar[dr]^{p} & k \ar[d] \\  & R/(T)}

in gr(C), where π : R ⟶ k is the homomorphism given by t ↦ 0 for t ∈ T and the homomorphism k ⟶ R/(T) is the canonical inclusion. The diagram commutes, since in both directions an element t ∈ T goes to 0. So we get Kern(π) = (T). But obviously, we have R = k ⊕ Kern(π).

1.3. Simplicial objects

Let I be an index set. A set N of subsets of I is called simplicial complex over I, if ∅ ∉ N; if for all i ∈ I, we have {i} ∈ N and if every nonempty subset of an element in N is again in N. For an element α of a simplicial complex N over I, containing n elements, set |α| := n − 1. Then for n ⩾ 0, the set N^{(n)} of all α ∈ N with |α| ⩽ n is again a simplicial complex over I. A simplicial complex N can be seen as category, where Hom(α, β) contains only the inclusion α ⊆ β if α ⊆ β and is empty in all other cases. Let A be a category. An N -object in A is a covariant functor N ⟶ A. The N -objects in A again form a category, denoted by A^{N}. If (C, N) is an admissible pair of categories and A = (A_{α})_{α∈N} an object of C^{N}, then we denote the category of N -objects M = (M_{α})_{α∈N} in M^{N} with M_{α} ∈ ob(M(A_{α})) by M^{N} (A). Let (C, M) be an admissible pair of categories and N a simplicial complex. Let ((F_{τ})_{τ∈T}, (G_{u})_{u∈U}) be a marking on (C, M). Then, for each pair (α, τ) in N × T, there is a functor F_{α,τ} : A ↦ F_{τ} (A_{α}). For a family (α_{i}, τ_{i}), i ∈ I of elements of N × T and A in C^{N}, there is a set-valued functor B ↦ ∏_{i∈I} F_{α_{i},τ_{i}}(B). We denote it by F_{I,A}.

Remark 1.10. Suppose that for each α ∈ N, the free A_{α}-algebra A′_{α} = A_{α}⟨e_{i}^{(α)}⟩_{α_{i}⊆α} in the free generators e_{i}^{(α)} ∈ F_{τ_{i}}(A′_{α}) exists. For α ⊆ β, let ρ_{αβ} : A′_{α} ⟶ A′_{β} be

the homomorphism in C over A_{α}, given by e_{i}^{(α)} ↦ e_{i}^{(β)}. Then A′ = (A′_{α})_{α∈N} is an algebra in C^{N}, and together with the family (e^{(α_{i})})_{i∈I}, it represents the functor F_{I,A}. We call it the free A-algebra in the free generators e_{i} := e_{i}^{(α_{i})} ∈ F_{α_{i},τ_{i}}(A′) and denote it by A⟨e_{i}⟩_{i∈I}.

For each pair (α, u) ∈ N × U, there is a functor G_{α,u} : M ↦ G_{u}(M_{α}). For a family (α_{i}, u_{i}), i ∈ I of elements of N ×U and A in C^{N}, there is a set-valued functor N ↦ ∏_{i∈I} G_{α_{i},u_{i}}(M). We denote it by G_{I,A}.

Remark 1.11. Fix a family (α_{i}, u_{i}), i ∈ I of elements of N × U and an algebra A in C^{N}. Suppose that for each α ∈ N, the free A_{α}-module M_{α} = ∐_{α_{i}⊆α} A_{α}e_{i}^{(α)} in

the free generators e_{i}^{(α)} ∈ G_{u_{i}}(M_{α}) exists. For α ⊆ β, let ρ_{αβ} : M_{α} ⟶ M_{β} be the homomorphism in M over A_{α}, given by e_{i}^{(α)} ↦ e_{i}^{(β)}. The collection M = (M_{α})_{α∈N} is a module in M^{N} and together with the family (e^{(α_{i})})_{i∈I} it represents the functor G_{I,A}. We call it the free A-module with free generators e_{i} := e_{i}^{(α_{i})} ∈ G_{α_{i},u_{i}}(A′) and denote it by ∐_{i∈I} Ae_{i}.

To distinguish the non-simplicial from the simplicial context, we call the first one affine.

1.4. Resolvents

Fix an admissible pair (C, M) of categories with marking (F, G). For a DG module K in gr(M)(R) with differential d, we define the i-th homology H_{M}^{i}(K) of K in M(R) as cokernel of the natural map Im(d^{i−1}) ⟶ Kern(d^{i}) (image and kernel formed in the category M(R)). If K is separated, i.e. the cokernels of the map d : K^{i} ⟶ K^{i+1} and of the induced maps K^{i} ⟶ Kern(d^{i+1}) and Kern(d^{i}) ⟶ K^{i} coincide with their cokernels formed in the category R-𝔐𝔬𝔡, then H_{M}^{i}(K) is as R-module isomorphic to the i − th cohomology of K, considered as complex in R − 𝔐𝔬𝔡. We call K acyclic, if H_{M}^{i}(K) = 0 for all i.

Remark 1.12. Suppose that (C, M) satisfies Axiom (S1) and all K^{i} are finite R-modules. K is acyclic if and only if K is acyclic as complex in R − 𝔐𝔬𝔡.

A DG resolution of an object B in M is a DG module M in gr(M) such that H_{M}^{i}(M) = 0, for i < 0, and H_{M}^{0}(M) = B.

We recall Definition I.8.1 of [1].

Definition 1.13. Let A ⟶ B be a homomorphism of DG objects in gr(C)^{N}. A resolvent of B over A is a free DG algebra R over A in gr(C)^{N} (with respect to the marking gr_{G}(F)) together with a morphism R ⟶ B of DG objects in gr(C)^{N} which is a surjective quasi-isomorphism on each α ∈ N.

In the present paper, we will mostly work in a Noetherian context, i.e. we will assume that the following Axiom is satisfied:

Axiom (N) Each algebra A in C is Noetherian and each finite module M in M(A) is a quotient of a finite free A-module.

If the good pair (C, M) satisfies Axioms (N) and (F2) and if A and B belong to gr′(C) and if A^{0} is a quotient of a g-finite B^{0}-module C in C^{N} such that each C_{α} is a finite free B_{α}^{0}-algebra, then such resolvents exist by Proposition I.8.7 and I.8.8 of [1] or by Remark 1.21.

The next proposition is of great importance for this work. In the algebraic context, the statement is well-known and was used by Quillen and others. The difference in the analytic context is that a free DG algebra over a ring k is not, in general, a complex of free k-modules as long as there are analytic algebra generators, i.e. free algebra generators of degree zero. Hence, even the affine version of the following statement seems to be new in this form.

Suppose that the marking G on M is trivial and that Axiom (N) is satisfied.

Proposition 1.14. Let A ⟶ B be a homomorphism of DG algebras in gr(C)^{N}. For two g-finite resolvents R_{1} and R_{2} of B over A, there exists a homomorphisms R_{1} ⟶ R_{2} in gr(C)^{N}, which is a homotopy equivalence over A.

Proof. We prove the affine case first and sketch the generalization to N -objects in gr(C)^{N}, afterwards. First affine case: Suppose that R_{1}^{0} = R_{2}^{0}. Set A′ := A ⊗_{A^{0}} R_{1}^{0}. Then R_{1} and R_{2} are resolvents of B over A′. By Proposition I.8.1 of [1], there is a quasi-isomorphism R_{1} ⟶ R_{2} in gr(C) over A′. Since R_{1}^{0} = R_{2}^{0} = A′^{0}, R_{1} and R_{2} are free A′-modules in gr(M). Hence the quasi-isomorphism is already a homotopy-equivalence. Second affine case: Suppose that R_{2}^{0} is a finite free algebra over R_{1}^{0} in C. By induction, we can restrict ourselves to the case where R_{2}^{0} = R_{1}^{0}⟨e⟩ is just a free algebra in one generator. Consider the free R_{1}^{0}-algebra R := R_{1}^{0}⟨e, f⟩ in gr(C), generated by a free generator e of degree 0 and a free generator f of degree −1. We define a differential on R by setting f ↦ e. By Remark 1.9, we have R_{1}^{0}⟨e⟩ = R_{0} ⊕ eR_{1}^{0}⟨e⟩. So by Axiom (S2), the differential gives an isomorphism fR_{1}^{0}⟨e⟩ ⟶ eR_{1}^{0}⟨e⟩. With this in mind, we can easily construct a contracting homotopy on R. Now R′_{1} := R_{1} ⊗_{R_{1}^{0}} R is homotopic over R_{1} to R_{1}. More precisely, the inclusion R_{1} ⟶ R′_{1} and the projection R′_{1} ⟶ R′ are homotopy equivalences. By the first case, there is a homotopy-equivalence R′_{1} ⟶ R_{2} in gr(C). General affine case: Let R_{3} be a free g-finite resolution of B over R_{1} ⊗_{A} R_{2}. Now R_{3} is free over R_{1} and R_{2} and by the second case, we get a homotopy-equivalence R_{1} ⟶ R_{3} ⟶ R_{2}. In the simplicial case, a free algebra in gr(C)^{N} over an algebra A in gr(C)^{N} is not a free module in gr(M)^{N} (A), even if all free algebra generators are of strictly negative degree. The point is that even A itself is not free as A-module. But a free algebra over A with free algebra generators of negative degree is as A-module in gr(M)^{N} a direct sum A ⊕ M with a free A-module M. To prove the simplicial case, we must first generalize the Comparison Theorem (see Theorem 2.2.6 of [20]) to (free) DG resolutions in gr(M)^{N} which is straightforward. Secondly, observe that for a DG algebra A in gr(C)^{N} and free A-modules M, N, each quasi-isomorphism Id ×f : A ⊕ M ⟶ A ⊕ N is even a homotopy equivalence. With those tools we can generalize the first case above and the second and third case go just as in the affine situation.

1.5. Double graded objects

Let (C, M) be an admissible pair of categories. We define the pair (gr^{2}(C), gr^{2}(M)) as follows: The objects of gr^{2}(C) are the double commutative graded^{5} rings A = ∐_{i,j⩽0} A^{i,j} with A^{0,0} in C and all A^{i,j} in M(A^{0,0}) such that the multiplication maps A^{i,j} × A^{k,l} ⟶ A^{i+j,k+k} belong to

Mult_{M(A^{0,0})}(A^{i,j} × A^{k,l}, A^{i+j,k+l}).

Following the ideas of Section 1.2, we can define Hom_{gr^{2}(C)}(A, B), for objects A, B in gr^{2}(C), the category gr^{2}(M), Hom_{gr^{2}(M)}(M, N) for objects M, N of gr^{2}(M) and Mult_{gr^{2}(M)(A)}(M_{1}, ..., M_{n}, N) for modules M_{1}, ..., M_{n}, N in gr^{2}(M)(A). We don’t make this definitions explicit here.

Remark 1.15. Let A be an object of gr^{2}(C) and M, N objects of gr^{2}(M). For (p, q) in ℤ × ℤ, set T^{p,q} := ∐_{i+j=p,k+l=q} M^{i,k} ⊗_{A^{0,0}}^{M} N^{j,l}. Then T := ∐_{p,q} T^{p,q} is a tensor product of A and B in gr^{2}(M)(A^{0,0}). T can be seen in two different ways as object of gr^{2}(M)(A). Consider the homomorphism u : A ⊗_{A^{0,0}} T ⟶ T in gr^{2}(M)(A^{0,0}), sending a⊗m⊗n to ma⊗n−m⊗an. u can be seen in two manners as homomorphism in gr^{2}(M)(A). Both of them induce the same A-module structure on T̄ := Cokern(u). We see that T̄ is a tensor product of M and N in gr^{2}(M)(A).

Remark 1.16. The pair (gr^{2}(C), gr(M)^{2}) is an admissible pair of categories.

Proof. Analogue to the proof of Proposition 6.9 of [1].

Convention: When we consider an object K of gr(M) as object of gr^{2}(M), we set K^{i,0} = K^{i} and K^{i,j} = 0 for j ≠ 0. In the same manner as above, we can define a marking (gr_{G}^{2}(F), gr^{2}(G)) on the pair (gr^{2}(C), gr^{2}(M)): Define the index set T′′ as T×{0, 0}∪U ×(ℤ^{⩽0}×ℤ^{⩽0})\(0, 0). For τ′′ = (τ, 0, 0) ∈ T′′ and A ∈ gr^{2}(C) set gr_{G}^{2}(F)_{τ′′}(A) := F_{τ} (A^{0,0}) and for τ′′ = (u, p, q) in T′′ with (p, q) ≠ (0, 0) set gr_{G}^{2}(F)_{τ′′}(A) := G_{u}(A^{p,q}). Define the index set U′′ as U × ℤ × ℤ. For u′′ = (u, p, q) ∈ U′′ and M ∈ gr^{2}(M), set gr(G)_{u′′}(M) := G_{u}(M^{p,q}).

Proposition 1.17. (1) Let A be an algebra in gr^{2}(C) and A′ = A⟨e_{i}⟩_{i∈I} a free algebra over A, with respect to the marking gr_{G}^{2}(F). Suppose that the bidegree of each e_{i} is different to 0, then the canonical homomorphism A[e_{i}]_{i∈I} ⟶ A′ in A-𝔄𝔩𝔤 is bijective.

(2) If (F, G) is good, then (gr_{G}^{2}(F), gr^{2}(G)) is good as well.

Proof. Analogue to the proof of Lemma I.7.6 of [1].

Definition 1.18. A DG algebra in gr^{2}(C) is an algebra A in gr^{2}(C) equipped with a (vertical) A^{0,0}-derivation^{6} v : A ⟶ A of bidegree (0, 1) with s^{2} = 0. A DDG Algebra in gr^{2}(C) is a DG algebra A in gr^{2}(C) equipped with a (horizontal) derivation h of bidegree (1, 0) that anti-commutes with v such that h^{2} = 0. A homomorphism of (D)DG algebras is a morphism in gr^{2}(C) that commutes with the vertical (and horizontal) differentials.

5i.e. for a ∈ A^{i,j} and b ∈ A^{k,l} we have ab = (−1)^{(i+j)(l+k)}ba 6i.e. for homogeneous a, b ∈ A we have v(ab) = v(a) + (−1)^{a}av(b)

The definition of (D)DG modules over D(DG) algebras is straightforward (pay attention to Koszul signs).

Remark 1.19. Let K = (K, h, v) be a DG algebra in gr^{2}(C). Consider a free algebra K⟨E⟩ over K in gr^{2}(C) with a set E = {e_{i} : i ∈ I} of free algebra generators with e_{i} ∈ gr_{G}^{2}(F)_{τ′′_{i}}(K⟨E⟩), for a certain τ′′_{i} ∈ T′′. For each i, if g(x_{i}) ≠ (0, 0) choose an element h_{i} ∈ G_{u_{i}}(K⟨E⟩^{g(x_{i})+(1,0)}) and an element v_{i} ∈ G_{u_{i}}(K⟨E⟩^{g(x_{i})+(0,1)}), where u_{i} is the first component of τ_{i} = (u_{i}, g(x_{i})). Setting h(e_{i}) := h_{i} and v(e_{i}) := v_{i}, we get an extension of the horizontal and the vertical derivation h and v of K. This extensions make K⟨E⟩ a DDG algebra, if and only if, for each i, we have

(1) h(v_{i}) + v(h_{i}) = 0 and

(2) h(h_{i}) = v(v_{i}) = 0.

Proof. Inductively, we can reduce the proof to the case where E consists of a single element e of bidegree (p, q). In this case, it is an easy calculation.

Definition 1.20. A DG resolvent of an algebra B in gr(C) is a free DG algebra A in gr^{2}(C) such that, for all i, the i-th row is a surjective DG module resolution of B^{i}. A DDG resolvent of a DG algebra B in gr(M) is a DDG algebra A in gr^{2}(M) which is a DG resolvent of B such that the map A^{∗,0} ⟶ B is a homomorphism of DG algebras in gr(C).

For a homomorphism A ⟶ B of DG algebras in gr(C), to get a resolvent R of B over A it is enough to construct a DDG resolvent K of B which is free over A as object of gr^{2}(C). Then we can choose R as total complex tot(K). This leads to the question of the existence of DDG resolvents. The following remark provides a positive answer:

Remark 1.21. Suppose that for the pair (C, M) the Axioms (N) and (F2) hold. Let K = (K, h, v) be a DDG algebra in gr^{2}(C) and u : K^{∗,0} ⟶ A a homomorphism of DG algebras in gr(C). Suppose that each A ∈ gr′(C) and that each K^{i,j} a finite K^{0,0}-module.

(1) If A^{0} is a quotient of a free K^{0,0}-algebra, then there exists a free DDG algebra L = K⟨F ⟩ over K, where F is a g-finite set of generators of bidegree (k, 0); k ⩽ 0, and a surjective homomorphism L^{∗,0} ⟶ A over K^{∗,0}.

(2) Suppose that u is surjective and that for a fixed p < 0, we have u^{p+1} = cokern(v^{p+1,−1}). There exists a free DDG algebra L = K⟨F ∪ G⟩ over K with finite sets F and G of generators of bidegree (p, −1) and (p + 1, −1), respectively, such that we still have u^{p+1} = cokern(v^{p+1,−1}), and additionally u^{p} = cokern(v^{p,−1}) holds.

(3) Fix p ⩽ 0 and q ⩽ −1. Suppose that we have H^{q+1}(K^{p+1,∗}) = 0. There exists a free DDG algebra L = K⟨F ∪ G⟩ over K spanned by finite sets F and G of generators of bidegree (p, q) and (p + 1, q), respectively, such that we still have H^{q+1}(K^{p+1,∗}) = 0, and additionally H^{q+1}(K^{p,∗}) = 0 holds.

Proof. The proofs of (1) - (3) are very similar, so we only do the proof of (3). We choose G such that there is an epimorphism π : ⨿_{g∈G}K^{0,0}g ⟶ Kern(v^{p+1,q+1}) ∩ Kern(h^{p+1,q+1}). Set v(g) := π(g) and h(g) := 0. We choose F such that there is an epimorphism π′ : ⨿_{f∈F} K^{0,0}f ⟶ Kern(v^{p,q+1}). Set v(f) = π′(f) and choose h(f) in ⨿K^{0,0}g in such a way that v(h(f)) = −h(v(f)).

Definition 1.22. For a g-finite free DG module M = ∐_{e∈E} Ae in gr(M) with differential d (this construction can be done more generally in gr(M)^{N}), we define the exterior algebra ∧ ∧_{A} M to be the free DDG algebra A⟨Ê⟩ in gr^{2}(C), where Ê contains for each e ∈ E a free algebra generator ê of bidegree (g(e), −1). The vertical differential of ∧ ∧_{A} M is set to be trivial, and the horizontal differential h is defined in such a way that the assignment e ↦ ê identifies M as DG module with the line A⟨Ê⟩^{∗,−1}. The total complex ∧_{A}M := tot(∧ ∧_{A} M) has the structure of a DG algebra in gr(C) and corresponds to the ordinary definition of the exterior algebra. In this situation, let ∧_{A}^{j}M be the DG module in gr(M) with (∧_{A}^{j}M)^{n} = A⟨Ê⟩^{(n,−j)}, for all j ⩾ 0.

In particular, we have ∧_{A}^{0}M ≅ A and ∧_{A}^{1}M ≅ M and

∧_{A}M = tot(∧ ∧_{A} M) = ∐_{j⩾0} ∧^{j}M[j].  (1.1)

1.6. The (cyclic) bar complex

For convenience of the reader and to motivate our definitions in Section 2, in this section, we state several statements about the (cyclic) bar complex. In the algebraic context they are well-known and they apply directly to admissible pairs of categories. Let (C, M) be an admissible pair of categories. Consider a homomorphism k ⟶ A of DG algebras in gr(C). The tensor product R := A ⊗_{k}^{gr(C)} A is a DG algebra with differential d^{R} defined by

d^{R}(a ⊗ a′) = d^{A}(a) ⊗ a′ + (−1)^{a}a ⊗ d^{A}(a′)

and the natural “multiplication” map μ : A ⊗_{k}^{C} A ⟶ A respects the differentials.

Let M be a DG A-bimodule in gr(M), which is a symmetrical k-bimodule. We can consider M as DG object of gr(M)(R), where the scalar multiplication R × M ⟶ M satisfies (a ⊗ a′, m) ↦ (−1)^{a′m}ama′, for homogeneous elements a, a′ ∈ A and m ∈ M. To see this, we have to apply Axioms (5.3), (5.5) and (5.6). The same axioms must be used to define the mappings in the sequel. For n = 0, 1, … set C_{n}^{cycl}(A, M) := M ⊗ A^{⊗n} and C_{n}^{bar}(A, M) := M ⊗ A^{⊗n} ⊗ A. (All tensor products are formed in the category gr(M)(k).) Consider the homomorphisms

d_{i} : M ⊗ A^{⊗n} ⟶ M ⊗ A^{⊗n−1},

sending elements a_{0} ⊗ ... ⊗ a_{n} to a_{0} ⊗ ... ⊗ a_{i} · a_{i+1} ⊗ ... ⊗ a_{n}, for i = 0, ..., n − 1, and the homomorphism d_{n}, sending homogeneous elements a_{0} ⊗ ... ⊗ a_{n} to (−1)^{a_{n}(a_{1}+...+a_{n−1})}a_{0}·a_{n}⊗a_{1}⊗...⊗a_{n−1}. Each d_{i} is a homomorphisms in gr(M)(A), if we regard the tensor products M ⊗A⊗...⊗A as A-modules by left-multiplication on the first factor. Remark that when M is only a A-right module, we consider it as an antisymmetrical A-bimodule by setting m · a := (−1)^{ma}a · m.

Set b′_{n−1} := d_{0}−...+(−1)^{n−1}d_{n−1} and b_{n} := b+(−1)^{n}d_{n}. Exactly as in the algebraic case (see paragraph III.2.1 of [1]), b defines a differential on C_{•}^{cycl}(A, M), i.e. b^{2} = 0. The pair (C_{•}^{cycl}(A, M), b) is called cyclic bar complex. b′ defines a differential on C_{•}^{bar}(A, M). The pair (C_{•}^{bar}(A, M), b′) is called (acyclic) bar complex. For later use, we set C^{bar}(A, M)^{−n} := C_{n}^{bar}(A, M) and C^{cycl}(A, M)^{−n} := C_{n}^{cycl}(A, M), for n ⩾ 0. Observe that C_{•}^{bar}(A, M) is even a complex in gr(M)(R), when we define the Rmodule structure on M ⊗ A^{⊗n} ⊗ A by

(a ⊗ a′) · (m ⊗ α ⊗ a_{n+1}) = (−1)^{a(a′+m+α)}a′m ⊗ α ⊗ a · a_{n+1}

for homogeneous elements a, a′, α and m. In the sequel, we write C_{•}^{cycl}(A) for C_{•}^{cycl}(A, A) and C_{•}^{bar}(A) for C_{•}^{bar}(A, A).

In gr(M)(k) there exist homomorphisms C_{n}^{bar}(A) ⟶ C_{n+1}^{bar}(A), sending elements a_{1} ⊗ ... ⊗ a_{n} to 1 ⊗ a_{1} ⊗ ... ⊗ a_{n}. They define a contracting homotopy for the bar complex. Hence the bar complex C^{bar}(A) is acyclic. By Theorem III.2.2 of [1], we can even define a DDG algebra structure on C^{bar}(A). Hence tot(C^{bar}(A)) is a DG algebra resolution of A over R. But it can only serve as resolvent in the algebraic case since: Attention: In the analytic case, tot(C^{bar}(A)) is not a free object in gr(C). Recall two well-known relations between the cyclic and acyclic bar complexes. We consider R as A-bimodule via a(a_{1} ⊗ a_{2}) = aa_{1} ⊗ a_{2} and (a_{1} ⊗ a_{2})a = a_{1} ⊗ a_{2}a.

Proposition 1.23. We have an isomorphism C_{•}^{cycl}(A, R) ⟶ C_{•}^{bar}(A) of complexes in gr(M)(A), which is in the n-th component given by

C_{n}^{cycl}(A, R) ⟶ C_{n}^{bar}(A)
(a ⊗ a′) ⊗ α ↦ (−1)^{a(a′+α)}a′ ⊗ α ⊗ a

with α ∈ A^{⊗n}. Furthermore, we have an isomorphism C_{•}^{cycl}(A, M) ⟶ M ⊗_{R} C_{•}^{bar}(A) of complexes in gr(M)(A), where the differential of the second complex is given by 1 ⊗ b′. The n-th component has the following form:

C_{•}^{cycl}(A, M) ⟶ M ⊗_{R} C_{•}^{bar}(A)
m ⊗ α ↦ m ⊗ 1 ⊗ α ⊗ 1.

Proof. Analogue to the proof of Proposition 1.1.13 of [15].

In the algebraic literature, the cyclic bar complex is often called Hochschild chain complex and the Hochschild cochain complex is defined as the complex C^{•}(A, M) = (C^{•}(A, M), β) where C^{0}(A, M) = M and C^{n}(A, M) = Hom_{k}(A^{⊗n}, M), for n = 1, 2, … The differential β is given by:

β(f)(a_{1}, …, a_{n+1}) = a_{1}f(a_{2}, …, a_{n+1}) − f(a_{1} · a_{2}, …, a_{n+1}) + …
+(−1)^{n}f(a_{1}, …, a_{n}a_{n+1}) + (−1)^{n+1}f(a_{1}, …, a_{n})a_{n+1}.

We will define the Hochschild complex in a more general way in Section 2.

Proposition 1.24. If M is a graded symmetric A-bimodule, then there exists an isomorphism of complexes

Hom_{k}(A^{⊗n}, M) ⟶ Hom_{A}(C_{n}^{cycl}(A), M),

where the differential on the left complex is β and the differential on the right complex is the one induced by the differential b on C_{•}^{cycl}(A). Furthermore, we have an isomorphism of complexes

Hom_{R}(C_{•}^{bar}(A), M) ⟶ Hom_{k}(A^{⊗n}, M),

sending an f : C_{n}^{bar} ⟶ M to the mapping a_{1} ⊗ … ⊗ a_{n} ↦ f(1 ⊗ a_{1} ⊗ … ⊗ a_{n} ⊗ 1).

Proof. Analogue to the proof of the corresponding statements in Section 1.5 in [15].

1.7. Regular sequences

In this section we want to define a regular sequence for the graded commutative context. In our definition the question if a sequence is regular won’t depend on the order of its elements. We suppose that the ground ring 𝕂 contains the rational numbers. We work with an admissible pair of categories (C, M), equipped with a marking ((F_{t})_{t∈T} , (G_{u})_{u∈U}), which induces the marking ((gr_{G}(F))_{τ′∈T′}, (gr(G))_{u′∈U′}) on (gr(C), gr(M)) and the marking ((gr_{G}^{2}(F))_{τ′′∈T′′}, (gr^{2})_{u′′∈U′′}) on (gr^{2}(C), gr^{2}(M)).

Definition 1.25. Let R be an algebra in gr(C). We call a g-finite subset X of R a handy sequence if for each x, there is an u(x) ∈ U such that x ∈ gr(G)_{(u(x),g(x))}(R) = G_{u(x)}(R^{g(x)}). When R = (R, s) is a DG algebra, then a handy sequence X ⊆ is called handy s-sequence if we have s(X) ⊆ (X). For a handy sequence X ⊆ R, let E be a set of free algebra generators, containing for each x ∈ X a generator e(x) ∈ gr_{G}^{2}(F)_{(u(x),g(x),−1)}(R⟨E⟩) of bidegree (g(x), −1). Then we call the free DG algebra K(X) := R⟨E⟩ in gr^{2}(C) over R, whose differential (of bidegree (0, −1)) is given by e(x) ↦ x, the Koszul complex of X over R.

For practical reasons, when we work with a handy sequence X = {x_{i} : i ∈ J}, we define an ordering on the index set J, subject to the condition g(x_{i}) ⩽ g(x_{j}), for i ⩽ j. Remark that for a handy sequence X ⊆ R and each subset Y ⊆ X, the quotient^{7} R/(Y) exists in gr(C). And if R is a DG algebra (R, s) and X is s-handy, then the quotient R/(X) is also a DG algebra.

Definition and Theorem 1.26. Suppose that ℚ ⊆ 𝕂. Let X ⊆ R be a handy sequence and let I be the ideal (X) ⊆ R. Suppose that for each subset Y ⊆ X, we have ∩_{n⩾1}I^{n}R/(Y) = 0. The set X is called a regular sequence, if one of the following equivalent conditions holds:

(i) Let T be a set of free algebra generators that contains for each x ∈ X, an element t(x) with g(t(x)) = g(x). The map R/I[T] ⟶ gr_{I}(R) = R/I ⊕I/I^{2} ⊕ ... in gr(ℚ − 𝔄𝔩𝔤)_{R/I}, sending t(x) to the class of x in I/I^{2} is an isomorphism of (differential) graded R/I-algebras.

7By “quotient”, we mean the cokernel in gr(M) of the embedding (X) ↪ R.

(ii) For each x ∈ X and for each ideal J ⊆ R, which is generated by a subset Y ⊆ X with x ∉ Y, we have: If g(x) is even, then x is no zero divisor in R/J. If g(x) is odd, then the annulator of x in R/J is just the ideal, generated by the class of x.

(iii) The Koszul complex K(X) is a DG resolvent of R/(X) over R. (iv) H^{−1}(K(X)) = 0.

Proof. The implication (iii)⇒(iv) is trivial. Proof of (i)⇒(ii) For an element r ∈ R, let n(r) be the greatest n such that r is contained in I^{n} and let in(r) be the class of r in I^{n(r)}/I^{n(r)+1} ⊆ gr_{I}(R). For elements r, r′ ∈ R, we have that:

in(r) · in(r′) = rr′ + I^{n(r)+n(r′)+1}.  (1.2)

Claim: A subset X ⊆ R satisfies condition (ii), if the subset {in(x) : x ∈ X} ⊆ gr_{I}(R) satisfies condition (ii). Proof of the claim: First step: For x ∈ X, if g(x) is even and in(x) is no zero divisor, then x is no zero divisor. If g(x) is odd and the annulator of in(x) in gr_{I}(R) is the ideal, generated by in(x), then the annulator of x in R is the ideal generated by x. The even case follows immediately by (1.2). In the odd case, let r be in the annulator of x, i.e rx = 0. By (1.2), we get in(x) · in(r) = 0. By the assumption, there is an a_{1} ∈ R, such that in(r) = in(x) · in(a_{1}). This implies that r_{1} := r − xa_{1} is in I^{n(r)+1} and n(r_{1}) ⩾ n(r) + 1. Since x^{2} = 0, we have r_{1}x = rx = 0, and in the same way we find a a_{2} ∈ R with r_{2} := r_{1} − xa_{2} ∈ I^{n(r−1)+1}. Inductively, for each m > n(r), we find a_{1}, ..., a_{k} such that r_{k} := r −x(a_{1} +...+a_{k}) ∈ I^{m}. Thus r belongs to ∩_{k⩾0}((x) + I^{k}), which, by the condition ∩_{n⩾1}I^{n}R/(x) = 0, equals (x).

Second step: For x ∈ X, if either g(x) is even and in(x) is no zero divisor, or g(x) is odd and the annulator of in(x) in gr_{I}(R) is (in(x)), then (x) ∩ I^{n(x)+n} = xI^{n}, for each n ⩾ 0. One inclusion and the even case are easy to see. Suppose that g(x) is odd and that rx is in I^{n(x)+n}. We have to find r′ ∈ I^{n} such that xr = xr′. If r ∈ I^{n}, we are done. Otherwise, we have n(r) < n and in(r) · in(x) = rx + I^{n(r)+n(x)+1} = 0. So there exists a y ∈ R such that in(r) = in(x) · in(y). This means that r_{1} := r − xy is in I^{n(r)+1} and we have r_{1}x = rx. Inductively, we find an r′ := r_{n−n(r)} such that r′ ∈ I^{n} and r′x = rx. As consequence, taking R̄ := R/(x) and Ī:= I/(x), we get an isomorphism

gr_{I}(R)/(in(x)) ≅ grĪ(R̄).

We deduce inductively that for R̄ := R/(x_{1}, ..., x_{s}) and Ī := I/(x_{1}, ..., x_{s}), we get an isomorphism

gr_{I}(R)/(in(x_{1}), ..., in(x_{s})) ≅ grĪ(R̄).

Last step: When g(x) is even, we have to show that x is no zero divisor in R/(x_{1}, ..., x_{s}). We know that in(x) is no zero divisor in gr_{I}(R)/(in(x_{1}), ..., in(x_{s})) ≅ grĪ(R̄). By the first step, the assumption follows. For the odd case, we use the analogue argument. This proves the claim.

When (i) is true, it is clear that {in(x) : x ∈ X}, which corresponds to the set T, satisfies condition (ii) and by the claim, X satisfies condition (ii). Proof of (ii)⇒(iii) We have to show that, for p ⩽ 0, the p-th row of the double complex K(X) is a DG resolution in M over R^{0} of the p-th component of R/(X). For this we can suppose that X is finite with g(x) ⩾ p, for all x ∈ X. Say X = {x_{1}, ..., x_{n}}. We have that K(X) = K(x_{1}) ⊗ ... ⊗ K(x_{n}). Each K(X)^{(p,q)} is obviously a finite R-module, so by Remark 1.12, we only have to show that K(X)^{(p,∗)} is a resolution of (R/(X))^{p} in the category of R-modules. We show it by induction. For n=1, we write x instead of x_{1} and e instead of e_{1}. Set m := g(x). Remark that if m is even, then the total degree m − 1 of e is odd, so in this case we have e^{2} = 0. If m is odd, then the total degree of e is even, so e^{2} ≠ 0. In the first case, K(x) is just the complex

0 …  0  0  …  0  0  …
↓  ↓  ↓  ↓  ↓
0 …  0  R^{0}e  …  R^{m−1}e  R^{m}e  …
↓  ↓  ↓  ↓  ↓
R^{0} … R^{m+1}  R^{m}  …  R^{2m−1}  R^{2m}  …
↓  ↓  ↓  ↓  ↓
R^{0} … R^{m+1} R^{m}/R^{0}x … R^{2m−1}/R^{m−1}x R^{2m}/R^{m}x ⋯

s is injective since x_{1} is no zero divisor in R, hence the rows are exact. In the second case, K(x) is the complex

0 …  0  0  …  0  R^{0}e^{2}  …
↓  ↓  ↓  ↓  ↓
0 …  0  R^{0}e  …  R^{m−1}e  R^{m}e  …
↓  ↓  ↓  ↓  ↓
R^{0} … R^{m+1}  R^{m}  …  R^{2m−1}  R^{2m}  …
↓  ↓  ↓  ↓  ↓
R^{0} … R^{m+1} R^{m}/R^{0}x … R^{2m−1}/R^{m−1}x R^{2m}/R^{m}x ⋯

In R^{i}, for i < m, there is no element that annulates x, so up to the row m − 1, the situation is as above. In the m-th row, the kernel of R^{m}e ⟶ R^{2m} is just R^{0}xe, so it coincides with the image of the map R^{0}e^{2} ⟶ R^{m}e. Remark that here - fore we use that 2 is invertible in R. Inductively we see that all rows are exact. Here - fore we use that all naturals are invertible. Now suppose that the statement is proved for n. Set L := K(x_{1}, ..., x_{n}) = R⟨e_{1}, ..., e_{n}⟩ and K(X) = K(x_{1}, …, x_{n+1}). We write x and e instead of x_{n+1} and e_{n+1}. K(x) is (as object of gr^{2}(M)(R)) a direct sum K_{0}⊕K_{−1}⊕K_{−2}⊕…, where in the case where x is even, we have K_{0} ≅ R, K_{−1} ≅ R[−m, 1] and K_{s} = 0 for s < −1 and in the odd case, we have K_{s} = R[sm, −s] for all s ⩽ 0. Hence, we have K_{q}^{p,q} ≅ R^{p+qm}, for all p, and in the even case for −1 ⩽ q ⩽ 0 and in the odd case for q ⩽ 0. The differential on K(x) is given by the maps d_{q} : K_{q}^{∗,q} ⟶ K_{q+1}^{∗,q+1}, where d_{q}^{p} : K_{q}^{p,q} ⟶ K_{q+1}^{p,q+1} is just the multiplication by x. We have

K(X)^{p,q} = (K(x) ⊗_{R} L)^{p,q}
= (K_{0} ⊗_{R} L)^{p,q} ⊕ (K_{−1} ⊗_{R} L)^{p,q} ⊕ …
≅ L^{p,q} ⊕ L^{p−m,q+1} ⊕ …,

where in the even case the sum has only two factors. Hence in the even case, for p ⩽ 0, the complex K(X)^{p,∗} is the total complex of the double complex

[figure: L^{p,−2}; δ; L^{p−m,−2}; δ; x; 0; L^{p,−1}; δ; L^{p−m,−1}; δ; x; 0; L^{p,0}; L^{p−m,0}; x; 0]

In the odd case, K(X)^{p,∗} is the total complex of the double complex

[figure: L^{p,−2}; δ; L^{p−m,−2}; δ; x; L^{p−2m,−2}; δ; x; L^{p,−1}; δ; L^{p−m,−1}; δ; x; L^{p−2m,−1}; δ; x; L^{p,0}; L^{p−m,0}; x; L^{p−2m,0}; x]

The first double complex is a DDG resolution in gr^{2}(M)(R^{0}) of the DG module

(R/(x_{1}, …, x_{n}))^{p} ← (R/(x_{1}, …, x_{n}))^{p} ← 0 ← …,

where the left arrow stands for multiplication by x. But this DG module is a resolution of (R/(x_{1}, …, x_{n}, x))^{p} over R^{0}, since g(x) is even. So K(X)^{p,∗} is a resolution of (R/(x_{1}, …, x_{n}, x))^{p}. The second double complex is a DDG resolution in gr^{2}(M)(R^{0}) of the DG module

(R/(x_{1}, …, x_{n}))^{p} ← (R/(x_{1}, …, x_{n}))^{p} ← (R/(x_{1}, …, x_{n}))^{p} ← …,

where the arrows stand for multiplication by x. But this DG module is a resolution of (R/(x_{1}, …, x_{n}, x))^{p} over R^{0}, since g(x) is odd. So K(X)^{p,∗} is a resolution of (R/(x_{1}, …, x_{n}, x))^{p}. For both cases, the induction step is done. Proof of (iv)⇒(i) Without restriction, we can suppose that C is the category of commutative ℚ-algebras. For each j ⩾ 0, we have to show that the j-th homogeneous component (R/I[T])_{j} in the T -grading of R/I[T] maps isomorphically to I^{j}/I^{j+1}. We will already make use of the implication (ii)⇒(iii). Set S := ℚ[T]. We consider R as S-algebra via the map t(x) ↦ x. Obviously T ⊆ S satisfies condition (ii), so by (iii), the Koszul complex K_{S}(T) is a DG resolution of ℚ over S. We consider the exact sequence

0 ⟶ (T)^{j}/(T)^{j+1} ⟶ S/(T)^{j+1} ⟶ S/(T)^{j} ⟶ 0

of graded S-modules. (T)^{j}/(T)^{j+1} is a free graded g-finite ℚ-vector space, which is a S-module via the canonical map S ⟶ ℚ. We write ∐_{i∈J} ℚe_{i} for it. Now ∐_{i∈J} K_{S}(T)e_{i} is a free resolution of ∐_{i∈J} ℚe_{i} over S. So we get

Tor_{1}^{S}((T)^{j}/(T)^{j+1}, R) = H^{−1}(∐_{i∈J}(K_{S}(T)e_{i} ⊗_{S} R)) = ∐_{i∈J} H^{−1}(K(X)e_{j}) = 0.

By the property of left derived functors, there is an exact sequence

0 ⟶ Tor_{1}^{S}(S/(T)^{j+1}, R) ⟶ Tor_{1}^{S}(S/(T)^{j}, R) ⟶
(T)^{j}/(T)^{j+1} ⊗_{S} R ⟶ S/(T)^{j+1} ⊗_{S} R ⟶ S/(T)^{j} ⊗_{S} R ⟶ 0

By induction on j and the exactness of the first line, we see that Tor_{1}^{S}(S/(T)^{j}, R) = 0 for any j ⩾ 0. The second line gives rise to a short exact sequence

0 ⟶ (R/I[T])_{j} ⟶ R/I^{j} ⟶ R/I^{j+1} ⟶ 0,

which implies the desired isomorphism.

Remark 1.27. The assumption ℚ ⊆ 𝕂 is used only to prove the implications (ii)⇒(iii) and (iv)⇒(i). The assumption that for each subset Y ⊆ X we have ∩_{n⩾1}I^{n}R/(Y) = 0 is used only to prove (i)⇒(ii). So if you want to get rid of it, use condition (ii) for the definition of regular sequences. It can be stated in a slightly modified manner, which depends on the order of the elements of X, then.

Definition 1.28. Let R be a DG algebra in gr(C)^{N}. Let (α_{i}, u_{i}, g_{i})_{i∈J} be a family in N × U′ and X = {x_{i} : i ∈ J} a family of elements with x_{i} ∈ G_{u_{i}}(R_{α_{i}}^{g_{i}}) such that, for β, β′ ⊆ α, the sets {ρ_{βα}(x_{i}) : α_{i} = β} and {ρ_{β′α}(x_{i}) : α_{i} = β′} are disjoint. Suppose that

X_{α} := ∪_{β⊆α}{ρ_{βα}(x_{i}) : α_{i} = β}

is a regular (resp. handy) (s_{α}-)sequence in R_{α}, for each α. Then X is called a regular (s-)sequence (resp. handy (s-)sequence) in R.

Corollary 1.29. If R = (R, s) is a DG algebra in gr(C)^{N} and X a handy s-sequence in R, then K(X) is a DG algebra in gr^{2}(C)^{N} and if X is regular, then K(X) is a DG resolution of R/(X) over R.

Remarks: When R carries the structure of a DG algebra (R, s), one would like the Koszul complex to carry the structure of a DDG module. In general, this is not the case. If X is an s-handy sequence then, since I = (X) is s-stable, then the algebra gr_{I}(R) has the structure of a DG algebra in gr(𝔄𝔩𝔤), such that each I^{n}/I^{n+1} is a DG submodule of gr_{I}(R). If, for example, R is already a free DG algebra in gr(ℚ − 𝔄𝔩𝔤) with a set X of free algebra generators, i.e. R = R/I[X], then the differential of gr_{I}(R) = R differs in general from the differential s. In this way we get a modified differential s̃ on R. In a similar way we get a modified differential, when R is a free DG algebra in gr(C), for any good pair of categories (C, M). This will play a role in Section 3. In geometric language, going over from s to s̃ is a deformation to the normal cone.

1.8. The universal module of differentials

Fix an admissible pair of categories (C, M). Let k ⟶ A be a morphism of DG algebras in gr(C). As in paragraph I.6.12 of [1], we define universal module Ω_{A/k} of k-differentials as the first homology of the complex C_{•}^{cycl}(A), i.e. the cokernel in gr(M)(A) of the map b_{2} : A ⊗_{k} A ⊗_{k} A ⟶ A ⊗_{k} A, sending a ⊗ b ⊗ c to ab ⊗ c − a ⊗ bc + (−1)^{bc}ac ⊗ b, for homogeneous elements a, b, c ∈ A. Ω_{A/k} is a DG module over A and there is an A-derivation d : A ⟶ Ω_{A/k} (i.e. a homomorphism of DG k-modules, which is a derivation), sending elements a ∈ A to the class of a⊗1. Ω_{A/k} is universal in the sense that, for each A-module M in gr(M), the natural map Hom_{gr(M)(A)}(Ω_{A/k}, M) ⟶ Der_{k}(A, M) is bijective. Set R := A⊗_{k}^{C} A and denote the kernel of the multiplication map μ : R ⟶ A in the category gr(M)(R) by I. (Attention: In general A is the cokernel of the inclusion I ↪ R only in the category R-𝔐𝔬𝔡.) A natural question is if Ω_{A/k} is isomorphic to the “quotient” I/I^{2}. But we already need several assumptions for the existence of I/I^{2} in the category gr(M)(A). An answer which is sufficient for our purpose is given by the following remark. The prove is left to the reader as exercise.

Remark 1.30. Suppose that (C, M) is a good pair of categories satisfying Axiom (S1) and that the marking G on M is canonical. Suppose that A ∈ gr′(C) (i.e. all A^{i} are finite A^{0}-modules) and that I is a g-finite R-module. Then we have A = R/I := Cokern(I ↪ R), I/I^{2} is in a natural way a module in gr(M)(A). Furthermore, the map A ⟶ I/I^{2}, sending a ∈ A to the class of a ⊗ 1 − 1 ⊗ a is a kderivation and the quotient I/I^{2} is a universal module of derivations. In particular I/I^{2} ≅ Ω_{A/k}.

Denote the differential of R = A ⊗_{k} A by s. The next remark is a consequence of Remark 1.30 and Definition and Theorem 1.26:

Remark 1.31. Take the assumptions of Remark 1.30. Suppose that the ideal I ⊆ R is generated by an s-regular sequence X ⊆ R. Then Ω_{A/k} is a free DG A-module in gr(M), generated by a set E = {e(x)| x ∈ X}, containing one free module generator for each element x ∈ X.

The definitions and statements of this subsection carry over directly to N -objects in gr(C) and gr(M).

2. Hochschild complex and Hochschild cohomology

In the (algebraic) literature, for an algebra homomorphism k ⟶ a, the notions Hochschild complex and cyclic bar complex of a over k are synonyms. However, as observed by Buchweitz and Flenner, there are reasons to define the Hochschild complex in the case where a and k are analytic algebras in a different manner. In this case, if a is flat over k, then the bar complex C^{bar}(a) is still a complex of flat a-modules but even if k is just the field ℂ, C^{bar}(a) is not a complex of projective a-modules. Thus, for the definition of Hochschild homology, the cyclic bar complex would do, but for the definition of Hochschild cohomology as cohomology of the a-dual of the cyclic bar complex, it is not a good choice. Thus, for a morphism a ⟶ k of algebras in good pairs of categories, we will give a different definition of the Hochschild complex ℍ(a/k) of a over k. We will see that in the flat case it coincides up to a quasi-isomorphism with the cyclic bar complex.

The definitions of this subsection were inspired by the article [6]. For simplicity, we restrict ourselves to the Noetherian context. Fix a simplicial complex N and a good pair of categories (C, M) with marking (F, G), where G is the canonical marking of M. Suppose that the Axioms (N) and (F2) are satisfied. Using a Čech construction (for more details, see paragraph I.10.1 of [1]), we get a functor Č^{•} : gr(M)^{N} ⟶ gr(𝕂 − 𝔐𝔬𝔡), sending a DG module M ∈ gr(M)^{N} to the total complex tot∏(C^{•}(M^{•})) of the double complex

[figure: C^{0}(M^{−1}); C^{1}(M^{−1}); C^{0}(M^{0}); C^{1}(M^{0})]

where C^{p}(M^{q}) = ∐_{|α|=p} M^{q} and the differentials as usual. This functor sends quasiisomorphisms to quasi-isomorphisms. We will write Ȟ^{n}(M) for H^{n}(Č^{•}(M)).

Let k ⟶ a be a finite morphism of N -objects in C, i.e. a is a quotient of a free k-algebra b in C^{N} such that, for each α ∈ N, the algebra b_{α} is a a free finite k_{α}- algebra. (More generally, we may assume that k ⟶ a is a morphism in gr(C)^{N}, as long as there exists a g-finite resolvent of a over k.) By Proposition 8.8 of [1], there exists a g-finite resolvent of a over k. Fix such a resolvent A. Set R := A ⊗_{k}^{gr(C)} A and consider A as algebra over R by the multiplication map μ : R ⟶ A. Let S be a free g-finite resolvent^{8} of A over R.

Definition 2.1. The simplicial Hochschild complex ℍ_{∗}(a/k) of a over k is the object represented by the complex S ⊗_{R} a in the homotopy category K^{−}(M^{N} (a)). The Hochschild complex ℍ(a/k) is the object represented by the complex Č^{•}(ℍ_{∗}(a/k)) in the derived category D(𝕂 − Mod).

Proposition 2.2. ℍ_{∗}(a/k) is a well defined object in K^{−}(M^{N} (a)) and consequently, ℍ(a/k) is a well-defined object in D(𝕂 − Mod).

8Again by [1], such a resolvent exists. We can even construct it in such a way that S^{0} = R^{0}.

Proof. For i = 1, 2, let A_{i} be a g-finite resolvent of a over k, R_{i} := A_{i} ⊗_{k} A_{i} and let S_{i} be a g-finite resolvent of A_{i} over R_{i}. We have to find a homotopy equivalence S_{1}⊗_{R_{1}}a ≃ S_{2}⊗_{R_{2}}a over a. By Proposition 1.14, there is a homomorphism A_{1} ⟶ A_{2} in gr(C)^{N} which is a homotopy equivalence over k. Hence,

R_{1} ≈ A_{1} ⊗_{k} a ≃ A_{2} ⊗_{k} a ≈ R_{2}.

By Proposition 1.14, the quasi-isomorphism R_{1} ≈ R_{2} is even a homotopy-equivalence over k. Thus we get a quasi-isomorphism

S_{1} ≅ S_{1} ⊗_{R_{1}} R_{1} ⟶ S_{1} ⊗_{R_{1}} R_{2}

over R_{1}. S_{2} and S′_{1} := S_{1} ⊗_{R_{1}} R_{2} are both resolvents of a over R_{2}. Hence, there is a homomorphism S′_{1} ⟶ S_{2} in gr(C)^{N}, which is a homotopy equivalence over R_{2}. We can tensorize both sides over R_{2} with a and still get a homotopy equivalence S_{1} ⊗_{R_{1}} a ⟶ S_{2} ⊗_{R_{2}} a.

Recall that the notion C^{bar}(a) = C^{bar}(a)^{•} stands for the complex C_{−•}^{bar}(a).

Proposition 2.3. Suppose that a is flat over k. There is an isomorphism ℍ_{∗}(a/k) ⟶ C^{cycl}(a) is the derived category D(a − 𝔐𝔬𝔡). More precisely, for each representation S ⊗_{R} a of ℍ_{∗}(a), there exists a morphism S ⊗_{R} a ⟶ C^{cycl}(a) in gr(C)^{N}, which is a quasi-isomorphism over a.

Proof. Set r := a ⊗_{k} a. Since a is flat over k, R is a resolvent of r and C^{bar}(a) a flat resolution of a over r. Another one is s := S ⊗_{R} r. By Theorem I.8.4 of [1], there is a morphism s ⟶ C^{bar}(a) in gr(C)^{N} over r. By flatness, we get a quasi-isomorphism

C^{cycl}(a) ≅ C^{bar}(a) ⊗_{r} a ≅ s ⊗_{r} a ≅ S ⊗_{R} a.

Definition 2.4. We define the n-th Hochschild homology of a over k as Ȟ^{−n}(ℍ_{∗}(a/k)).

Definition 2.5. Let M be an object of M^{N} over a. We define the Hochschild cochain complex of a over k with values in M to be the complex

Hom_{a}^{N} (ℍ_{∗}(a/k), M),

with the differential induced by the differential of ℍ_{∗}(a/k). We define the Hochschild cohomology HH(a/k, M) of a over k with values in M to be the cohomology of the Hochschild cochain complex.

Proposition 2.6. The Hochschild cochain complex is well defined up to homotopy equivalence.

Proof. This is a consequence of Lemma I.3.7 of [1] and Proposition 2.2.

3. Decomposition of Hochschild (co) homology

For the whole section, let (C, M) be a good pair of categories with marking (F, G), such that G is canonical and suppose that Axioms (N) and (F2) are satisfied. Fix an algebra k in C^{N} and a free g-finite DG algebra A over k in gr(C)^{N}. Set R := A ⊗_{k} A and denote the differential of R by s. Let I be the kernel of the multiplication R ⟶ A. The Hochschild complex of A over k is represented by S ⊗_{R} A, for each g-finite resolvent S of A over R. The HKR theorem for free DG algebras is due to the following key observation: If (C, M) is the pair (C^{(0)}, M^{(0)}), then R is as graded algebra isomorphic to R̃ := gr_{I} R, but the isomorphism does not respect the differentials. This makes the free DG case more difficult then the classical case. In contrast, the natural map R̃ ⟶ A does respect the differentials and one may ask, if for a g-finite resolvent S̃ of A over R̃, the Hochschild complex is represented by S̃ ⊗R̃ A in the homotopy category. In fact, this is true by the algebraic version of Theorem 3.7. The HKR-type theorem is a consequence of this theorem, since by a Koszul construction, we will find an appropriate S̃, such that S̃ ⊗R̃ A = ∧_{A}Ω_{A/k}. In the analytic case, R and gr_{I} R can’t be identified as graded algebras, hence, for the general theory, we have to modify the construction of R̃, slightly.

3.1. Balanced and convex markings

Let k be an algebra in gr(C) and A := k⟨T ⟩ a free algebra over k in gr(C) with a g-finite set T of free generators t ∈ F_{τ(t)}(R^{g(t)}). Then A ⊗_{k} A is a free algebra over k with two free algebra generators t_{1} = t ⊗ 1 and t_{2} = 1 ⊗ t, for each t ∈ T. For t ∈ T, set t^{+} := \frac{1}{2}(t_{1} + t_{2}) and t^{−} := \frac{1}{2}(t_{1} − t_{2}). Let T^{+} be the set of all t^{+} and T^{−} be the set of all t^{−}. We say that the marking F on C is balanced, if for each τ ∈ T and each A in C and each t ∈ F_{τ} (A), we have −t ∈ F_{τ} (A). We say that the marking F is convex if for each τ ∈ T, each A in C, each t_{1}, t_{2} ∈ F_{τ} (A) and each a, b ∈ 𝕂 with a + b = 1 we have at_{1} + bt_{2} ∈ F_{τ} (A).

Remark 3.1. If the marking gr_{G}(F) on gr(C) is balanced and convex, we have A ⊗_{k} A ≅ k⟨T^{+} ∪ T^{−}⟩.

Example 3.2. (1) The trivial marking on C is balanced and convex. (2) If C is the category C^{(1)} of (local) analytic algebras and M the category M^{(1)} of DFN-modules over C^{(1)}, then the marking F on C (see Example 1.5) is balanced and convex.

Proof. The first example is trivial. For the second example, we show that if a free generator t is in F_{τ} (R), then t^{+} and t^{−} are in F_{τ} (R ⊗_{B} R). Here, τ stands for a positive real number and F_{τ} (R) is the set of all r ∈ R such that, for each character ξ ∈ X (R), we have |ξ(r)| ⩽ τ. Now, t_{1} = t ⊗ 1 and t_{2} = 1 ⊗ t belong to F_{τ} (R ⊗_{B} R), so for each character ξ ∈ X (R ⊗_{B} R), we have |ξ(t_{1})| ⩽ τ and |ξ(t_{2})| ⩽ τ. Hence |ξ(t^{+})| = |\frac{1}{2}(ξ(t_{1}) + ξ(t_{2}))| ⩽ τ and |ξ(t^{−})| = |\frac{1}{2}(ξ(t_{1}) − ξ(t_{2}))| ⩽ τ. The case of local analytic algebras is clear, since maximal ideals are additively closed.

3.2. Deformation to the normal cone

For the rest of this section, suppose that the marking F on C is balanced and convex. Take A and R as in the beginning of this section. We have R ≅ k⟨T^{+} ∪T^{−}⟩. Since T^{−} is s-stable and g-finite, it is a regular s-sequence. Each element r of R has a unique decomposition r = ř + ṙ + r̂ with ř ∈ Ř := k⟨T^{+}⟩, ṙ ∈ Ṙ := ∑_{t∈T^{−}} tk⟨T^{+}⟩ and r̂ ∈ R̂ := ∑_{t,t′∈T^{−}} tt′k⟨T^{+}⟩. We define a R^{0}-derivation s̃ on R, setting s̃(t) := (s(t))^{∨}, for t ∈ T^{+}, and s̃(t) := (s(t))^{·} for t ∈ T^{−}. The philosophy of this modification is that s̃ preserves the T^{−}-degree of homogeneous elements in R. More precisely we have s̃(Ř) ⊆ Ř, s̃(Ṙ) ⊆ Ṙ and s̃(R̂) ⊆ R̂, in contrast to s(Ṙ) ⊆ Ṙ ⨿ R̂ and s(R̂) ⊆ R̂.

Proposition 3.3. s̃ is a differential, i.e. s̃^{2} = 0.

Proof. First remark that for a ∈ k⟨T^{+}⟩, we have that (s(a))^{∨} = s̃(a). To prove this we can suppose that a is of the form a_{0}t_{1} · . . . · t_{n} with a_{0} ∈ k⟨T^{+,0}⟩ and t_{i} ∈ T^{+,<0}. The proof of this case is immediate.

Now suppose that t is in T^{+}. We have that s(t) = s̃(t) + rest, where rest is in Ṙ ⨿ R̂. Thus s^{2}(t) = s̃^{2}(t) + rest′ where rest′ is in Ṙ ⨿ R̂. Since s̃^{2}(t) is in Ř and s^{2}(t) = 0, we get s̃^{2}(t) = 0. Similarly, we see that s̃^{2}(t) = 0 for t ∈ T^{−}, which proves the Proposition.

According to Section 1.7, we write X for the regular s̃-sequence T^{−}. We will see that for R̃ = (R, s̃), the Koszul complex (K(X), v) has the structure (K(X), h, v) of a DDG algebra, so its total complex is a resolvent of A = R̃/(X) over R̃: We denote by E the set of free algebra generators, containing for each x_{i} ∈ X an element e_{i} of bidegree (g(x_{i}), −1). Here s̃(x_{i}) is a sum of the form ∑ a_{j}x_{j}, where no a_{j} belongs to the ideal (X). In fact all a_{j} belong to B = k⟨T^{+}⟩. There is exactly one choice for the element h_{i}, which shall be the image of e_{i} by the horizontal differential h of R⟨E⟩, namely h_{i} = ∑ a_{j}e_{j}. We have that 0 = s̃^{2}(x_{i}) = s̃(∑ a_{j}x_{j}) = ∑_{j,k} a_{ij}a_{jk}x_{k} and the coefficients a_{ij}a_{jk} belong to k⟨T^{+}⟩. So, we get h^{2}(e_{i}) = h(∑ a_{ij}e_{j}) = ∑_{j,k} a_{ij}a_{jk}e_{k} = 0. I.e. the hypothesis of Remark 1.19 is satisfied. Thus the Koszul complex (K(X), v), equipped with the horizontal differential h, is a DDG resolvent in gr^{2}(C) of A = R/(X) over R̃ = (R, s̃). And the total complex S̃ of K(X) is a resolvent of A over R̃.

The reason for this construction is that there is a nice description of the tensor product

S̃ ⊗R̃ A = tot(K(X)) ⊗R̃ A = tot(K(X) ⊗R̃ A).

Namely:

Remark 3.4. The double complex K(X) ⊗R̃ A is isomorphic to ∧ ∧_{A}^{·} Ω_{A/k}.

Proof. This follows immediately by Proposition 1.30 and Definition 1.22.

The going over from s to s̃ is natural, i.e. when (R, s) = (R_{α}, s_{α})_{α∈N} is a DG algebra in gr(C)^{N}, then R̃ := (R_{α}, s̃_{α})_{α∈N} is again a DG algebra in gr(C)^{N} and Remark 3.4 keeps true in the simplicial case.

3.3. Construction of the resolvent S

To get a resolvent S of A over R, which is good for our purpose, we have to work harder. The strategy is to construct again a DDG resolvent K in gr^{2}(C)^{N}, which is free over the double graded object K(X) and such that the projection K ⟶ K(X) is a morphism of DG algebras in gr^{2}(C)^{N} (of course, it will not be a morphism of DDG algebras), and such that the induced map K⊗_{R}A ⟶ K(X)⊗R̃A is a morphism of DDG algebras in gr^{2}(C)^{N}, inducing an isomorphism on vertical homology. When we have managed to realize this, with S := tot(K) we get a resolvent of A over R and a quasi-isomorphism

S ⊗_{R} A ⟶ S̃ ⊗R̃ A  (3.1)

over A.

First, we explain heuristically the construction of K: We take the Koszul-complex K(X) with its vertical differential v. The problem is to define a horizontal differential h on it, since here in general there are no good candidates for the values h(e) of h on the free generators e ∈ E. When we have s(x) = ∑ a_{xy}y, to get commutative diagrams, h(e(x)) must be up to a vertical cocycle ∑ a_{xy}e(y). In general, h(∑ a_{xy}e(y)) is not zero. Inductively, for e we add free algebra- generators f of bidegree (g(x) + 1, −1) with v(f) = 0, in such a way that we can find candidates for h(e) in K(X)^{g(x)+1,−1} + ∑ R^{0}f. When this is done for all e ∈ E, we get a DDG structure on the extension K(X)⟨F ⟩ = R⟨E ∪ F ⟩. This extension is not any more a resolvent. To get a resolvent again, we apply the construction of Remark 1.21. Now we start with the construction of K. First we consider the affine situation. For each x ∈ X, we fix a finite family (a_{xy})_{y∈Y}, for a finite subset Y ⊆ X, such that s(x) = ∑_{y∈Y} a_{xy}y. Set K(X) := R⟨E⟩ as double graded algebra.

Proposition 3.5. There is a g-finite family F = ∪_{p⩽0}F^{p} of free algebra generators with g(f) = (p, −1), for f ∈ F^{p}, and a DDG algebra structure (L, h, v) on L := R⟨E ∪ F ⟩ such that

(1) v(e(x)) = x

(2) v(f) = 0

(3) h(f) is in the ideal (X ∪ F), generated by X and F.

(4) h(e(x)) = ∑_{y} a_{xy}e(y) + γ(x) with γ(x) ∈ (F).

Proof. We construct a decreasing sequence L_{k} = (L_{k}, h_{k}, v_{k}) of free DDG algebras over R and a family {γ(x) : x ∈ X ∧g(x) ⩾ k−1} with γ(x) ∈ L_{k}^{k,−1}, for x ∈ X^{k−1}, such that the following conditions hold:

(a) L_{0} = R⟨E^{0}⟩.

(b) L_{k−1} = L_{k}⟨E^{k−1} ∪ F^{k−1}⟩ is a free DDG algebra over L_{k}, where F^{k−1} is a finite set of algebra generators of bidegree (k − 1, −1) and we have that v_{k−1}(e(x)) = x (as in the Koszul-construction) and v_{k−1}(f) = 0.

(c) h_{k−1} maps the submodule ⨿_{f∈F^{k−1}}R^{0}f of L_{k−1}^{k−1,−1} surjectively onto Kern(h_{k}^{k,−1}) ∩ Kern(v_{k}^{k,−1}) ⊆ L_{k}^{k,−1}.

(d) For all i ⩾ k − 1, the sequence L_{k−1}^{i,−1} ⟶ R^{i} ⟶ A^{i} in M(R^{0}) is exact.

(e) For x ∈ X^{k−1}, we have γ(x) ∈ ⨿_{f∈F^{k}}R^{0}f and h_{k}^{k,−1}(γ(x)) = h_{k}^{k,−1}(∑_{y} a_{xy}e(y))

For L_{0} = R⟨E^{0}⟩, we have already seen that by setting v(e(x)) := x, we get an exact sequence L_{0}^{0,−1} ⟶ R^{0} ⟶ A^{0}. Now suppose that L_{k} and {γ(x) : g(x) ⩾ k} is already constructed. We choose finitely many free algebra generators f of bidegree (k − 1, −1) such that there exists an epimorphism

π : ⨿R^{0}f ⟶ Kern(h_{k}^{k,−1}) ∩ Kern(v_{k}^{k,−1}) ⊆ (X ∪ F).

To explain the inclusion: The vertical differential on the subalgebra K(X) is exact. Thus a homogeneous element of K(X)⟨F ⟩, which is in the kernel of v, is a sum of an element in the image of v and an element in the ideal (F). We set h_{k−1}(f) := π(f) and v_{k−1}(f) := 0. For x ∈ X^{k−1}, to see that there exists a good candidate for γ(x), we only have to show that h_{k}(∑_{y} a_{xy}e(y)) belongs to Kern(h_{k}^{k+1,−1}) ∩ Kern(v_{k}^{k+1,−1}). But we have h_{k}(∑_{y} a_{xy}e(y)) = ∑_{y} s(a_{xy})e(y) + ∑_{y} ±a_{xy}[∑_{z} a_{yz}e(z)+γ(y)] = (∑_{y} s(a_{xy})e(y)+∑_{y,z} ±a_{xy}a_{yz}e(z))+∑_{y} ±a_{xy}γ(y). The first term maps vertically to s^{2}(x) which is zero, the second factor maps obviously vertically to zero. Finally, we can set h_{k−1}^{k−1,−1}(e(x)) := ∑ a_{xy}e(y) − γ(x) and v_{k}^{k−1,−1}(e(x)) := x. This gives the desired L_{k−1}. We set L := lim L_{k}.

The free DDG algebra L = R⟨E ∪ F ⟩ has the following properties:

(1) L^{0,∗} = K(X)^{0,∗}, hence this is a resolvent of A^{0} over R^{0}.

(2) L^{0,p} = R^{p}, for all p ⩽ 0.

(3) The sequence L^{p,−1} ⟶ L^{p,0} ⟶ A^{p} ⟶ 0 is exact, for all p ⩽ 0.

(4) The inclusion K(X) ↪ L and the projection L ⟶ K(X) are homomorphisms of DG algebras over (R, 0), so in the category of DG modules in gr(M) there is a decomposition L = K(X) ⨿ L′.

Proposition 3.6. There is a g-finite family G = ∪_{p⩽0,q⩽−2}G^{p,q} of free algebra generators and extensions of h and v on K := L⟨G⟩, such that

(1) The i-th row of K is a R^{0}-module resolution of A^{i}.

(2) v(g) ∈ (F ∪ G)

(3) h(g) ∈ (G)

Proof. We can construct the free DDG resolvent K of A over R with the method of Remark 1.21.

Comparing the values of h on the free generators e with its values by the differential h̃ of the Koszul-complex over R with the modified differential, we see that h̃(e) − h(e) ∈ ⨿R^{0}f + ∑_{x∈X} xK.

Consequence: Consider the projection π : K = R⟨E∪F ∪G⟩ ⟶ R⟨E⟩ = K(X) (a priori only as algebra homomorphism in gr^{2}(C)). By Proposition 3.5 and 3.6, π respects the vertical differential. By the construction of s̃ and Proposition 3.5,

π ⊗ 1 : K ⊗_{R} A ⟶ K(X) ⊗R̃ A

is a homomorphism of DDG algebras in gr^{2}(C) over A.

Now we can prove the (affine case of the) crucial result of this chapter. It says that to construct the Hochschild complex it is enough to work with a resolvent of A over R̃.

Theorem 3.7. Let S := tot(K) and S̃ := tot(K(X)). There exists a homotopyequivalence

S ⊗_{R} A ⟶ S̃ ⊗R̃ A

over A.

Proof. We have seen that the projection π : K ⟶ K(X) is a homomorphism of DG Algebras in gr^{2}(C) over (R, 0). Since both double complexes are free resolutions, for each p, the restriction K^{p,∗} ⟶ K(X)^{p,∗} is a homotopy equivalence over R^{0}. For each p, the restriction (K ⊗_{R} A)^{p,∗} ⟶ (K(X) ⊗R̃ A)^{p,∗} of π ⊗ 1 is a homotopy equivalence. Hence, π ⊗ 1 induces a quasi-isomorphism

S ⊗_{R} A = tot(K) ⊗_{R} A = tot(K ⊗_{R} A) ⟶ tot(K(X) ⊗R̃ A)
= tot(K(X)) ⊗R̃ A = S̃ ⊗R̃ A.

But a quasi-isomorphism of free algebras is already a homotopy-equivalence.

We have to explain that the construction of K also works in the simplicial case. Suppose that k is an object of gr(C)^{N} and A is a free algebra over k in gr(C)^{N}. Take A = k⟨T ⟩, where each t ∈ T is associated to a pair (α_{t}, τ_{t}, g_{t}) ∈ N × T × ℤ_{⩽0}. Now T^{+} and X := T^{−} are sets of free generators in the simplicial sense. Write X = {x_{i} : i ∈ I} and (α_{i}, τ_{i}, g_{i}) for the triple associated to x_{i}. For α ∈ N set

X_{α} := {ρ_{αα_{i}}(x_{i}) : α_{i} ⊆ α}.

In the sequel, we will simply write x_{i} for the element ρ_{αα_{i}}(x_{i}) of R_{α}. Let E = {e_{i} : i ∈ I} be a family of free algebra generators containing for each x_{i} ∈ X an e_{i} of degree (g(x), −1), belonging to the simplex α_{i}. We form the free algebra K(X) = R⟨E⟩ in gr^{2}(C)^{N}. Set E_{α} := {ρ_{αα_{i}}(e_{i}) : α_{i} ⊆ α}. We have K(X)_{α} = R_{α}⟨E_{α}⟩. For each x = x_{i} in X, we fix a family a_{xy}; y ∈ Y with Y ⊆ X_{α_{i}} and a_{xy} ∈ R_{α_{i}}, such that s_{α}(x) = ∑_{y} a_{xy}y.

Proposition 3.8. There exists a g-finite family F = {f_{j} : j ∈ J} of free algebra generators, where f_{j} belongs to α_{j} and is of bidegree (g_{j}, −1) and a DDG algebra structure (L, h, v) on L = R⟨E ∪ F ⟩ over R, such that for all α and all x ∈ X_{α} and all f ∈ F_{α} := {f_{j} : α_{j} ⊆ α}, the following conditions hold:

(1) v_{α}(e(x)) = x

(2) v_{α}(f) = 0
(3) h_{α}(f) ∈ (X_{α} ∪ F_{α})
(4) h_{α}(e(x)) = ∑_{y} a_{xy}e(y) + γ(x) for a γ(x) in (F_{α}).

Proof. We reduce the Proposition by induction on the following statement: Suppose that there exists a family F^{(n)} of free simplicial algebra generators and a DDG algebra structure on (R⟨E⟩_{α})_{|α|⩽n}⟨F^{(n)}⟩, such that the conditions (i)-(iv) hold for all α ∈ N^{(n)}, then there exists a family F^{(n+1)} and a DDG algebra structure on (R⟨E⟩_{α})_{|α|⩽n+1}⟨F^{(n+1)}⟩ such that the conditions (i)-(iv) hold for all α ∈ N^{(n+1)}. The case n = 0 as well as the induction step can be done easily as in the affine case.

Proposition 3.9. There is a g-finite family G = {g_{j} : j ∈ J} of free algebra generators, where g_{j} belongs to α_{j} and is of bidegree g_{j} and a DDG algebra structure on K = L⟨G⟩ over L such that for all α ∈ N and all x ∈ X_{α} and all g ∈ G_{α} the following conditions hold:

(1) K is a DDG resolvent of A over R.

(2) v_{α}(g) ∈ (G_{α} ∪ F_{α})

(3) h_{α}(g) ∈ (G_{α}).

Proof. With the same method as above, we reduce the statement to the affine case.

Using Propositions 3.8 and 3.9, we can generalize Theorem 3.7 to the simplicial context.

3.4. A HKR-type theorem Theorem 3.10. For each free DG algebra A in gr′(C)^{N}, we have a homotopy equivalence

∧Ω_{A/k} ⟶ ℍ(A/k)

over A.

Proof. Glue together Remark 3.4 and Theorem 3.7 and use Proposition 1.14.

In the textbook [15], one can find a plan for the proof of Theorem 3.10. This plan doesn’t work in the analytic situation, since the fact that the cyclic bar complex is a complex of free modules is involved, which is false for analytic algebras. Recall that for an algebra a over k in C^{N} with resolvent A in gr′(C)^{N}, the cotangent complex 𝕃(a/k) of a over k is defined as the class of Ω_{A} ⊗_{A} a in the homotopy category K(M^{N} (a)). By Theorem III.2.4 of [1], the homotopy class does not depend on the resolvent A. Choosing A in Theorem 3.10 as resolvent of a, we immediately get the announced generalization of the classical Hochschild-Kostant-Rosenberg theorem:

Theorem 3.11. Consider a homomorphism k ⟶ a in C^{N}. Suppose that ℚ ⊆ k. There exists a homotopy equivalence

∧𝕃_{a/k} ⟶ ℍ(a/k)

in gr(C)^{N} over a.

An algebraic version of this theorem can be found in Quillen’s article [18]. Quillen’s proof does not work in our situation, since the category M is not, in general, Abelian.

Corollary 3.12. If a is already free over k (in this case there is no need to assume that ℚ ⊆ a) and A = a, then Ω_{a/k} is an object of C_{a}^{N} and we get isomorphisms

∧_{a}^{n}Ω_{a/k} ≅ H_{n}(ℍ(a/k))

Dually, with T_{A/k} := Hom_{A}(Ω_{A/k}, A) we get

H^{n}(Hom_{a}(ℍ(a/k), a)) ≅ ∧^{n}T_{a/k}.

3.5. Decomposition of Hochschild (co-) homology Let M be a module in M^{N} (a).

Theorem 3.13. We have the following decomposition of Hochschild (co)homology:

HH_{n}(a/k) ≅ ∏_{i−j=n} Ȟ^{i}(∧^{j}𝕃(a/k)),  (3.2)
HH^{n}(a/k, M) ≅ ∐_{i+j=n} H^{i}(Hom_{a}(∧_{a}^{j}𝕃(a/k), M)).  (3.3)

Proof. The first isomorphism is a direct consequence of Theorem 3.11. We show the second one:

HH^{n}(a/k, M) = H^{n}(Hom_{a}(ℍ(a/k), M)) ≅ H^{n}(Hom_{a}(∧𝕃(a/k), M))

≅ H^{n}(Hom_{A}(tot(∧ ∧^{·} Ω_{A/k}), M)) ≅ H^{n}(Hom_{A}(∐_{j⩾0} ∧_{A}^{j}Ω_{A/k}[j], M))

≅ H^{n}(∏_{j} Hom_{A}(∧_{A}^{j}Ω_{A/k}[j], M)) ≅ ∏_{j⩾0} H^{n−j}(Hom_{A}(∧^{j}Ω_{A}, M))

≅ ∐_{i+j=n} H^{i}(Hom_{A}(∧_{A}^{j}Ω_{A/k}, M)).

The first equality holds by definition. The second one follows by Theorem 3.11. The other equalities are elementary.

Remark that tangent cohomology is a direct factor on the right hand-side of isomorphism (3.2).

4. Application to complex spaces and Noetherian schemes

In this section, all schemes and complex spaces are supposed to be paracompact and separated. For more details on many of the constructions, we refer to [5] and [6]. First, we will sketch the correlation between the the theory of coherent sheaves on schemes or complex spaces and the theory of N -objects in good pairs of categories. The main tools that we need here are:

(1) Instead of considering a space X, we consider the simplicial scheme, associated to an affine covering of X. By an affine subspace, we mean an open affine subscheme in the case of schemes and a Stein compact^{9} in the case of complex spaces. There are functors that make simplicial modules out of sheaves of modules and functors doing the inverse.

(2) Let X be a complex space or a Noetherian scheme. For affine subsets U ⊆ X, we use the equivalence of categories of coherent O_{U}-modules and finite modules over the ring Γ(U, O_{X}). (Remember that Γ(U, O_{X}) is Noetherian, when X is a complex space.) This equivalence is given by Cartan’s theorem A in the analytic case and by Exercise II.2.4 of [12] in the algebraic case.

More generally, let X be a ringed space and (X_{i})_{i∈I} a locally finite covering of X. The nerf N of this covering is the set of all subsets α ⊆ I, such that ∩_{i∈α}X_{i} ≠ ∅. N is a simplicial scheme in the sense of Section 1.3. Further, there is a contravariant functor from N in the category of ringed spaces, mapping an object α to the object X_{α} := ∩_{i∈α}X_{i}. For α ⊆ β, denote the inclusion X_{β} ⟶ X_{α} by p_{αβ}. Such a functor is called simplicial scheme of ringed spaces. Let X_{∗} = (X_{α})_{α∈N} be a simplicial scheme of ringed spaces. Following [7], we define the category of O_{X_{∗}}-modules as follows: Its objects are families F_{∗} = (F_{α})_{α∈N} with F_{α} in Mod(X_{α}), together with compatible maps p_{αβ}^{∗}F_{α} ⟶ F_{β}. For O_{X_{∗}}-modules F, G, we set Hom_{X_{∗}}(F, G) to be the set of compatible families f_{α} : F_{α} ⟶ G_{α}. We denote this category by Mod(X_{∗}). The full subcategory of those F_{∗}, where each F_{α} is coherent is denoted by Coh(X_{∗}).

Definition 4.1. Let A and B be simplicial schemes over the index sets A_{0} and B_{0}. Suppose that X_{∗} = (X_{α})_{α∈A} and Y_{∗} = (Y_{β})_{β∈B} are simplicial schemes of ringed spaces. A morphism f : X_{∗} ⟶ Y_{∗} consists of a mapping τ : A_{0} ⟶ B_{0}, such that for α ∈ A, we get τ(α) ∈ B, and a family of compatible maps f_{α} : X_{α} ⟶ Y_{τ(α)}.

As in [7], we can form the adjoint functors

f^{∗} :Mod(Y_{∗}) ⟶ Mod(X_{∗}) and
f_{∗} :Mod(X_{∗}) ⟶ Mod(Y_{∗}).

For F in Mod(Y_{∗}) and α ∈ A, we have (f^{∗}F)_{α} := f_{α}^{∗}F_{τ(α)}. The construction of f_{∗} is more complicated. For the general case, we refer to [7]. We need only the following particular case:

Remark 4.2. Let F_{∗} be an object of Mod(X_{∗}). Here, for elements β ∈ B of the form β = τ(α), we have

(f_{∗}F)_{β} = f_{α∗}F_{α}.

Hence, if the map τ : A_{0} ⟶ B_{0} is surjective, then the construction of f_{∗} becomes very simple.

Examples 4.3. (1) If X is a Noetherian scheme or a complex space and (X_{i})_{i∈I} is a covering by affine subspaces, then by the separated condition, all X_{α} are affine. Now let (C, M) be the good pair (C^{(0)}, M^{(0)}) or (C^{(1)}, M^{(1)}) (see Example 1.1). Then, a_{∗} := (Γ(X_{α}, O_{X_{α}}))_{α∈N} is an N -object in C and there is

9Remember that a Stein compact is a compact subset of a complex space, having a base of neighborhoods, consisting of Stein spaces. A Stein compact is only a pseudocomplex space.

a 1:1-correspondence between the objects of Coh(X_{∗}) and the N -objects M_{∗} in M over a_{∗}, such that each M_{α} is finite over a_{α}.

(2) If X is a complex space and the covering (X_{i})_{i∈I} is locally finite and chosen in such a way that each X_{i} admits a closed embedding into a polydisc P_{α}, then we get another simplicial scheme of Stein compacts: Set P_{α} := ∏_{i∈α} P_{i}. For α ⊆ β, we have the projection P_{β} ⟶ P_{α}. This makes P_{∗} = (P_{α})_{α∈N} a simplicial scheme of Stein compacts and there is a closed embedding X_{∗} ⟶ P_{∗}.

(3) Let X be a scheme of finite type over a Ring 𝕂 and (X_{i})_{i∈I} an open affine covering of X. We can construct a new simplicial scheme: Set a_{α} := Γ(X_{α}, O_{X_{α}}), for α ∈ N. For each α, there is a free, finitely generated algebra 𝕂[T] that maps surjectively onto a_{α}. We get a closed embedding X_{α} ⟶ Spec(𝕂[X]) =: P_{α}. As above, we get a simplicial scheme P_{∗} and a closed embedding X_{∗} ⟶ P_{∗}.

The inclusions j_{α} : X_{α} ⟶ X give rise to a map j : X_{∗} ⟶ X of simplicial schemes of ringed spaces. Next, we will study the adjoint functors j_{∗} and j^{∗}: j^{∗} is just the exact functor, mapping an O_{X}-module F to the O_{X_{∗}}-module (F|_{X_{α}})_{α∈N}. To describe j_{∗}, we consider the Čech -functor: For an O_{X_{∗}}-module F_{∗}, set

Č^{p}(F_{∗}) := ∏_{|α|=p} j_{α∗}F_{α}

and define a differential on Č^{•}(F_{∗}) in the usual sense. Then, j_{∗}F_{∗} is just H^{0}(Č^{•}(F_{∗})). j_{∗}j^{∗} is the identity functor. One can prove the adjointness of j^{∗} and j_{∗} directly by a gluing argument. Since j^{∗} is an exact functor and j_{∗} is right adjoint to j^{∗}, we see that j_{∗} transforms injective objects in Mod(X_{∗}) into injective objects in Mod(X). For each α ∈ N, we define a functor p_{α}_{∗} : 𝔐𝔬𝔡(X_{α}) ⟶ 𝔐𝔬𝔡(X_{∗}) via

(p_{α}_{∗})_{β} := \{ \begin{matrix}p_{βα}_{∗}F_{α} for & β ⊆ α \\ 0 for & all other cases \end{matrix}

By Proposition 2.26 of [5], each O_{X_{∗}}-module admits an injective resolution by modules of the form ∏_{α∈N} p_{α∗}I_{α} with injective O_{X_{α}}-modules I_{α}. We will use the following properties of the functor Č^{•}:

Remark 4.4. (1) For p ⩾ 0, the functor Č^{p} is exact.

(2) If F_{α} is an O_{X_{α}}-module, then Č^{•}(p_{α∗}F_{α}) is a resolution of j_{∗}(p_{α∗}F_{α}).

(3) If F is an O_{X}-module, then Č^{•}(j^{∗}F) is a resolution of F.

We generalize a part of Proposition 2.28 of [5] for the case where X is only assumed to be a ringed space and X_{∗} is the simplicial scheme of ringed spaces associated to an open or closed covering (X_{i})_{i∈I} of X:

Proposition 4.5. The functor j^{∗} : D(X) ⟶ D(X_{∗}) embeds D(X) as a full and exact subcategory into D(X_{∗}) and Č^{•} = Rj_{∗} is an exact right adjoint. In particular, for F, G ∈ D(X) and M_{∗} ∈ D(X_{∗}), there are functorial isomorphisms

Ext_{X}^{k}(F, G) ≅ Ext_{X_{∗}}^{k}(j^{∗}F, j^{∗}G) and
Ext_{X_{∗}}^{k}(j^{∗}F, M_{∗}) ≅ Ext_{X}^{k}(F, Č^{•}(M)).

If all the maps p_{αβ}^{∗}(M_{α}) ⟶ M_{β}, for α ⊆ β in N, are quasi-isomorphisms, then the natural map

j^{∗}Č^{•}(M_{∗}) ⟶ M_{∗}

is a quasi-isomorphism, and in consequence, for all n, there are isomorphisms

Ext_{X_{∗}}^{n}(M_{∗}, j^{∗}F) ≅ Ext_{X}^{n}(Č^{•}(M_{∗}), F).

Proof. For the proof that Č^{•} is the right derived functor of j_{∗}, we use an injective resolution I_{∗} of an O_{X_{∗}}-module F_{∗} of the same form as above. We have

(Rj_{∗})(F_{∗}) = (j_{∗}I_{∗})^{•} = ∏ j_{∗}(p_{α∗}I_{α})^{•} ≈ ∏ Č^{•}(p_{α∗}I_{α}) = Č^{•}(I_{∗}^{•}) ≈ Č^{•}(F_{∗}).

We only prove the first formula for Ext. Here, I_{∗}^{•} denotes an injective resolution of j^{∗}G.

Ext_{X_{∗}}^{n}(j^{∗}F, j^{∗}G) = H^{n}(Hom_{X_{∗}}(j^{∗}F, I_{∗}^{•})) = H^{n}(Hom_{X}(F, j_{∗}I_{∗}^{•})) =
Ext_{X}^{n}(F, j_{∗}I_{∗}^{•}) = Ext_{X}^{n}(F, (Rj_{∗})(j^{∗}G)) =
Ext_{X}^{n}(F, Č^{•}(j^{∗}G)) = Ext_{X}^{n}(F, G).

In the sequel, let X be a complex space or a scheme of finite type over a Noetherian ring. The structure sheaf O_{X} defines an N -Object a = a_{∗} in C. In the algebraic case each O_{X} -module F defines an N -object F = F_{∗} in M over a. In the analytic case each coherent O_{X} -module F defines an N -object F = F_{∗} in M over a. Here, (C, M) stands for (C^{(0)}, M^{(0)}) in the algebraic case and for (C^{(1)}, M^{(1)}) in the analytic case (see Example 1.1). We make the following convention to avoid the distinction between analytic and algebraic tensor products:

Convention: Let f : X_{∗} ⟶ Y_{∗} be a morphism of simplicial schemes of Stein compacts and let F, G be graded objects in Mod(X_{∗}), coherent in each degree. By F ⊗_{O_{Y}} G, we mean the object in Mod(X_{∗}), which is given by the sheafification of the object T_{∗} in gr(C)^{N}, defined as follows: For α ∈ N, set B_{α} := Γ(Y_{τ(α)}, O_{Y_{τ(α)}}), F_{α} := Γ(X_{α}, F_{α}) and G_{α} := Γ(X_{α}, G_{α}). Then, F_{α} and G_{α} are modules over B_{α} via the comorphism of f_{α}. Set T_{α} := F_{α} ⊗_{B_{α}} G_{α}. This defines a simplicial DG algebra T_{∗}. In the same manner, we define the tensor product F ⊗_{R} G, when F and G are modules over a sheaf of O_{X_{∗}}-modules R, coherent in each degree.

4.1. Hochschild-cohomology for complex spaces and Noetherian schemes

Let f : X ⟶ Y be a morphism of complex spaces or a morphism of finite type of Noetherian schemes.

A resolvent of X over Y is a collection of the following things:

(1) The simplicial scheme Y_{∗} associated to a local finite affine covering (Y_{j})_{j∈J} of Y;

(2) the simplicial scheme X_{∗} = (X_{α})_{α∈N} associated to a local finite affine covering (X_{ji})_{j∈J,i∈I_{j}} of X. This covering is chosen in such a way that for a fixed j ∈ J, the family (X_{ji})_{i∈I_{j}} is a covering of f^{−1}(Y_{j}); (3) a simplicial scheme P_{∗} = (P_{α})_{α∈N} with the same index category; (4) a commutative diagram

\xymatrix{X_{∗} \ar[r]^{ι} \ar[d]^{f̄} & P_{∗} \ar[dl]^{g} \\ Y_{∗}}

Here, f̄ = (f̄, τ) is the induced map of simplicial schemes; ι is a closed embedding and g is a smooth map^{10}; (5) a free resolution A_{∗} of O_{X_{∗}} as sheaf of DG-algebras on P_{∗} with A_{∗}^{0} = O_{P_{∗}}, such that in each degree there is only a finite number of free algebra generators. If A_{∗} ⟶ B_{∗} is a morphism of sheaves of DG-algebras, coherent in each degree, on a simplicial space X_{∗}, where each X_{α} is affine, then, going over to global sections, we can construct a free resolution S_{∗} of B_{∗} := (Γ(X_{α}, B_{α}))_{α∈N} over A_{∗} := (Γ(X_{α}, A_{α}))_{α∈N}, at least when B_{∗}^{0} is a quotient of a free algebra over A_{∗}^{0} in gr(C)^{N}. This follows by Proposition I.8.8 of [1]. Sheafifying S_{∗}, we get a free resolution S_{∗} of B over A. Using this remark, it is easy to deduce the existence of resolvents in both situations we are going to consider.

Example 4.6. Suppose that X is smooth and Y is just the single point Spec(ℂ). For i ∈ I, we can choose P_{i} = X_{i}. Then, X_{α} is a diagonal in P_{α} and A can be chosen to be a Koszul resolution of a = (Γ(X_{α}, O_{X_{α}}))_{α∈N} over A^{0} = (Γ(P_{α}, O_{P_{α}}))_{α∈N}. In this case, one can prove that for each α, Ω_{A_{α}} is a module resolution of Ω_{a_{α}}. It follows that for α ⊆ β, the restriction maps 𝕃_{α}(a/ℂ) ⟶ 𝕃_{β}(a/ℂ) are quasi-isomorphisms. Consequently, the canonical map 𝕃(X) ⟶ Ω_{X} is a quasi-isomorphism.

Let (X_{∗}, Y_{∗}, P_{∗}, A_{∗}) be a resolvent of the morphism f : X ⟶ Y. Set R := A ⊗_{O_{Y∗}} A and let S be a free resolution of A over R. The following definition coincides for complex spaces with the corresponding definition in [6]:

Definition 4.7. The simplicial Hochschild complex of X over Y is the object in the derived category D(X_{∗}) of O_{X_{∗}}-modules, represented by

ℍ_{∗}(X/Y) := S ⊗_{R} O_{X_{∗}}.

The Hochschild complex of X over Y is defined as the object in D(X), represented by

ℍ(X/Y) := Č^{•}(ℍ_{∗}(X/Y)).

When Y is just the simple point, we will write ℍ(X) instead of ℍ(X/Y).

10This means that for each α ∈ N and each p ∈ P_{α} the stalk O_{P_{α},p} is free (in the analytic case as local analytic algebra) over O_{Y_{τ(α)},y}.

To show the independence of the Hochschild complex of the choice of the resolvent, we have to use the following version of Lemma I.13.7 of [1]:

Lemma 4.8. Let f : X ⟶ X′ be a flat homomorphism of complex spaces (resp. schemes) and (X_{i})_{i∈I} and (X′_{i})_{i∈I′} be compact locally finite coverings of X and X′ by Stein compacts (resp. open affine subsets). Let τ : I ⟶ I′ be a mapping, such that f(X_{i}) ⊆ X_{τ(i)} for all i ∈ I. Denote the associated simplicial schemes by X_{∗} and X′_{∗}. Then, f defines a homomorphism (f̄, τ) of simplicial schemes of ringed spaces. Let G^{•} be a complex in Coh(X′) such that, for α ⊆ β, the restriction map p_{αβ}^{∗}G_{α}^{•} ⟶ G_{β}^{•} is a quasi-isomorphism. Then, the canonical homomorphism

f^{∗}Č(G^{•}) ⟶ Č(f̄^{∗}G^{•})

is a quasi-isomorphism.

Proposition 4.9. The definition of ℍ(X/Y) depends neither on the resolvent (Y_{∗}, X_{∗}, P_{∗}, A_{∗}) nor on the choice of the resolvent S.

Proof. Let (Y_{∗}, X_{∗}, P_{∗}, A_{∗}) and (Ỹ_{∗}, X̃_{∗}, P̃_{∗}, Ã_{∗}) be two resolvents, S a free resolution of A over A ⊗ A and S̃ a resolvent of Ã over Ã ⊗ Ã. We have to show that there is a quasi-isomorphism

Č(S̃ ⊗R̃ OX̃_{∗}) ⟶ Č(S ⊗_{R} O_{X_{∗}}).

First case: Suppose that Y_{∗} = Ỹ_{∗}, X_{∗} = X̃_{∗} and P_{∗} = P̃_{∗}. By Proposition 2.2, there is a quasi-isomorphism

S̃ ⊗R̃ OX̃_{∗} ⟶ S ⊗_{R} O_{X_{∗}}

in Mod(X_{∗}). Applying the Čech functor this case is proved. General case: Let Y′_{∗} be the simplicial scheme associated to the covering {Y_{j}} ∪ {Y′_{j}}, and let X′_{∗} be the simplicial scheme associated to the covering {X_{ij}} ∪ {X′_{ij}}. We construct P′_{∗} in the canonical way and can find a resolvent A′, such that (Y′_{∗}, X′_{∗}, P′_{∗}, A′_{∗}) forms another resolvent of f : X ⟶ Y. There is a commutative diagram

\xymatrix{X_{∗} \ar[r]^{h} \ar[d]^{f_{∗}} & X′_{∗} \ar[d]^{f′_{∗}} \\ Y_{∗} \ar[r] & Y′_{∗}}

By the first case, there is a quasi-isomorphism

h^{∗}(S′ ⊗_{R′} O_{X′_{∗}}) ≈ S ⊗_{R} O_{X_{∗}}.

By Lemma 4.8, there is a quasi-isomorphism

Č(S′ ⊗_{R′} O_{X′_{∗}}) ≈ Č(h^{∗}(S′ ⊗_{R′} O_{X′_{∗}})).

Hence, we get Č(S ⊗_{R} O_{X_{∗}}) ≈ Č(S′ ⊗_{R′} O_{X′_{∗}}). In the same way we get Č(S̃ ⊗R̃ OX̃_{∗}) ≈ Č(S′ ⊗_{R′} O_{X′_{∗}}).

As in [6], we define the n-th Hochschild cohomology of X over Y with values in the sheaf F as Ext_{X}^{n} (ℍ(X /Y), F). We define the n-th Hochschild homology of X over Y as H^{−n}(X, ℍ(Y/Y)). At least, in the case where F is coherent, we want to show that this definition is equal to the following one, which seems to be more natural from the viewpoint of good pairs of categories:

Definition 4.10. [alternative] Suppose that F is coherent. Let a be the algebra (Γ(X_{α}, O_{X_{α}}))_{α∈N} in C^{N}, let k be the algebra ((Γ(Y_{τ(α)}, O_{Y_{τ(α)}}))_{α∈N} in C^{N}. To the map f : X ⟶ Y, there is associated a homomorphism k ⟶ a in C^{N}. Let F be the module (Γ(X_{α}, F_{α}))_{α∈N}. We define the n-th Hochschild cohomology of X over Y with values in F as

HH^{n}(X /Y, F) := H^{n}(Hom_{a}(ℍ_{∗}(a/k), F)).

We define the n-th Hochschild homology of X over Y as

HH_{n}(X /Y) := Ȟ^{−n}(ℍ_{∗}(a/k)).

We see directly, that the Hochschild cohomology is concentrated in non-negative degrees, whereas Hochschild homology, in general has positive and negative degrees.

Remark 4.11. For M_{∗} := ℍ_{∗}(X /Y), the assumption of the second part of Proposition 4.5 is satisfied, i.e., for α ⊆ β, the maps p_{αβ}^{∗}(M_{α}) ⟶ M_{β} are quasiisomorphisms.

Proof. See Lemma 1.7 of [6].

Corollary 4.12. For coherent O_{X} -modules F, the two definitions of Hochschild (co)homology coincide, i.e.

HH^{n}(X /Y, F) ≅ Ext_{X}^{n} (ℍ(X /Y), F)  and
HH_{n}(X/Y) ≅ H^{n}(X, ℍ(X/Y)).

Proof. Since ℍ_{∗}(X /Y) is a complex of free O_{X_{∗}}-modules, by Proposition 4.5 we get

Ext_{X}^{n} (ℍ(X /Y), F) = Ext_{X_{∗}}^{n}(ℍ_{∗}(X /Y), j^{∗}F) =
H^{i}(Hom_{a}(ℍ_{∗}(a/k), F_{∗})) = HH^{i}(X /Y, F).

The second isomorphism is obtained as follows:

H(X, ℍ(X/Y)) ≅ H(tot^{Π} Γ(X, Č^{•}(j^{∗}ℍ(X/Y)))) ≅
H(tot^{Π} Γ(X, Č^{•}(j^{∗}Č^{•}ℍ_{∗}(X/Y))) ≅ H(tot^{Π} Γ(X, Č^{•}ℍ_{∗}(X/Y))) ≅
H(Č^{•}ℍ_{∗}(a/k)) = Ȟ(ℍ_{∗}(a/k)).

In the third step, we have made use of Remark 4.11.

In the absolute case, i.e. in the case where Y = Spec ℂ, Definition 4.7 is up to quasi-isomorphism equivalent to the definition proposed by Weibel/ Geller [21].

Proposition 4.13. Let X be a Noetherian scheme of finite type over a field or a complex space. Let C^{cycl}(X) be the complex of sheaves in 𝔐𝔬𝔡(X) associated to the presheaf U ↦ C^{cycl}(Γ(U, O_{X})). (In the analytic case, the cyclic bar complex is formed, using the analytic tensor product, of course.) There exists a quasiisomorphism of sheaves

ℍ(X) ⟶ C^{cycl}(X).

Proof. Choose a resolvent (X_{∗}, P_{∗}, A_{∗}) of X. Let S be a resolvent of A over R = A ⊗ A. Let a, A, R and S be the simplicial algebras in gr(C)^{N} corresponding to O_{X_{∗}}, A, R and S. By Remark 4.4, there is an quasi-isomorphism

Č(j^{∗}C^{cycl}(X)) ⟶ C^{cycl}(X).

Now, j^{∗}C^{cycl}(X) corresponds to C^{cycl}(a). Set r := a ⊗_{k} a. In the absolute case, R is a resolvent of r over k, hence S ⊗_{R} r is a resolvent of a over r. Thus, there is a quasi-isomorphism

S ⟶ C^{bar}(a).

Since C^{bar}(a) is a complex of flat r-modules, after tensoring over r with a, we get a quasi-isomorphism S ⊗_{r} a ⟶ C^{cycl}(a), i.e. a quasi-isomorphism

S ⊗_{R} O_{X_{∗}} ⟶ j^{∗}C^{cycl}(X)

in 𝔐𝔬𝔡(X_{∗}). Applying the Čech functor, we get the desired result.

4.2. The decomposition Theorem

The quasi-isomorphism ∧𝕃_{a/k} ⟶ ℍ(a/k) in gr(M)^{N} over a in Theorem 3.11 defines a quasi-isomorphism

∧𝕃_{∗}(X/Y) ⟶ ℍ_{∗}(X/Y)

in Mod(X_{∗}). Since the Čech -functor is exact, we get the following HKR-type theorem:

Theorem 4.14. There is an isomorphism

∧𝕃(X/Y) ⟶ ℍ(X/Y)

in the derived category D(X).

Corollary 4.15. There are natural decompositions

HH^{n}(X/Y, M) ≅ ∐_{p+q=n} Ext_{X}^{p}(∧^{q}𝕃(X/Y), M)
HH_{n}(X/Y) ≅ ∏_{p−q=n} H^{q}(X, ∧^{p}𝕃(X/Y)).

For complex spaces, this is just Theorem 4.2 of [6]. There is another nice description of Hochschild cohomology of complex spaces or Noetherian schemes over a field K in any characteristic:

Remark 4.16. HH^{n}(X) = Ext_{X^{2}}(O_{X}, O_{X}).

Proof. We use the letter 𝕂 for the field K or for the complex numbers, depending on the context. With the notations as above, we get:

HH^{n}(X) = H^{n}(Hom_{a}(S ⊗_{R} a, a)) = H^{n}(Hom_{R}(S, a)) =
H^{n}(Hom_{a⊗_{𝕂}a}(S ⊗_{R} (a ⊗_{𝕂} a), a)) = H^{n}(Hom_{O_{X^{2}_{∗}}} (S ⊗_{R} O_{X_{∗}^{2}}, O_{X_{∗}}))
= Ext_{O_{X^{2}_{∗}}}^{n} (O_{X_{∗}}, O_{X_{∗}}) = Ext_{X^{2}}(O_{X}, O_{X}).

Here, we have used that S ⊗_{R} O_{X_{∗}^{2}} is a free resolution of O_{X_{∗}} over O_{X_{∗}^{2}}.

4.3. Hochschild-cohomology for manifolds and smooth varieties Theorem 4.17. Let X be a complex analytic manifold or a smooth scheme of finite type over a field 𝕂 of characteristic zero. There is the following decomposition of Hochschild (co)homology:

HH^{n}(X) ≅ ∐_{i+j=n} H^{i}(X , ∧^{j}T_{X})  (4.1)
HH_{n}(X) ≅ ∏_{i−j=n} H^{j}(X, ∧^{i}Ω_{X}).  (4.2)

Proof. For complex analytic manifolds, we work with a fixed covering by Stein compacts and its associated simplicial scheme X_{∗}. For the case of smooth schemes of finite type over 𝕂, we work with an open affine covering by schemes of the form Spec(A), where A is a finitely generated 𝕂-algebra. Denote the associated simplicial scheme also by X_{∗}. By Proposition 4.11, Theorem 3.11 and Example 4.6, there are quasi-isomorphisms

j^{∗}(ℍ(X)) = j^{∗}C(ℍ_{∗}(X)) ≈ ℍ_{∗}(X) ≈ ∧𝕃_{∗}(X) ≈ ∧_{O_{X∗}} Ω_{X_{∗}} = j^{∗}(∧_{O_{X}} Ω_{X})

of O_{X}-modules. j_{∗}j^{∗} is the identity functor, so there exists a quasi-isomorphism^{11} of O_{X}-modules

ℍ(X) ≈ ∧_{O_{X}} Ω_{X}.  (4.3)

We consider ∧_{O_{X}} Ω_{X} as complex in negative degrees, so ∧Ω_{X} = ∐_{j⩾0} ∧^{j}Ω_{X}[j] and

HH^{n}(X) = Ext_{X}^{n}(ℍ(X), O_{X}) ≅ ∐_{j⩾0} Ext_{X}^{n−j}(∧^{j}Ω_{X}, O_{X}).

By Theorem 7.3.3 of [10], there exists a (bounded) spectral sequence with

E_{2}^{p,q} = H^{p}(X, Ext_{X}^{q}(∧^{j}Ω_{X}, O_{X})),

converging to Ext_{X}(∧^{j}Ω_{X}, O_{X}). But ∧^{j}Ω_{X} is a locally free O_{X}-module, so Ext_{X}^{q}(∧^{j}Ω_{X}, O_{X}) is zero for q > 0 and Hom_{X} (∧^{j}Ω_{X}, O_{X}) for q = 0. Hence, the spectral sequence degenerates at once and we get

Ext_{X}^{q}(∧^{j}Ω_{X}, O_{X}) = H^{q}(X, Hom_{X}(∧^{j}Ω_{X}, O_{X})).

11Remark that for smooth schemes in positive characteristic, by [19] Lemma 2.4, there are natural isomorphisms ∧_{O_{X}}^{n} Ω_{X} ⟶ H_{n}(ℍ(X)), but I don’t know if they are induced by a quasiisomorphism of complexes.

There is a natural isomorphism of sheaves

∧^{j}T_{X} = ∧^{j}Hom_{X}(Ω_{X}, O_{X}) ⟶ Hom_{X}(∧^{j}Ω_{X}, O_{X}),

which, by Proposition 7, p. 154 of [2], is an isomorphism. As consequence,

HH^{n}(X) ≅ ∐_{j⩾0} H^{n−j}(X, ∧^{j}T_{X}) ≅ ∐_{i+j=n} H^{i}(X, ∧^{j}T_{X}).

The second equality is a direct consequence of (4.3).

The decomposition (4.1) for analytic manifolds was announced in Kontsevich’s famous paper [14]. For smooth schemes it was proved in a different way by Yekutieli [23]. A similar statement for quasi-projective smooth schemes is due to Gerstenhaber/ Schack [9] and Swan [19]. For smooth schemes, decomposition (4.2) was proved in a different way by Weibel [22].

References

[1] Jürgen Bingener, Siegmund Kosarew: Lokale Modulräume in der analytischen Geometrie, Band 1 und 2, Aspekte der Mathematik, Vieweg, Braunschweig 1987

[2] Bourbaki: Eléments de mathématiques. Algèbre, Chapitre 3: Algèbre Multilineaire, Hermann, Paris (1958)

[3] Bourbaki: Eléments de mathématiques. Algèbre, Chapitre 10: Algèbre homologique, Paris etc., Masson (1980)

[4] Bourbaki: Eléments de mathématiques. Théories spectrales, Hermann, Paris 1967

[5] Ragnar-Olaf Buchweitz and Hubert Flenner: Semiregularity maps and deformations, preprint arXiv:math. AG/9912245

[6] Ragnar-Olaf Buchweitz and Hubert Flenner: Hochschild (co-)homology of complex spaces, in preparation

[7] Hubert Flenner: Über Deformationen holomorpher Abbildungen, Habilitationsschrift, Osnabrück 1978

[8] M. Gerstenhaber; S.D. Schack: Algebraic cohomology and deformation theory, in ”Deformation Theory of Algebras and Structures and Applications” Michiel Hazewinkel and Murray Gerstenhaber, eds., Kluwer (1988), 11–264

[9] M. Gerstenhaber; S.D. Schack: A Hodge-type decomposition for commutative algebra cohomology, J. Pure and Appl. Alg. 48 (1987), 229–247.

[10] Roger Godement: Topologie algebrique et théorie des faisceaux, Hermann, Paris 1963, 3rd edition (1973)

[11] H. Grauert; R. Remmert: Theorie der Steinschen Räume, Grundlehren der mathematischen Wissenschaften, Springer 1977

[12] Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, 52. New York-Heidelberg-Berlin: Springer Verlag 1983 [13] G. Hochschild; B. Kostant; A. Rosenberg: Differential forms on regular

affine algebras, Trans. AMS 102 (1962), 383-408

[14] Maxim Kontsevich: Deformation quantization of Poisson manifolds I, preprint q-alg/9709040

[15] Jean-Louis Loday: Cyclic Homology, Grundlehren der Math. Wissenschaften 301, Springer Verlag 1992

[16] Palamodov, V.P.: Deformations of complex spaces, Russian Math. Surveys 31 (1976), 129-197

[17] Albrecht Pietsch: Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin (1969)

[18] Daniel Quillen: On the (co-) homology of commutative rings, Proc. Symp. Pure Math. 17 (1970), 65-87

[19] Richard G. Swan: Hochschild cohomology of quasiprojective schemes, Journal of Pure and Applied Algebra 110 (1996) 57-80

[20] Charles A. Weibel: An introduction to homological algebra, Cambridge studies in advanced mathematics 38, 1994

[21] Charles A. Weibel; Susan C. Geller: Etale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991), 368-388

[22] Charles A. Weibel: The Hodge filtration and cyclic homology, preprint

[23] Amnon Yekutieli: The Continuous Hochschild Cochain Complex of a Scheme, Canad. J. Math., Vol. 54 (6), 2002 pp. 1319-1337

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Frank Schuhmacher frank.schuhmacher@ujf-grenoble.fr

Institut Fourier UMR 5582 BP 74 38402 Saint Martin d’Hères France