Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Afﬁne Schemes and Quasi-Free Algebras

: The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the ﬂat-dimension difference and its global dimension. Our result is used to show that for a smooth afﬁne scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n -forms Ω n ( X , M ) . Further restricting the notion of smoothness, we use our result to show that most k -algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k = C .


Introduction
Non-commutative geometry is a rapidly developing area of contemporary mathematical research that studies non-commutative algebras using formal geometric tools. The field traces its most evident origins back to the results of [1], which show that any compact Hausdorff space can be fully reconstructed, and largely understood, from its associated C -algebra of functions C(X). However, the trend of understanding geometric properties via algebraic dual theories is echoed throughout mathematics; with notable examples coming from the duality between finitely generated algebras and affine schemes (see [2]), the description of any smooth manifold M through its commutative algebra C ∞ (M), and ultimately culminating with the work of [3,4] describing the duality relationship between algebra and geometry in full generality.
Though a large portion of the interest in non-commutative geometry stems from its connections with physics, see [5][6][7]. A. Connes largely made these connections through the cyclic cohomology theory of [8], a generalized de Rham cohomology theory for noncommutative spaces, which closely tied through the Connes complex to one of the central tools of non-commutative geometry and the central object of study of this paper, namely Hochschild (co)homology.
Hochschild (co)homology, originally introduced in [9], is a cohomology theory for non-commutative k-algebras. Since its introduction, it has become a key tool and object of study in non-commutative geometry since the results of [10] (and more recently generalized in [11] to characteristic p fields); which identifies the Hochschild homology of commutative k-algebras over a characteristic 0 field k, to the module of Khäler differentials over their associated affine scheme. Likewise, the result identifies Hochschild's cohomology theory with the modules of derivations and, therefore, with the tangential structure over the commutative algebra's associated affine scheme. Likewise, in these cases, Pointcaré dualitylike results can also be entirely formulated between these structures and the Hochschild (co)homology theories as shown in [12].
This article focuses on a fundamental non-commutative geometric invariant derived from the Hochschild (co)homology, namely its (co)homological dimension. We focus on the interplay between this (co)homological invariant of commutative k-algebras over general commutative rings k, and its implications on various notions of smoothness of its associated dual non-commutative space; such as the quasi-freeness (or formal smoothness) of [13,14], or more generally, the vanishing of their higher modules of differential forms as seen in [12].
The relationship between the Hochschild (co)homology theory and smoothness has seen study in the case where k is a field in [15,16]. However, the general case is still far from understood and this is likely due to it requiring a more subtle treatment offered by the less-standard tools of relative homological-algebraic (see [17,18] for example). Indeed, this paper proposes a set of lower-estimates of this invariant, which can be easily computed from local data of any commutative k-algebra over a commutative ring k with unity.
The paper's main results are used to show that for any smooth affine scheme X there must exist a vector bundle on X with section M and a suitably small natural number n for which the module of algebraic differential n-forms with values in M, denoted by Ω n (X, M) is non-trivial. Our results are also used to derive simple tests for a k-algebras' quasi-freeness. This latter application extends known results of [14] in the special case where k = C. Using this result, we conclude that typical k-algebras are not quasi-free. Concrete applications are considered within the scope of arithmetic geometry.

Organization of the Paper
The paper is organized as follows. Section 2 contains the paper's main theorems as well as its non-commutative geometric questions consequences. Each result is followed by examples which unpack the general implications in the context of algebraic geometry. Appendix A contains detailed background material in the relative homological algebraic tools required for the paper's proofs is included after the paper's conclusion. Likewise, the paper's proofs and any auxiliary technical lemma is also relegated to Appendices B-D.

Main Result
From here on out, A will always be a commutative k-algebra. The remainder of this paper will focus on establishing the following result. An analogous statement was made in [14] that all affine algebraic varieties over C of dimension at greater than 1 fail to have a quasi-free C-algebra of functions. Once, the assumption that k = C is relaxed, we find an analogous claim is true; however, the analysis is more delicate. Our principle result is the following.
Theorem 1 (Lower-Bound on Hochschild Cohomological Dimension). Let A be a commutative k-algebra and m be a non-zero maximal ideal in A such that A m is has finite k i −1 [m] -flat dimension and D(k i −1 [m] ) is finite. Then: Theorem 1 allows for an easily computable lower-bound on the Hochschild cohomological dimension of nearly any commutative k-algebra A, granted that it is smooth in the classical sense at-least at one point. The next result, obtains an even simpler criterion under the additional assumption that A is k-flat. Theorem 2. Let k be of finite global dimension, A be a k-algebra which is flat as a k-module. Then M: Example 2. Let k be of finite global dimension, A be a k-algebra which is flat as a k-module. Then, for every A-module M, if x 1 , .., x n is a regular sequence in A then: Furthermore if A is commutative and Cohen-Macaulay at a maximal ideal m then: Next, we consider the implications of our dimension-theoretic formulas within the scope of algebraic geometry from the non-commutative geometric vantage-point.

Non-Triviality of Higher Differential Forms
The paper's provides a homological argument showing that a smooth affine scheme must have some non-trivial module of higher-differential forms. These begin with the non-triviality of the Hochschild homology modules.
To show our result, we begin by recalling the terminology introduced in [12]. Recall that a k-algebra is satisfies We also recall that an A-bimodule M is invertible if and only if there exits another Abimodule, which we denote by Corollary 1 (Non-Triviality of Hochschild Homology Modules). Let k be a commutative ring and X be a d-dimensional smooth affine scheme over k whose coordinate ring satisfies Pointcaré duality in dimension d and is invertible. Then, there is an A-bimodule M and some On applying the Hochschild-Kostant-Rosenberg Theorem to Corollary 1, we immediately obtain the claimed result. Recall that Ω n (X, M) denotes the algebraic differential n-forms on the affine scheme X with coefficients in the vector bundle whose section is the k[A]-bimodule M.

Corollary 2.
Let k be a commutative ring and X be a d-dimensional smooth affine scheme over k whose coordinate ring satisfies Pointcaré duality in dimension d and is invertible. Then, there exists a some 0 ≤ n ≤ d − f d A (M) + D(k) and a vector bundle whose section is the k[A]-module M for which the algebraic differential n-forms for which Proof. Since Z[x 1 , ...x n ] is Cohen-Macaulay at the maximal ideal (x 1 , ...x n , p) and is of Krull dimension n + 1 = Krull(Z[x 1 , ...x n ]). Moreover, one computes that D(Z) = 1. Whence by point 2 of Theorem 2: Z[x 1 , .., x n ] fails to be Quasi-free if 2 ≤ Krull(Z[x 1 , ...x n ]) − D(Z) = (n + 1) − 1 = n.
The contributions of the paper are now summarized.

Conclusions
This paper's main result derived a general lower bound on the Hochschild cohomological dimension of an arbitrary commutative k-algebra A over a general commutative ring k. Theorem 1 derived, the lower-bound for this (co)homological invariant was expressed in terms of other (co)homological dimension-theoretic invariants, namely the flat dimension over A, the global dimension of A, and the flat dimension of A over k; where each quantity was appropriately localized. Examples 1 and 2, built on these results to lower-bound the Hochschild cohomological dimension purely in terms of easily computable quantities, such as the Krull dimension, when A was Cohen-Macaulay. Theorem 2 then expresses a non-localized analog of Theorem 1 wherein no commutativity of A was required.
The paper's results have then been applied the results to purely geometric questions. First, the dimension-theoretic formula was used in Corollary 2 to show infer the nontriviality of certain higher algebraic differential forms of any smooth affine scheme with values in a vector bundle with a non-trivial section. The dual result was also considered in Corollary 1 where dimension-theoretic conditions were obtained for the non-vanishing of some of the Hochschild homology modules under Pointcaré duality in the sense of [12].
Next, using the general (co)homological dimension-theoretic estimates, a result of [14], which showed that most commutative affine k-algebras fail to be smooth in the noncommutative sense formalized by quasi-freeness, was extended from the simple case where k was a field to the general case where k is simply a commutative ring. Specifically, in Corollaries 3 and 4, easily applicable dimension-theoretic tests for the non-quasi-freeness (non-formal smoothness) of a commutative k-algebra over a general ring k were derived. The tools are simple and only require a simple computation involving the Krull dimension of A, the flat-dimension of k at one point, and the base ring's global dimension to identify if A's associated non-commutative space is quasi-free or not.  Acknowledgments: I would like to thank Abraham Broer for the numerous helpful and insightful conversations.

Conflicts of Interest:
The author declares no conflict of interest.

Appendix A. Background
This appendix contains the necessary background material for the formulation of this paper's main results. We refer the reader in further reading to the notes of [21].

Appendix A.1. Relative Homological Algebra
The results in this paper are formulated using the relative homological algebra, see [17] for example. The theory is analogous to standard homological algebra; see [22] for example, but in this case, one builds the entire theory relative to a suitable subclass of epi(resp. mono)-morphisms. In our case, these are defined as follows.
For any k-algebra A, an epimorphism in A Mod is an E k A -epimorphism if and only if 's underlying morphism of k-modules is a k-split epimorphism in k Mod. The class of these epimorphisms is denoted E k A .
Definition A2 (E k A -Exact sequence). An exact sequence of A-modules: ...
is said to be E k A -exact if and only if for every integer i the there exists a morphism of k-modules In particular, a short exact sequence of A-modules which is E k A -exact is called an E k A -short exact sequence.
Definition A3 (E k A -Projective module). If A is a k-algebra and P is an A-module, then P is said to be E k A -projective if and only if for every E k A -short exact sequence: the sequence of k-modules: Remark A2. This definition is equivalent to requiring that P verify the universal property of projective modules only on E k A -epimorphisms [23].
E k A -projective A-modules have analogous properties to projective A-modules. For example, E k A -projective A-modules admit the following characterization.
Proposition A1. For any A-module P the following are equivalent: There exists a k-module F, an A-module Q and an isomorphism of A-modules φ : Remark A3. If F is a free k-module, some authors call A ⊗ k F an E k A -free module. In fact this gives an alternative proof that A e ⊗ k A ⊗n ∼ = A ⊗n+2 is E k A e -free for every n ∈ N.) Proof. See [23] pages 261 for the equivalence of 1, 2 and 3 and page 277 for the equivalence of 1 and 4.
For a homological algebraic theory to be possible, one needs enough projective (resp. injective) objects. The next result shows that there are indeed enough E k A -projectives in A Mod.

Proposition A2 (Enough E k A -projectives). If A is a k-algebra and M is an A-module then there exists an
Since there are enough projective objects, then one can build a resolution of any A-module by E k A -projective modules.
Definition A4 (E k A -projective resolution). If M is an A e -module then a resolution P of M is called an E k A -projective resolution of M if and only if each P i is an E k A -projective module and P is an E k A -exact sequence.

Example A3. The augmented bar complexĈB (A) of A is an E k A e -projective resolution of A.
Remark A4. A nearly completely analogous argument to Example A3 shows that for any (A, A)-bimodule M, M ⊗ AĈ B (A) is an E k A e -projective resolution of M, see for details [24].
Following [18], the E k A -relative derived functors of the tensor product and the Hom Afunctors are introduced, as follows.
Definition A5. E k A -relative Tor If N is a right A-module, M is an A-module and P is an E k A -projective resolution of N then the k-modules H (P ⊗ A M) are called the E k A -relative Tor k-modules of N with coefficients in the A-module M and are denoted by Tor n Let H (resp. H ) denote the (co)homology functor from the category of chain (co)complexes on an A-module to the category of A-modules. The E k A -relative Tor functors are defined as follows.
Example A4. The E k A -relative Tor functors may differ from the usual (or "absolute") Tor functors. For example consider all the Z-algebra Z, any Z-modules N and M are E Z Z -projective. In particular, this is true for the Z-modules Z and Z/2Z. Therefore Tor n E Z Z (Z, Z/2Z) vanish for every positive n, however Tor n [22].
Similarly there are E k A -relative Ext functors.
If N is and M are A-modules and P is an E k A -projective resolution of N then the k-modules H (Hom A (P , M)) are called the E k A -relative Ext k-modules of N with coefficients in the A-module M and are denoted by Ext n The E k A -relative homological algebra is indeed well defined, since both the definitions of E k A -relative Ext and E k A -relative Tor are independent of the choice of E k A -projective resolution.
Theorem A1 (E k A -Comparison theorem). If P and P are E k A -projective resolutions of an Amodule N then for any A-module M there are natural isomorphisms: and if P and P are E k A -projective resolutions of a right A-module N then: Proof. Nearly identical to the usual comparison theorem, see [23].
Example A5. The Ext Z and E Z Z -relative Ext may differ. For example, one easily computes Ext 1 Analogous to the fact that for any A-module P, P is projective if and only if Ext 1 A (P, N) ∼ = 0 for every A-module N there is the following result, which can be found in ( [18], Chapter IX).

Proposition A3. P is an E k
A -projective module if and only if for every A-module N: Using the theory of relative (co)homology, we are now in-place to review the Hochschild cohomology theory over general k-algebras.

Appendix A.2. Hochschild (Co)homological Dimension
Since CB (A) is an E k A e -projective resolution of A then Theorem A1 and the definition of the Ext E k A e (A, −) functors imply that the Hochschild cohomology of A with coefficients in of [9], denoted by HH (A, N), can be expressed using the Ext E k A e . We maintain this perspective throughout this entire article.
Proposition A4. For every A e module N there are k-module isomorphisms, natural in N: Taking short E k A e -exact sequences to isomorphic long exact sequences.
Definition A7 (Hochschild Homology). The Hochschild homology HH (A, N) of a k-algebra A with coefficient in the (A, A)-bimodule N is defined as: where P is an E k A e -projective resolution of A.
Following the results of [10], the Hochschild cohomology has become the central tool for obtaining non-commutative algebraic geometric analogues of classical commutative algebraic geometric notions. The one of central focus in this paper, is the Hochschild cohomological dimension, Definition A8 (Hochschild cohomological dimension). The Hochschild cohomological dimension of a k-algebra A is defined as: where N # is the ordered set of extended natural numbers.
The Hochschild cohomological dimension may be related to the following cohomological dimension.

Definition A9 (E k
A -projective dimension). If n is a natural number and M is an A-module then M is said to be of E k A -projective dimension at most n if and only if there exists a deleted E k A -projective resolution of M of length n. If no such E k A -projective resolution of M exists then M is said to be of The following is a translation of a classical homological algebraic result into the setting of E k A e -projective dimension, Ω n (A/k) and Hochschild cohomology. Here, Ω n (A/k) Ker(b n−1 ) and b n−1 is the (n − 1) th differential in the augmented Bar resolution of A; see [24] for details on the augmenter Bar complex.
Theorem A2. For every natural number n, the following are equivalent: Therefore for every A e -module M: (Ω n (A/k), M) ∼ = 0.
(2 ⇒ 1) If A admits an E k A e -projective resolution P of length n then Theorem A1 implies there are natural isomorphisms of A e -modules: Since P is of length n all the maps p j : P j+1 → P j are the zero maps therefore so are the maps p j : Hom A e (P j ) → Hom A e (P j+1 ). Whence (A12) entails that for all j > n + 1 Ext Next, the non-commutative geometric object focused on in this paper is reviewed.

Appendix A.3. Quasi-Free Algebras
Many of the properties of an algebra are summarized by its Hochschild cohomological dimension, see [10,17] for example. However, this article focuses on the following noncommutative analogue of smoothness of [13], introduced by [14].
Remark A5. Due to their lifting property, the quasi-free k-algebras are considered a non-commutative analogue to smooth k-algebras; that is k-algebras for which Ω A|k is a projective A-module.
This notion of smoothness has played a key role in a number of places in noncommutative algebraic geometry, especially in the cyclic (co)homology of [25].
Definition A10 (Quasi-free k-algebra). A k-algebra for which all k-Hochschild extensions of A by an (A, A)-bimodule lift is called a quasi-free k-algebra.
Corollary A1. For a k-algebra A, the following are equivalent: A is quasi-free.
One typically construct quasi-free algebras using Morita equivalences. However, the next proposition, which extends a result of [14] to the case where k need not be a field, may also be used without any such restrictions on k.

Proposition A5. If A is a quasi-free k-algebra and P is an E k A e -projective (A, A)-bimodule then T A (P) is a quasi-free A-algebra.
Proof. Differed until the appendix.
Proof. Since all free Z-modules are projective Z-modules and all projective Z-modules are Z is a quasi-free Z-algebra.
Example A7. If A is a quasi-free k-algebra then T A (Ω 1 (A/k)) is a quasi-free A-algebra.
Proof. By Corollary A1 if A is quasi-free Ω 1 (A/k) must be an E k A e -projective (A, A)bimodule; whence Proposition A5 applies.
Next, we overview some relevant dimension-theoretic notions and terminology.

Appendix A.4. Classical Cohomological Dimensions
We remind the reader of a few important algebraic invariants which we will require. The reader unfamiliar with certain of these notions from commutative algebra and algebraic geometry is referred to [2,26] or to [19].

Definition A11 (A-Flat Dimension). If A is a commutative ring then the A-flat dimension f d A (M) of an A-module M is the extended natural number n, defined as the shortest length of a resolution of M by A-flat A-modules. If no such finite n exists n is taken to be ∞.
We will require the following result, whose proof can be found in [24]. Proposition A6. If n is a positive integer and if there exists a regular sequence x 1 , .., x n in A of length n then: n = f d A (A/(x 1 , .., x n )).
One more ingredient related to the flat dimension will soon be needed.

Definition A12. A-Projective Dimension
If A is a commutative ring and M is an A-module then the A-projective dimension pd A (M) of M is the extended natural number n, defined as the shortest length of a deleted A-projective resolution of M. If no such finite n exists n is taken to be ∞.

Lemma A1. If A is a commutative ring and M is an A-module then f d A (M) ≤ pd A (M).
Proof. Since all A-projective A-modules are A-flat, then any A-projective resolution is a A-flat resolution.
Lemma A2. If A is a commutative ring then for any A-module M the following are equivalent: • The A-projective dimension of M is at most n. Proof. Nearly identical to the proof of Theorem A2, see page 456 of [22] for details. (A15)

Definition A13 (Cohen-Macaulay at an Ideal). A commutative ring A is said to be Cohen-
The following modification of the global dimension of a k-algebra, does not ignore the influence of k on a k-algebra A, as will be observed in the next section of this paper.

Definition A15. E k -Global dimension
The E k -global Dimension D E k (A) of a k-algebra A is defined as the supremum of all the E k A -projective dimensions of its A-modules. That is:

Appendix B. Proofs
This appendix contains certain technical lemmas or auxiliary results that otherwise detracted from the overall flow of the paper.

Appendix C. Technical Lemmas
We make use of the following result appearing in a technical note of Hochschild circa 1958, see [27].
Theorem A3 ( [27]). If k is of finite global dimension, A is a k-algebra which is flat as a k-module and M is an A-module then: is a deleted E k A -projective resolution of an A-module M then for every A-module N and for every positive integer n there are isomorphisms natural in N: Proof. By definition the truncated sequence is exact: where η is the canonical map satisfying d n = ker(d n ) • η (arising from the universal property of ker(d n )). Moreover, since (A30) is E k A -exact, d n is k-split; whence η must be k-split. Moreover, for every j ≥ n + 1, d j was by assumption k-split therefore (A20) is E k A -exact and since for every natural number m > n P m is by hypothesis E k A -projective then (A20) is an augmented E k A -projective resolution of the A-module Ker(d n ). For every natural number m, relabel: Q m := P m+n and p m := d n+m . (A21) By Theorem A1, for all N ∈ A Mod and all m ∈ N, we have that: Therefore, the result follows.

Appendix D. Auxiliary Results
Proof of Proposition A5. Let be a k-Hochschild extension of T A (P) by M. We use the universal property of T A (P) to show that there must exist a lift l of (A23). Let p : T A (P) → A be the projection k-algebra homomorphism of T A (P) onto A. p is k-split since the k-module inclusion i : A → T A (P) is a section of p; therefore p is an E k A e -epimorphism and is a k-Hochschild extension of A by the (A, A)-bimodule Ker(p • π). Since A is a quasifree k-algebra there exists a k-algebra homomorphism l 1 : A → B lifting p • π. Hence B inherits the structure of an (A, A)-bimodule and π may be viewed as an (A, A)-bimodule homomorphism. Moreover, l 1 induces an A-algebra structure on B.
Let f : P → T A (P) be the (A, A)-bimodule homomorphism satisfying the universal property of the tensor algebra on the (A, A)-bimodule P. Since π : B → A is an E k A eepimorphism and since P is an E k A e -projective (A, A)-bimodule, Proposition A1 implies that that there exists an (A, A)-bimodule homomorphism l 2 : P → B satisfying π • l 2 = f .
Since l 2 : P → B is an (A, A)-bimodule homomorphism to a A-algebra the universal property of the tensor algebra T A (P) on the (A, A)-bimodule P, see [28], implies there is an A-algebra homomorphism l : T A (P) → B whose underlying function satisfies: l • f = l 2 .
Therefore l • π • l 2 = l 2 ; whence l • π = 1 T A (P) ; that is l is a A-algebra homomorphism which is a section of π, that is l lifts π.
Appendix D.1. Proof of Theorem 1 Our first lemma is a generalization of the central theorem of [27]; which does not rely on the assumption that A is k-flat. Lemma A3. If k is of finite global dimension and A is a k-algebra which is of finite flat dimension as a k-module, then for every A-module M: The proof of Lemma A3 relies on the following lemma.
Lemma A4. If A is a k-algebra such that f d k (A) < ∞ then: Proof. For every k-module M and every A-module N there is a convergent third quadrant spectral sequence (see [22], page 667): Hom A (A, N)).
Moreover, the adjunction − ⊗ k A Hom A (A, −) extends to a natural isomorphism: Therefore there is a convergent third-quadrant spectral sequence: If pd A (N) < ∞, then the result is immediate. Therefore assume that: and in the latter case Finally, the result follows since f d k (A) is finite and, therefore, can be subtracted unambiguously.
Lemma A5. If A is a k-algebra then for any k-module M there is an E k A -exact sequence: where α be the map defined on elementary tensors (a ⊗ k m) in A ⊗ k M as a ⊗ k m → a · m.
Proof. α is k-split by the map β : Lemma A6. If M and N are A-modules then: Proof.
(A33) Therefore Ext n A (M ⊕ N, X) vanishes only if both Ext n A (M, X) and Ext n A (N, X) vanish. Lemma A2 then implies: pd A (M) ≤ pd(M ⊕ N).

Proof of Lemma A3
Proof.
Since k's global dimension is finite hence (A34) implies: . The proof will proceed by induction on d.
Proposition A1 implies that (A36) is A-split therefore M is a direct summand of the A-module A ⊗ k M. Hence Lemma A6 implies: Lemma A4 together with (A37) imply: Definition A15 and (A38) together with the assumption that pd E k A (M) = 0 imply: Since k's global dimension and f d k (A) are finite then (A39) implies: Step: d > 0 Suppose the result holds for all A-modules K such that pd E k A (K) + D(k) + f d k (A) = d for some integer d > 0. Again appealing to Lemma A5, there is an E k A -exact sequence: Proposition A1 implies A ⊗ k M is E k A -projective; whence (A41) implies: Since Ker(α) is an A-module of strictly smaller E k A -projective dimension than M the induction hypothesis applies to Ker(α) whence: The proof will be completed by demonstrating that: pd A (M) ≤ pd A (Ker(α)) + 1. For any N ∈ A Mod Ext A (−, N) applied to (A41) gives way to the long exact sequence in homology, particularly the following of its segments are exact: ; thus for every positive integer n ≥ D(k) (in particular d is at least n): whence ∂ n must be an isomorphism. Therefore Lemma A2 implies pd A (M) is at most equal to pd A (Ker(α)) + 1. Therefore: Finally since k is of finite global dimension and A is of finite k-flat dimension then (A48) implies: thus concluding the induction.
We will also require the following result.
Remark A6. Let A be a k-algebra, i : k → A the morphism defining the k-algebra A and m a maximal ideal in A. For legibility the E A m -projective dimension of an A m -module N will be abbreviated by pd E m,k (N) (instead of writing pd Lemma A7. If A is a commutative k-algebra and m is a non-zero maximal ideal in A then for every A-module M: where i : k → A is the inclusion of k into A.

Proof. Since m is a prime ideal in
be an E k A -projective resolution of an A-module M. The exactness of localization [26] implies: is exact. It will now be verified that (A52) is a E m,k -projective resolution of the A mmodule M m .
The d n ⊗ A A m are k i −1 [m] -split Since (A51) was k-split then for every i ∈ N there existed a k-module homomorphism s i : P n−1 → P n (where for convenience write P −1 : -algebra A m may be viewed as a k i −1 [m] -module therefore the maps: -module homomorphisms; moreover they must satisfy: implies there exists some A-module Q and some k-module X satisfying: Therefore we have that: Since A, k and k i −1 [m] are commutative rings the tensor products − ⊗ A −, − ⊗ k − and − ⊗ k i −1 [m] − are symmetric [22], hence (A55) implies: Since A is a subring of A m then (A56) implies: ⊗ k X) may be viewed as a k i −1 [m] -module with action· defined as: All the homological dimensions discussed to date are related as follows: Proposition A10. If A is a commutative k-algebra and m be a non-zero maximal ideal in A such that A m has finite k i −1 [m] -flat dimension and D(k i −1 [m] ) is finite then there is a string of inequalities: Proof.
• By definition: Since the global dimension of k i −1 [m] was assumed to be finite (A60) implies: Lemma A8. If A is a commutative k-algebra and M and N be A-modules, then there are natural isomorphisms: Ext n In particular (A63) implies that for every n in N there is an isomorphism which is natural in the first input: whence if b n+1 : A ⊗n+3 → A ⊗n+2 is the n th map in the Bar complex (recall Example A3) and for legibility denote Hom A Mod A (b n , Hom k (M, N)) by β n . The naturality of the maps ψ n imply the following diagram of k-modules commutes: • Therefore for every n in N: > is a chain complex. Moreover, the commutativity of (A65) implies that: (A69) (A71) We may now prove Theorem 1.

• For any
The right hand side of (A89) is precisely the definition of the Hochschild cohomological dimension. Therefore D E k (A) ≤ HCdim(A|k) Proposition A10 applied to (A90), which draws out the conclusion.
The right hand side of (A89) is precisely the definition of the Hochschild cohomological dimension. Therefore D E k (A) ≤ HCdim(A|k) (A90) Proposition A10 applied to (A90) then draws out the conclusion. Proposition A6 implies that: (x 1 , .., x n )).
Therefore (1) applied to the A-module A/(x 1 , .., x n together with (A91) imply: If Ω 1 (A/k) is generated by a regular sequence x 1 , .., x n then Proposition A6 implies: However by definition of Ω 1 (A/k) as the kernel of µ A : A ⊗ k A/Ω 1 (A/k) ∼ = A. Therefore: n = f d A e (A).
Lemma A1 together with Lemma A3 imply: Since D(k) is finite then (A95) entails: for every A-bimodule M. In particular, (A103) holds for the A-bimodule M of Corollary 1.