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

Abstract

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 ${\Omega }^n(X,M)$. 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 $k=\mathbb{C}$.

1. 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 $C^{\star }$-algebra of functions $C\left(X\right)$. 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 $C^{\infty }\left(M\right)$, 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 non-commutative 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 duality-like 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 ${\Omega }^n(X,M)$ 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 $k=\mathbb{C}$. 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 Appendix B, Appendix C and Appendix D.

2. 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 $\mathbb{C}$ of dimension at greater than 1 fail to have a quasi-free $\mathbb{C}$-algebra of functions. Once, the assumption that $k=\mathbb{C}$ 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 $\mathfrak{m}$ be a non-zero maximal ideal in A such that $A_{\mathfrak{m}}$ is has finite $k_{i^{-1}\left[\mathfrak{m}\right]}$-flat dimension and $D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)$ is finite. Then:

$$\begin{array}{cc}\hfill f{d}_{A_{\mathfrak{m}}}\left(M_{\mathfrak{m}}\right)-D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)-f{d}_{k_{\mathfrak{m}}}\left(A_{\mathfrak{m}}\right)\le & HCdim\left(A\right|k)\hfill \end{array}$$

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:

$$f{d}_A\left(M\right)-D\left(k\right)\le HCdim\left(A\right|k).$$

(1)

Example 1.

Let A be a commutative k-algebra and $\mathfrak{m}$ be a non-zero maximal ideal in A such that $A_{\mathfrak{m}}$ is has finite $k_{i^{-1}\left[\mathfrak{m}\right]}$-flat dimension, $D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)$, and A is Cohen-Macaulay at some maximal ideal $\mathfrak{m}$. Then

$$Krull\left(A_{\mathfrak{m}}\right)-D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)-f{d}_{\mathfrak{m}}\left(A_{\mathfrak{m}}\right)\le HCdim\left(A\right|k).$$

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:

$$n-D\left(k\right)\le HCdim\left(A\right|k).$$

(2)

Furthermore if A is commutative and Cohen-Macaulay at a maximal ideal $\mathfrak{m}$ then:

$$Krull\left(A_{\mathfrak{m}}\right)-D\left(k\right)\le HCdim\left(A\right|k).$$

(3)

Next, we consider the implications of our dimension-theoretic formulas within the scope of algebraic geometry from the non-commutative geometric vantage-point.

2.1. 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 Pointcaré duality in dimension d if the dualising module${\omega }_A\triangleq Ex{t}_{{\mathcal{E}}_A^k}^d(A,A)$ satisfies $Ex{t}_{{\mathcal{E}}_A^k}^i(A,k)=0$ for every $i\ne d$ and if in addition $p{d}_{{\mathcal{E}}_A^k}\left({\omega }_A\right)<\infty$. We also recall that an A-bimodule M is invertible if and only if there exits another A-bimodule, which we denote by $M^{-1}$, for which $M{\otimes }_A{M}^{-1}\cong M^{-1}{\otimes }_{A}M\cong A$ in ${}_{A}Mo{d}_A$.

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 $0\le n\le d-f{d}_A\left(M\right)+D\left(k\right)$ satisfying

$$H{H}_n(A,M)\ncong 0.$$

On applying the Hochschild-Kostant-Rosenberg Theorem to Corollary 1, we immediately obtain the claimed result. Recall that ${\Omega }^n(X,M)$ denotes the algebraic differential n-forms on the affine scheme X with coefficients in the vector bundle whose section is the $k\left[A\right]$-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 $0\le n\le d-f{d}_A\left(M\right)+D\left(k\right)$ and a vector bundle whose section is the $k\left[A\right]$-module M for which the algebraic differential n-forms for which

$${\Omega }^n(X,M)\ncong 0.$$

Next, we use Theorem 1 to demonstrate the rarity of commutative quasi-free k-algebras.

2.2. Quasi-Free Algebras are Uncommon

Corollary 3

(Krull Dimension-Theoretic Criterion for Quasi-Freeness). Let A be a commutative k-algebra and $\mathfrak{m}$ be a non-zero maximal ideal in A such that $A_{\mathfrak{m}}$ is has finite $k_{i^{-1}\left[\mathfrak{m}\right]}$-flat dimension, $D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)$, and A is Cohen-Macaulay at some maximal ideal $\mathfrak{m}$. Then, A is not Quasi-free if

$$Krull\left(A_{\mathfrak{m}}\right)\le 2+D\left(k_{i^{-1}\left[\mathfrak{m}\right]}\right)-f{d}_{\mathfrak{m}}\left(A_{\mathfrak{m}}\right).$$

Let us also consider the simpler form implied by Theorem 2.

Corollary 4.

If k is of finite global dimension, A is a k-algebra which is flat as a k-module, and if $A_{\mathfrak{m}}$’s Krull dimension is at least $2+D\left(k\right)$ then A is not Quasi-free.

We unpack Theorem 2 in the context of classical algebraic and arithmetic geometry.

Examples

To build intuition before proceeding, we consider a counter-intuitive consequence. Namely, that most examples of smooth commutative algebras fail to be quasi-free, even when $k\ne \mathbb{C}$. This makes smoothness, in the sense of [14], very rare in the non-commutative category. The following example from arithmetic geometry is of interest.

Let be an affine algebraic $\mathbb{C}$-variety $V\left(A\right)$. For any point x in $V\left(A\right)$ the ideal generated by the collection of regular functions on $V\left(A\right)$ vanishing at the point x is denoted by $\mathcal{I}\left(x\right)$; in fact $\mathcal{I}\left(x\right)$ is a maximal ideal in A [19]. Moreover, for any affine-algebraic variety $V\left(A\right)$ there exists a point x such that $A_{\mathcal{I}\left(x\right)}$ is regular. Since every regular local $\mathbb{C}$-algebra is Cohen Macaulay at its maximal ideal, then A is Cohen-Macaulay at $\mathcal{I}\left(x\right)$. Since $\mathbb{C}$ is a field it is a regular local ring of Krull dimension 0; the Auslander-Buchsbaum-Serre theorem thus implies $D\left(k\right)=Krull\left(k\right)=0$, moreover $A_{\mathcal{I}\left(x\right)}$ is a $\mathbb{C}$-vector space whence it is a $\mathbb{C}$-free and so is a $\mathbb{C}$-flat module. Therefore Theorem 2 applies if $Krull\left(A\right)\ge 2$. We summarize this finding as follows.

Corollary 5.

If X is an affine $\mathbb{C}$-variety and $k\left[A\right]$’s Krull dimension is greater than 1 then the $\mathbb{C}$-algebra A is not quasi-free

Remark 1.

Corollary 5 implies that any affine algebraic $\mathbb{C}$-variety which is not a disjoint union of curves or points has a coordinate ring which fails to be quasi-free over $\mathbb{C}$.

Example 3.

The $\mathbb{C}$-algebra $\mathbb{C}{[x_{1,1},x_{1,2},x_{2,1},x_{2,2}]}_{\left(det\right)}$ is not quasi-free.

Proof. 

$\mathbb{C}{[x_{1,1},x_{1,2},x_{2,1},x_{2,2}]}_{\left(det\right)}$ is of Krull dimension $4>1$ [20] therefore Theorem 2 applies. □

Corollary 6

(Arithmetic Polynomial-Algebras). The $\mathbb{Z}$-algebra $\mathbb{Z}[x_1,..,x_n]$ fails to be quasi-free for values of $n>1$.

Proof. 

Since $\mathbb{Z}[x_1,\dots x_n]$ is Cohen-Macaulay at the maximal ideal $(x_1,\dots x_n,p)$ and is of Krull dimension $n+1=Krull\left(\mathbb{Z}[x_1,\dots x_n]\right)$. Moreover, one computes that $D\left(\mathbb{Z}\right)=1$. Whence by point 2 of Theorem 2: $\mathbb{Z}[x_1,..,x_n]$ fails to be Quasi-free if $2\le Krull\left(\mathbb{Z}[x_1,\dots x_n]\right)-D\left(\mathbb{Z}\right)=(n+1)-1=n$. □

The contributions of the paper are now summarized.

3. 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 non-triviality 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 non-commutative 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.