Dimension theory (algebra) explained

In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension.

The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebraic sets, most of the definitions of the dimension are equivalent. For general commutative rings, the lack of geometric interpretation is an obstacle to the development of the theory; in particular, very little is known for non-noetherian rings. (Kaplansky's Commutative rings gives a good account of the non-noetherian case.)

Throughout the article,

\dim denotes Krull dimension of a ring and \operatorname{ht} the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings.

Basic results

Let R be a noetherian ring or valuation ring. Then$$\backslash dim\; R[x]\; =\; \backslash dim\; R\; +\; 1.$$If R is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal

ak{p} in R,$$\backslash operatorname(\backslash mathfrak\; R[x])\; =\; \backslash operatorname(\backslash mathfrak).$$$$\backslash operatorname(\backslash mathfrak)\; =\; \backslash operatorname(\backslash mathfrak)\; +\; 1$$ for any prime ideal ak{q} \supsetneq ak{p} R[x] in R[x] that contracts to ak{p} .This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of \operatorname{Spec} R[x] \to \operatorname{Spec} R , one cannot have a chain of primes ideals of length \ge 2 .

Since an artinian ring (e.g., a field) has dimension zero, by induction one gets a formula: for an artinian ring R,$$\backslash dim\; R[x\_1,\; \backslash dots,\; x\_n]\; =\; n.$$

Local rings

Fundamental theorem

Let

(R, ak{m}) be a noetherian local ring and I a ak{m} -primary ideal (i.e., it sits between some power of ak{m} and ak{m} ). Let F(t) be the Poincaré series of the associated graded ring $\backslash operatorname\_I\; R\; =\; \backslash bigoplus\_0^\backslash infty\; I^n\; /\; I^$. That is,$$F(t)\; =\; \backslash sum\_0^\backslash infty\; \backslash ell(I^n\; /\; I^)\; t^n$$where \ell refers to the length of a module (over an artinian ring (\operatorname{gr} _{I} R) _{0} = R/I ). If x _{1,} ... , x _{s} generate I, then their image in I/I ^{2} have degree 1 and generate \operatorname{gr} _{I} R as R/I -algebra. By the Hilbert–Serre theorem, F is a rational function with exactly one pole at t =1 of order d \le s . Since$$(1-t)^\; =\; \backslash sum\_0^\backslash infty\; \backslash binom\; t^j,$$we find that the coefficient of t ^{n} in F(t) = ( 1- t) ^{d} F(t) ( 1 - t) ^{-d} is of the form$$\backslash sum\_0^N\; a\_k\; \backslash binom\; =\; (1\; -\; t)^d\; F(t)\backslash big|\_\; +\; O(n^).$$That is to say, \ell(I ^{n} / I ^{n+1} ) is a polynomial P in n of degree d - 1 . P is called the Hilbert polynomial of \operatorname{gr} _{I} R .

We set

d(R) = d . We also set \delta(R) to be the minimum number of elements of R that can generate an ak{m} -primary ideal of R. Our ambition is to prove the fundamental theorem:$$\backslash delta(R)\; =\; d(R)\; =\; \backslash dim\; R.$$Since we can take s to be \delta(R) , we already have \delta(R) \ge d(R) from the above. Next we prove d(R) \ge \dim R by induction on d(R) . Let ak{p} _{0} \subsetneq … \subsetneq ak{p} _{m} be a chain of prime ideals in R. Let D = R/ ak{p} _{0} and x a nonzero nonunit element in D. Since x is not a zero-divisor, we have the exact sequence$$0\; \backslash to\; D\; \backslash overset\backslash to\; D\; \backslash to\; D/xD\; \backslash to\; 0.$$The degree bound of the Hilbert-Samuel polynomial now implies that d(D) > d(D/xD) \ge d(R/ ak{p} _{1)} . (This essentially follows from the Artin-Rees lemma; see Hilbert-Samuel function for the statement and the proof.) In R/ ak{p} _{1} , the chain ak{p} _{i} becomes a chain of length m - 1 and so, by inductive hypothesis and again by the degree estimate,$$m-1\; \backslash le\; \backslash dim\; (R/\backslash mathfrak\_1)\; \backslash le\; d(R/\backslash mathfrak\_1)\; \backslash le\; d(D)\; -\; 1\; \backslash le\; d(R)\; -\; 1.$$The claim follows. It now remains to show \dim R \ge \delta(R). More precisely, we shall show:(Notice: (x _{1,} ... , x _{d)} is then ak{m} -primary.) The proof is omitted. It appears, for example, in Atiyah–MacDonald. But it can also be supplied privately; the idea is to use prime avoidance.

Consequences of the fundamental theorem

Let

(R, ak{m}) be a noetherian local ring and put k = R/ ak{m} . Then\dim R \le \dim _{k} ak{m}/ ak{m} ^{2} , since a basis of ak{m}/ ak{m} ^{2} lifts to a generating set of ak{m} by Nakayama. If the equality holds, then R is called a regular local ring.\dim \widehat{R} = \dim R , since \operatorname{gr}R = \operatorname{gr}\widehat{R} .

• (Krull's principal ideal theorem) The height of the ideal generated by elements

x _{1,} ... , x _{s} in a noetherian ring is at most s. Conversely, a prime ideal of height s is minimal over an ideal generated by s elements. (Proof: Let ak{p} be a prime ideal minimal over such an ideal. Then s \ge \dim R _{ak{p}=} \operatorname{ht} ak{p} . The converse was shown in the course of the proof of the fundamental theorem.)

Proof: Let

x _{1,} ... , x _{n} generate a ak{m} _{A} -primary ideal and y _{1,} ... , y _{m} be such that their images generate a ak{m} _{B/ak{m}} _{A} B -primary ideal. Then { ak{m} ^{s} _{B}} \subset (y _{1,} ... , y _{m)} + ak{m} _{A} B for some s. Raising both sides to higher powers, we see some power of ak{m} _{B} is contained in (y _{1,} ... , y _{m,} x _{1,} ... , x _{n)} ; i.e., the latter ideal is ak{m} _{B} -primary; thus, m + n \ge \dim B . The equality is a straightforward application of the going-down property. Q.E.D.

Proof: If

ak{p} _{0} \subsetneq ak{p} _{1} \subsetneq … \subsetneq ak{p} _{n} are a chain of prime ideals in R, then ak{p} _{iR[x]} are a chain of prime ideals in R[x] while ak{p} _{nR[x]} is not a maximal ideal. Thus, \dim R + 1 \le \dim R[x] . For the reverse inequality, let ak{m} be a maximal ideal of R[x] and ak{p} = R \cap ak{m} . Clearly, R[x] _{ak{m}=} R _{ak{p} }[x]_\mathfrak.Since R[x] _{ak{m} } / \mathfrak R_ R[x]_ = (R_/\mathfrakR_)[x]_ is then a localization of a principal ideal domain and has dimension at most one, we get 1 + \dim R \ge 1 + \dim R _{ak{p}\ge} \dim R[x] _{ak{m}} by the previous inequality. Since ak{m} is arbitrary, it follows 1 + \dim R \ge \dim R[x] . Q.E.D.

Nagata's altitude formula

Proof: First suppose

R' is a polynomial ring. By induction on the number of variables, it is enough to consider the case R' = R[x] . Since R is flat over R,$$\backslash dim\; R\text{'}\_\; =\; \backslash dim\; R\_\; +\; \backslash dim\; \backslash kappa(\backslash mathfrak)\; \backslash otimes\_R\; \_.$$By Noether's normalization lemma, the second term on the right side is:$$\backslash dim\; \backslash kappa(\backslash mathfrak)\; \backslash otimes\_R\; R\text{'}\; -\; \backslash dim\; \backslash kappa(\backslash mathfrak)\; \backslash otimes\_R\; R\text{'}/\backslash mathfrak\text{'}\; =\; 1\; -\; \backslash operatorname\_\; \backslash kappa(\backslash mathfrak\text{'})\; =\; \backslash operatorname\_R\; R\text{'}\; -\; \backslash operatorname\; \backslash kappa(\backslash mathfrak\text{'}).$$Next, suppose R' is generated by a single element; thus, R' = R[x]/I . If I = 0, then we are already done. Suppose not. Then R' is algebraic over R and so \operatorname{tr.deg} _{R} R' = 0 . Since R is a subring of R, I \cap R = 0 and so \operatorname{ht} I = \dim R[x] _{I} = \dim Q(R)[x] _{I} = 1 - \operatorname{tr.deg} _{Q(R)} \kappa(I) = 1 since \kappa(I) = Q(R') is algebraic over Q(R) . Let ak{p} ^{\prime} denote the pre-image in R[x] of ak{p}' . Then, as \kappa( ak{p} ^{\prime} ) = \kappa( ak{p}) , by the polynomial case,$$\backslash operatorname\; =\; \backslash operatorname\; \backslash le\; \backslash operatorname\; -\; \backslash operatorname\; =\; \backslash dim\; R\_\; -\; \backslash operatorname\_\; \backslash kappa(\backslash mathfrak\text{'}).$$Here, note that the inequality is the equality if R is catenary. Finally, working with a chain of prime ideals, it is straightforward to reduce the general case to the above case. Q.E.D.

See also: Quasi-unmixed ring.

Homological methods

Regular rings

Let R be a noetherian ring. The projective dimension of a finite R-module M is the shortest length of any projective resolution of M (possibly infinite) and is denoted by

\operatorname{pd} _{R} M . We set \operatorname{gl.dim} R = \sup \{ \operatorname{pd} _{R} M \mid Misafinitemodule \} ; it is called the global dimension of R.

Assume R is local with residue field k.

Proof: We claim: for any finite R-module M,$$\backslash operatorname\_R\; M\; \backslash le\; n\; \backslash Leftrightarrow\; \backslash operatorname^R\_(M,\; k)\; =\; 0.$$By dimension shifting (cf. the proof of Theorem of Serre below), it is enough to prove this for

n = 0 . But then, by the local criterion for flatness, \operatorname{Tor} ^{R} _{1 (M,} k) = 0 ⇒ M flat ⇒ M free ⇒ \operatorname{pd} _{R(M)} \le 0 . Now,$$\backslash operatorname\; R\; \backslash le\; n\; \backslash Rightarrow\; \backslash operatorname\_R\; k\; \backslash le\; n\; \backslash Rightarrow\; \backslash operatorname^R\_(-,\; k)\; =\; 0\; \backslash Rightarrow\; \backslash operatorname\_R\; -\; \backslash le\; n\; \backslash Rightarrow\; \backslash operatorname\; R\; \backslash le\; n,$$completing the proof. Q.E.D.

Remark: The proof also shows that

\operatorname{pd} _{R} K = \operatorname{pd} _{R} M - 1 if M is not free and K is the kernel of some surjection from a free module to M.

Proof: If

\operatorname{pd} _{R} M = 0 , then M is R-free and thus M ⊗ R _{1} is R _{1} -free. Next suppose \operatorname{pd} _{R} M > 0 . Then we have: \operatorname{pd} _{R} K = \operatorname{pd} _{R} M - 1 as in the remark above. Thus, by induction, it is enough to consider the case \operatorname{pd} _{R} M = 1 . Then there is a projective resolution: 0 \to P _{1} \to P _{0} \to M \to 0 , which gives:$$\backslash operatorname^R\_1(M,\; R\_1)\; \backslash to\; P\_1\; \backslash otimes\; R\_1\; \backslash to\; P\_0\; \backslash otimes\; R\_1\; \backslash to\; M\; \backslash otimes\; R\_1\; \backslash to\; 0.$$But \operatorname{Tor} ^{R} _{1 (M,} R _{1)} = {} _{f} M = \{ m \in M \mid fm = 0 \} = 0 . Hence, \operatorname{pd} _{R} (M ⊗ R _{1)} is at most 1. Q.E.D.

Proof: If R is regular, we can write

k = R/(f _{1,} ... , f _{n)} , f _{i} a regular system of parameters. An exact sequence 0 \to M \overset{f}\to M \to M _{1} \to 0 , some f in the maximal ideal, of finite modules, \operatorname{pd} _{R} M < inf ty , gives us:$$0\; =\; \backslash operatorname^R\_(M,\; k)\; \backslash to\; \backslash operatorname^R\_(M\_1,\; k)\; \backslash to\; \backslash operatorname^R\_i(M,\; k)\; \backslash overset\backslash to\; \backslash operatorname^R\_i(M,\; k),\; \backslash quad\; i\; \backslash ge\; \backslash operatorname\_R\; M.$$But f here is zero since it kills k. Thus, \operatorname{Tor} ^{R} _{i +1} (M _{1,} k) \simeq \operatorname{Tor} ^{R} _{i(M,} k) and consequently \operatorname{pd} _{R} M _{1} = 1 + \operatorname{pd} _{R} M . Using this, we get:$$\backslash operatorname\_R\; k\; =\; 1\; +\; \backslash operatorname\_R\; (R/(f\_1,\; \backslash dots,\; f\_))\; =\; \backslash cdots\; =\; n.$$The proof of the converse is by induction on \dim R . We begin with the inductive step. Set R _{1} = R/f _{1} R , f _{1} among a system of parameters. To show R is regular, it is enough to show R _{1} is regular. But, since \dim R _{1} < \dim R , by inductive hypothesis and the preceding lemma with M = ak{m} ,

The basic step remains. Suppose

\dim R = 0 . We claim \operatorname{gl.dim}R = 0 if it is finite. (This would imply that R is a semisimple local ring; i.e., a field.) If that is not the case, then there is some finite module M with 0 < \operatorname{pd} _{R} M < inf ty and thus in fact we can find M with \operatorname{pd} _{R} M = 1 . By Nakayama's lemma, there is a surjection F \to M from a free module F to M whose kernel K is contained in ak{m} F . Since \dim R = 0 , the maximal ideal ak{m} is an associated prime of R; i.e., ak{m} = \operatorname{ann}(s) for some nonzero s in R. Since K \subset ak{m} F , s K = 0 . Since K is not zero and is free, this implies s = 0 , which is absurd. Q.E.D.

Proof: Let R be a regular local ring. Then

\operatorname{gr}R \simeq k[x _{1,} ... , x _{d]} , which is an integrally closed domain. It is a standard algebra exercise to show this implies that R is an integrally closed domain. Now, we need to show every divisorial ideal is principal; i.e., the divisor class group of R vanishes. But, according to Bourbaki, Algèbre commutative, chapitre 7, §. 4. Corollary 2 to Proposition 16, a divisorial ideal is principal if it admits a finite free resolution, which is indeed the case by the theorem. Q.E.D.

Depth

Let R be a ring and M a module over it. A sequence of elements

x _{1,} ... , x _{n} in R is called an M-regular sequence if x _{1} is not a zero-divisor on M and x _{i} is not a zero divisor on M/(x _{1,} ... , x _{i-1} )M for each i = 2 , ... , n . A priori, it is not obvious whether any permutation of a regular sequence is still regular (see the section below for some positive answer.)

Let R be a local Noetherian ring with maximal ideal

ak{m} and put k = R/ ak{m} . Then, by definition, the depth of a finite R-module M is the supremum of the lengths of all M-regular sequences in ak{m} . For example, we have \operatorname{depth} M = 0 \Leftrightarrow ak{m} consists of zerodivisors on M \Leftrightarrow ak{m} is associated with M. By induction, we find$$\backslash operatorname\; M\; \backslash le\; \backslash dim\; R/$$for any associated primes ak{p} of M. In particular, \operatorname{depth} M \le \dim M . If the equality holds for M = R, R is called a Cohen–Macaulay ring.

Example: A regular Noetherian local ring is Cohen–Macaulay (since a regular system of parameters is an R-regular sequence.)

In general, a Noetherian ring is called a Cohen–Macaulay ring if the localizations at all maximal ideals are Cohen–Macaulay. We note that a Cohen–Macaulay ring is universally catenary. This implies for example that a polynomial ring

k[x _{1,} ... , x _{d]} is universally catenary since it is regular and thus Cohen–Macaulay.

Proof: We first prove by induction on n the following statement: for every R-module M and every M-regular sequence

x _{1,} ... , x _{n} in ak{m} ,The basic step n = 0 is trivial. Next, by inductive hypothesis, \operatorname{Ext} ^{n -1} _{R} (N, M) \simeq \operatorname{Hom} _{R(N,} M/(x _{1,} ... , x _{n-1} ) M) . But the latter is zero since the annihilator of N contains some power of x _{n} . Thus, from the exact sequence 0 \to M \overset{x _{1}} \to M \to M _{1} \to 0 and the fact that x _{1} kills N, using the inductive hypothesis again, we get$$\backslash operatorname^n\_R(N,\; M)\; \backslash simeq\; \backslash operatorname^\_R(N,\; M/x\_1\; M)\; \backslash simeq\; \backslash operatorname\_R(N,\; M/(x\_1,\; \backslash dots,\; x\_n)\; M),$$proving . Now, if n < \operatorname{depth} M , then we can find an M-regular sequence of length more than n and so by we see \operatorname{Ext} ^{n(N,} _{R} M) = 0 . It remains to show \operatorname{Ext} ^{n(N,} _{R} M) \ne 0 if n = \operatorname{depth} M . By we can assume n = 0. Then ak{m} is associated with M; thus is in the support of M. On the other hand, ak{m} \in \operatorname{Supp}(N). It follows by linear algebra that there is a nonzero homomorphism from N to M modulo ak{m} ; hence, one from N to M by Nakayama's lemma. Q.E.D.

The Auslander–Buchsbaum formula relates depth and projective dimension.

Proof: We argue by induction on

\operatorname{pd} _{R} M , the basic case (i.e., M free) being trivial. By Nakayama's lemma, we have the exact sequence 0 \to K \overset{f}\to F \to M \to 0 where F is free and the image of f is contained in ak{m} F . Since \operatorname{pd} _{R} K = \operatorname{pd} _{R} M - 1 , what we need to show is \operatorname{depth} K = \operatorname{depth} M + 1 .Since f kills k, the exact sequence yields: for any i,$$\backslash operatorname\_R^i(k,\; F)\; \backslash to\; \backslash operatorname\_R^i(k,\; M)\; \backslash to\; \backslash operatorname\_R^(k,\; K)\; \backslash to\; 0.$$Note the left-most term is zero if i < \operatorname{depth} R . If i < \operatorname{depth} K - 1 , then since \operatorname{depth} K \le \operatorname{depth} R by inductive hypothesis, we see \operatorname{Ext} ^{i(k,} _{R} M) = 0 . If i = \operatorname{depth} K - 1 , then \operatorname{Ext} ^{i +1} _{R} (k, K) \ne 0 and it must be \operatorname{Ext} ^{i(k,} _{R} M) \ne 0 . Q.E.D.

As a matter of notation, for any R-module M, we let$$\backslash Gamma\_(M)\; =\; \backslash \; =\; \backslash .$$One sees without difficulty that

\Gamma _{ak{m} } is a left-exact functor and then let H ^{j} _{ak{m} } = R^j \Gamma_ be its j-th right derived functor, called the local cohomology of R. Since \Gamma _{ak{m} }(M) = \varinjlim \operatorname_R(R/\mathfrak^j, M), via abstract nonsense,$$H^i\_(M)\; =\; \backslash varinjlim\; \backslash operatorname^i\_R\; (R/^j,\; M).$$This observation proves the first part of the theorem below.Proof: 1. is already noted (except to show the nonvanishing at the degree equal to the depth of M; use induction to see this) and 3. is a general fact by abstract nonsense. 2. is a consequence of an explicit computation of a local cohomology by means of Koszul complexes (see below). \square

Koszul complex

See main article: Koszul complex.

Let R be a ring and x an element in it. We form the chain complex K(x) given by

K(x) _{i} = R for i = 0, 1 and K(x) _{i} = 0 for any other i with the differential$$d:\; K\_1(R)\; \backslash to\; K\_0(R),\; \backslash ,\; r\; \backslash mapsto\; xr.$$For any R-module M, we then get the complex K(x, M) = K(x) ⊗ _{R} M with the differential d ⊗ 1 and let \operatorname{H} _{*(x,} M) = \operatorname{H} _{*(K(x,} M)) be its homology. Note:$$\backslash operatorname\_0(x,\; M)\; =\; M/xM,$$$$\backslash operatorname\_1(x,\; M)\; =\; \_x\; M\; =\; \backslash .$$

More generally, given a finite sequence

x _{1,} ... , x _{n} of elements in a ring R, we form the tensor product of complexes:$$K(x\_1,\; \backslash dots,\; x\_n)\; =\; K(x\_1)\; \backslash otimes\; \backslash dots\; \backslash otimes\; K(x\_n)$$and let \operatorname{H} _{*(x} _{1,} ... , x _{n,} M) = \operatorname{H} _{*(K(x} _{1,} ... , x _{n,} M)) its homology. As before,$$\backslash operatorname\_0(\backslash underline,\; M)\; =\; M/(x\_1,\; \backslash dots,\; x\_n)M,$$$$\backslash operatorname\_n(\backslash underline,\; M)\; =\; \backslash operatorname\_M((x\_1,\; \backslash dots,\; x\_n)).$$

We now have the homological characterization of a regular sequence.

A Koszul complex is a powerful computational tool. For instance, it follows from the theorem and the corollary$$\backslash operatorname^i\_(M)\; \backslash simeq\; \backslash varinjlim\; \backslash operatorname^i(K(x\_1^j,\; \backslash dots,\; x\_n^j;\; M))$$(Here, one uses the self-duality of a Koszul complex; see Proposition 17.15. of Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.)

Another instance would be

Remark: The theorem can be used to give a second quick proof of Serre's theorem, that R is regular if and only if it has finite global dimension. Indeed, by the above theorem,

\operatorname{Tor} ^{R} _{s(k,} k) \ne 0 and thus \operatorname{gl.dim} R \ge s . On the other hand, as \operatorname{gl.dim} R = \operatorname{pd} _{R} k , the Auslander–Buchsbaum formula gives \operatorname{gl.dim} R = \dim R . Hence, \dim R \le s \le \operatorname{gl.dim} R = \dim R .

We next use a Koszul homology to define and study complete intersection rings. Let R be a Noetherian local ring. By definition, the first deviation of R is the vector space dimension$$\backslash epsilon\_1(R)\; =\; \backslash dim\_k\; \backslash operatorname\_1(\backslash underline)$$where

\underline{x} = (x _{1,} ... , x _{d)} is a system of parameters. By definition, R is a complete intersection ring if \dim R + \epsilon _{1(R)} is the dimension of the tangent space. (See Hartshorne for a geometric meaning.)

Injective dimension and Tor dimensions

Let R be a ring. The injective dimension of an R-module M denoted by

\operatorname{id} _{R} M is defined just like a projective dimension: it is the minimal length of an injective resolution of M. Let \operatorname{Mod} _{R} be the category of R-modules.

Proof: Suppose

\operatorname{gl.dim} R \le n . Let M be an R-module and consider a resolution$$0\; \backslash to\; M\; \backslash to\; I\_0\; \backslash overset\backslash to\; I\_1\; \backslash to\; \backslash dots\; \backslash to\; I\_\; \backslash overset\backslash to\; N\; \backslash to\; 0$$where I _{i} are injective modules. For any ideal I,$$\backslash operatorname^1\_R(R/I,\; N)\; \backslash simeq\; \backslash operatorname^2\_R(R/I,\; \backslash operatorname(\backslash phi\_))\; \backslash simeq\; \backslash dots\; \backslash simeq\; \backslash operatorname\_R^(R/I,\; M),$$which is zero since \operatorname{Ext} ^{n +1} _{R} (R/I, - ) is computed via a projective resolution of R/I . Thus, by Baer's criterion, N is injective. We conclude that \sup \{ \operatorname{id} _{R} M | M \} \le n . Essentially by reversing the arrows, one can also prove the implication in the other way. Q.E.D.

The theorem suggests that we consider a sort of a dual of a global dimension:$$\backslash operatorname\; =\; \backslash inf\; \backslash \; .$$It was originally called the weak global dimension of R but today it is more commonly called the Tor dimension of R.

Remark: for any ring R,

\operatorname{w.gl.dim} R \le \operatorname{gl.dim} R .

Dimensions of non-commutative rings

Let A be a graded algebra over a field k. If V is a finite-dimensional generating subspace of A, then we let

f(n) = \dim _{k} V ^{n} and then put$$\backslash operatorname(A)\; =\; \backslash limsup\_\; .$$It is called the Gelfand–Kirillov dimension of A. It is easy to show \operatorname{gk}(A) is independent of a choice of V. Given a graded right (or left) module M over A one may similarly define the Gelfand-Kirillov dimension {gk}(M) of M.

Example: If A is finite-dimensional, then gk(A) = 0. If A is an affine ring, then gk(A) = Krull dimension of A.

Example: If

A _{n} = k[x _{1,} ..., x _{n,} \partial _{1,} ..., \partial _{n]} is the n-th Weyl algebra then \operatorname{gk}(A _{n)} = 2 n.

See also

• Multiplicity theory
• Bass number
• Perfect complex
• amplitude

References

• • Part II of .
• Chapter 10 of .
• Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.
• Book: Matsumura . H. . Commutative Ring Theory . M. Reid . Cambridge Studies in Advanced Mathematics . 8 . 1987 . Cambridge University Press . 978-0-521-36764-6 . 10.1017/CBO9781139171762.
  • Book: Weibel, Charles A. . Charles Weibel . An Introduction to Homological Algebra . 1995 . Cambridge University Press .