Dimension theory (algebra)

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 the 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, formula_1 denotes Krull dimension of a ring and formula_2 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
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 formula_4 in "R",
This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of formula_10, one cannot have a chain of primes ideals of length formula_11.
Since an artinian ring (e.g., a field) has dimension zero, by induction one gets a formula: for an artinian ring "R",
Local rings.
Fundamental theorem.
Let formula_13 be a noetherian local ring and "I" a formula_14-primary ideal (i.e., it sits between some power of formula_14 and formula_14). Let formula_17 be the Poincaré series of the associated graded ring formula_18. That is,
where formula_20 refers to the length of a module (over an artinian ring formula_21). If formula_22 generate "I", then their image in formula_23 have degree 1 and generate formula_24 as formula_25-algebra. By the Hilbert–Serre theorem, "F" is a rational function with exactly one pole at formula_26 of order formula_27. Since
we find that the coefficient of formula_29 in formula_30 is of the form
That is to say, formula_32 is a polynomial formula_33 in "n" of degree formula_34. "P" is called the Hilbert polynomial of formula_24.
We set formula_36. We also set formula_37 to be the minimum number of elements of "R" that can generate an formula_14-primary ideal of "R". Our ambition is to prove the fundamental theorem:
Since we can take "s" to be formula_37, we already have formula_41 from the above. Next we prove formula_42 by induction on formula_43. Let formula_44 be a chain of prime ideals in "R". Let formula_45 and "x" a nonzero nonunit element in "D". Since "x" is not a zero-divisor, we have the exact sequence
The degree bound of the Hilbert-Samuel polynomial now implies that formula_47. (This essentially follows from the Artin-Rees lemma; see Hilbert-Samuel function for the statement and the proof.) In formula_48, the chain formula_49 becomes a chain of length formula_50 and so, by inductive hypothesis and again by the degree estimate,
The claim follows. It now remains to show formula_52 More precisely, we shall show:
(Notice: formula_57 is then formula_14-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 formula_13 be a noetherian local ring and put formula_60. Then
Proof: Let formula_69 generate a formula_70-primary ideal and formula_71 be such that their images generate a formula_72-primary ideal. Then formula_73 for some "s". Raising both sides to higher powers, we see some power of formula_74 is contained in formula_75; i.e., the latter ideal is formula_74-primary; thus, formula_77. The equality is a straightforward application of the going-down property. formula_78
Proof: If formula_79 are a chain of prime ideals in "R", then formula_80 are a chain of prime ideals in formula_8 while formula_82 is not a maximal ideal. Thus, formula_83. For the reverse inequality, let formula_14 be a maximal ideal of formula_8 and formula_86. Clearly, formula_87.
Since formula_88 is then a localization of a principal ideal domain and has dimension at most one, we get formula_89 by the previous inequality. Since formula_14 is arbitrary, it follows formula_91. formula_78
Nagata's altitude formula.
 {R'/\mathfrak{p}'} \le \dim R_{\mathfrak{p}} + \operatorname{tr.deg}_{R} {R'}</math>
Proof: First suppose formula_93 is a polynomial ring. By induction on the number of variables, it is enough to consider the case formula_94. Since "R" is flat over "R",
By Noether's normalization lemma, the second term on the right side is:
Next, suppose formula_93 is generated by a single element; thus, formula_98. If "I" = 0, then we are already done. Suppose not. Then formula_93 is algebraic over "R" and so formula_100. Since "R" is a subring of "R", formula_101 and so formula_102
since formula_103 is algebraic over formula_104. Let formula_105 denote the pre-image in formula_8 of formula_107. Then, as formula_108, by the polynomial case,
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. formula_78
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 formula_111. We set formula_112; 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",
By dimension shifting (cf. the proof of Theorem of Serre below), it is enough to prove this for formula_114. But then, by the local criterion for flatness, formula_115
Now, 
completing the proof. formula_78
Remark: The proof also shows that formula_118 if "M" is not free and formula_119 is the kernel of some surjection from a free module to "M".
Proof: If formula_120, then "M" is "R"-free and thus formula_121 is formula_122-free. Next suppose formula_123. Then we have: formula_118 as in the remark above. Thus, by induction, it is enough to consider the case formula_125. Then there is a projective resolution: formula_126, which gives: 
But formula_128 Hence, formula_129 is at most 1. formula_78
Proof: If "R" is regular, we can write formula_131, formula_132 a regular system of parameters. An exact sequence formula_133, some "f" in the maximal ideal, of finite modules, formula_134, gives us:
But "f" here is zero since it kills "k". Thus, formula_136 and consequently formula_137. Using this, we get:
The proof of the converse is by induction on formula_139. We begin with the inductive step. Set formula_140, formula_141 among a system of parameters. To show "R" is regular, it is enough to show formula_122 is regular. But, since formula_143, by inductive hypothesis and the preceding lemma with formula_144,
The basic step remains. Suppose formula_146. We claim formula_147 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 formula_148 with formula_149 and thus in fact we can find "M" with formula_125. By Nakayama's lemma, there is a surjection formula_151 from a free module "F" to "M" whose kernel "K" is contained in formula_152. Since formula_146, the maximal ideal formula_14 is an associated prime of "R"; i.e., formula_155 for some nonzero "s" in "R". Since formula_156, formula_157. Since "K" is not zero and is free, this implies formula_158, which is absurd. formula_78
Proof: Let "R" be a regular local ring. Then formula_160, 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. formula_78
Depth.
Let "R" be a ring and "M" a module over it. A sequence of elements formula_69 in formula_163 is called an "M"-regular sequence if formula_164 is not a zero-divisor on formula_148 and formula_166 is not a zero divisor on formula_167 for each formula_168. "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 formula_14 and put formula_60. Then, by definition, the depth of a finite "R"-module "M" is the supremum of the lengths of all "M"-regular sequences in formula_14. For example, we have formula_172 consists of zerodivisors on "M" formula_173 is associated with "M". By induction, we find
for any associated primes formula_4 of "M". In particular, formula_176. 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 formula_177 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 formula_69 in formula_14,
The basic step "n" = 0 is trivial. Next, by inductive hypothesis, formula_181. But the latter is zero since the annihilator of "N" contains some power of formula_182. Thus, from the exact sequence formula_183 and the fact that formula_164 kills "N", using the inductive hypothesis again, we get
proving (*). Now, if formula_186, then we can find an "M"-regular sequence of length more than "n" and so by (*) we see formula_187. It remains to show formula_188 if formula_189. By (*) we can assume "n" = 0. Then formula_14 is associated with "M"; thus is in the support of "M". On the other hand, formula_191 It follows by linear algebra that there is a nonzero homomorphism from "N" to "M" modulo formula_14; hence, one from "N" to "M" by Nakayama's lemma. formula_78
The Auslander–Buchsbaum formula relates depth and projective dimension.
Proof: We argue by induction on formula_111, the basic case (i.e., "M" free) being trivial. By Nakayama's lemma, we have the exact sequence formula_195 where "F" is free and the image of "f" is contained in formula_152. Since formula_197 what we need to show is formula_198.
Since "f" kills "k", the exact sequence yields: for any "i",
Note the left-most term is zero if formula_200. If formula_201, then since formula_202 by inductive hypothesis, we see formula_203 If formula_204, then formula_205 and it must be formula_206 formula_78
As a matter of notation, for any "R"-module "M", we let
One sees without difficulty that formula_209 is a left-exact functor and then let formula_210 be its "j"-th right derived functor, called the local cohomology of "R". Since formula_211, via abstract nonsense,
This observation proves the first part of the theorem below.
(M) = 0, i < n \}</math>.
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). formula_78
Koszul complex.
Let "R" be a ring and "x" an element in it. We form the chain complex "K"("x") given by formula_218 for "i" = 0, 1 and formula_219 for any other "i" with the differential
For any "R"-module "M", we then get the complex formula_221 with the differential formula_222 and let formula_223 be its homology. Note:
More generally, given a finite sequence formula_69 of elements in a ring "R", we form the tensor product of complexes:
and let formula_228 its homology. As before,
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
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, formula_232 and thus formula_233. On the other hand, as formula_234, the Auslander–Buchsbaum formula gives formula_235. Hence, formula_236.
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
where formula_238 is a system of parameters. By definition, "R" is a complete intersection ring if formula_239 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 formula_240 is defined just like a projective dimension: it is the minimal length of an injective resolution of "M". Let formula_241 be the category of "R"-modules.
Proof: Suppose formula_242. Let "M" be an "R"-module and consider a resolution
where formula_244 are injective modules. For any ideal "I",
which is zero since formula_246 is computed via a projective resolution of formula_25. Thus, by Baer's criterion, "N" is injective. We conclude that formula_248. Essentially by reversing the arrows, one can also prove the implication in the other way. formula_78
The theorem suggests that we consider a sort of a dual of a global dimension:
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", formula_251.
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 formula_252 and then put
It is called the Gelfand–Kirillov dimension of "A". It is easy to show formula_254 is independent of a choice of "V".
Example: If "A" is finite-dimensional, then gk("A") = 0. If "A" is an affine ring, then gk("A") = Krull dimension of "A".
See also: Goldie dimension, Krull–Gabriel dimension.