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, denotes Krull dimension of a ring and 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 in R,
for any prime ideal in that contracts to . This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of , one cannot have a chain of primes ideals of length .
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 be a noetherian local ring and I a -primary ideal (i.e., it sits between some power of and ). Let be the Poincaré series of the associated graded ring . That is,
where refers to the length of a module (over an artinian ring ). If generate I, then their image in have degree 1 and generate as -algebra. By the Hilbert–Serre theorem, F is a rational function with exactly one pole at of order . Since
we find that the coefficient of in is of the form
That is to say, is a polynomial in n of degree . P is called the Hilbert polynomial of .
We set . We also set to be the minimum number of elements of R that can generate an -primary ideal of R. Our ambition is to prove the fundamental theorem:
Since we can take s to be , we already have from the above. Next we prove by induction on . Let be a chain of prime ideals in R. Let 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 . (This essentially follows from the Artin-Rees lemma; see Hilbert-Samuel function for the statement and the proof.) In , the chain becomes a chain of length and so, by inductive hypothesis and again by the degree estimate,
The claim follows. It now remains to show More precisely, we shall show:
(Notice: is then -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 be a noetherian local ring and put . Then
, since a basis of lifts to a generating set of by Nakayama. If the equality holds, then R is called a regular local ring. , since . (Krull's principal ideal theorem) The height of the ideal generated by elements 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 be a prime ideal minimal over such an ideal. Then . The converse was shown in the course of the proof of the fundamental theorem.)
Proof: Let generate a -primary ideal and be such that their images generate a -primary ideal. Then for some s. Raising both sides to higher powers, we see some power of is contained in ; i.e., the latter ideal is -primary; thus, . The equality is a straightforward application of the going-down property. Q.E.D.