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).
(Kaplansky's Commutative rings gives a good account of the non-noetherian case.)
refers to the length of a module (over an artinian ring
-primary ideal of R. Our ambition is to prove the fundamental theorem:
and x a nonzero nonunit element in D. Since x is not a zero-divisor, we have the exact sequence
(This essentially follows from the Artin–Rees lemma; see Hilbert–Samuel function for the statement and the proof.)
where the equality holds if either (a) R is universally catenary and R' is finitely generated R-algebra or (b) R' is a polynomial ring over R. Proof:[2] First suppose
; it is called the global dimension of R. Assume R is local with residue field k. Lemma —
(This would imply that R is a semisimple local ring; i.e., a field.)
Corollary — A regular local ring is a unique factorization 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.
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.
Then, by definition, the depth of a finite R-module M is the supremum of the lengths of all M-regular sequences in
Proof: We first prove by induction on n the following statement: for every R-module M and every M-regular sequence
Theorem — Let M be a finite module over a noetherian local ring R. If
be its j-th right derived functor, called the local cohomology of R. Since
Then 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).
of elements in a ring R, we form the tensor product of complexes:
Theorem — Suppose R is Noetherian, M is a finite module over R and
We next use a Koszul homology to define and study complete intersection rings.
is defined just like a projective dimension: it is the minimal length of an injective resolution of M. Let
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,
Proposition — A ring has weak global dimension zero if and only if it is von Neumann regular.
Let A be a graded algebra over a field k. If V is a finite-dimensional generating subspace of A, then we let
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
If A is an affine ring, then gk(A) = Krull dimension of A. Bernstein's inequality — See [1] Example: If