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Math Notes

University Mathematics

ConceptComplete

Cohen-Macaulay Rings

Cohen-Macaulay rings occupy a central position in commutative algebra, sitting between the class of all Noetherian rings and the class of regular rings. They are characterized by the equality of depth and dimension.

Depth and Regular Sequences

Definition

Let (R,m)(R, \mathfrak{m}) be a Noetherian local ring and MM a finitely generated RR-module. A sequence x1,…,xr∈mx_1, \ldots, x_r \in \mathfrak{m} is a regular sequence (or MM-sequence) if xix_i is a non-zero-divisor on M/(x1,…,xi−1)MM/(x_1, \ldots, x_{i-1})M for each i=1,…,ri = 1, \ldots, r, and M/(x1,…,xr)M≠0M/(x_1, \ldots, x_r)M \neq 0. The depth of MM is depth⁡(M)=sup⁡{r:there exists a regular sequence x1,…,xr on M}\operatorname{depth}(M) = \sup\{r : \text{there exists a regular sequence } x_1, \ldots, x_r \text{ on } M\}

Depth is always bounded above by dimension: depth⁡(M)≤dim⁡(M)\operatorname{depth}(M) \leq \dim(M) for any finitely generated module MM over a Noetherian local ring.

Definition

A Noetherian local ring (R,m)(R, \mathfrak{m}) is Cohen-Macaulay if depth⁡(R)=dim⁡(R)\operatorname{depth}(R) = \dim(R). A Noetherian ring RR is Cohen-Macaulay if RpR_\mathfrak{p} is Cohen-Macaulay for all prime ideals p\mathfrak{p}.

Examples and Non-Examples

ExampleCohen-Macaulay rings
  1. Every regular local ring is Cohen-Macaulay (a regular system of parameters is a regular sequence).
  2. Every 11-dimensional reduced local ring is Cohen-Macaulay.
  3. k[x,y]/(xy)k[x, y]/(xy) (the coordinate ring of two intersecting lines) is Cohen-Macaulay.
  4. k[x,y,z]/(xz,yz)k[x, y, z]/(xz, yz) (a line meeting a plane at one point) is not Cohen-Macaulay: dim⁡=2\dim = 2 but depth⁡=1\operatorname{depth} = 1.

Characterizations

Theorem8.3Cohen-Macaulay Characterizations

For a Noetherian local ring (R,m)(R, \mathfrak{m}) of dimension dd, the following are equivalent:

  1. RR is Cohen-Macaulay
  2. Every system of parameters is a regular sequence
  3. There exists a system of parameters that is a regular sequence
  4. For every ideal II generated by a regular sequence, ht⁡(I)=\operatorname{ht}(I) = number of generators of II
RemarkUnmixedness

A key property of Cohen-Macaulay rings is unmixedness: if (R,m)(R, \mathfrak{m}) is Cohen-Macaulay and I=(x1,…,xr)I = (x_1, \ldots, x_r) with ht⁡(I)=r\operatorname{ht}(I) = r, then all associated primes of R/IR/I have height exactly rr (no embedded primes). This makes Cohen-Macaulay rings the natural setting for intersection theory in algebraic geometry.