Generalized divisors and reflexive sheaves

We introduce the notions of depth and of when a local ring (or module over a local ring) is “S2”. These notions are found in most books on commutative algebra, see for example [Mat89, Section 16] or [Eis95, Section 18]. Another excellent book that is focussed on these ideas is [BH93]. We won’t be focussing on the commutative algebra, but one should be aware, at the very least, that this background exists. In particular, we won’t really use any of the theory on Cohen-Macaulay rings. However, since one ought to build up the same machinery in order to define S2, we include these definitions as well. All rings will be assumed to be noetherian. Definition 1.1. Let (R,m) be a local ring and let M be a finite R-module (which means finitely generated). An element r ∈ R is said to be M-regular if rx 6 = 0 for all x ∈M, x 6 = 0 (in other words, r is not a zero divisor on M). A sequence of elements r1,..., rn ∈ R is said to be M-regular if (i) ri is a regular M/(r1,..., ri−1) element for all i ≥ 1 (in particular r1 is M-regular) (ii) (r1,..., rn)M (M (that is, the containment is proper). Remark 1.2. If condition (i) is satisfied, but condition (ii) is not necessarily satisfied then such a sequence is called weakly M-regular. Remark 1.3. Notice that condition (ii) implies in a regular sequence, all ri ∈ m ∈ R. Remark 1.4. One can (and often does) consider the notion of a regular sequence without the hypothesis R is local or that M if finitely generated. However, in that case these notions are not as well behaved (for example, whether or not a sequence is regular depends on the order of the sequence without these hypotheses). Definition 1.5. Let (R,m) be a local ring and suppose that M is a finite R-module. We say that M has depth k (or in some books, the grade of m on M is k) if the maximal length of an M-regular sequence is k. We state a few basic properties of depth below. Please see any of the above references for proofs. Proposition 1.6. Let (R,m) be a local ring and suppose that M if a finite R-module. Then (a) If we define a maximal M-regular sequence in the obvious way (it can’t be extended to a longer M-regular sequence), then every maximal M-regular sequence has the same length. (b) Modules have a notion of dimension (dimM = dim(A/AnnRM)) and one has depthM ≤ dimM. In particular, R is a finite R-module so we have depthR ≤ dimR