Betti numbers of modules over Noetherian rings with ... - IPM

Chapter 1: Preliminaries 12
1.4 Local cohomology and Local Duality
define
Let I be an ideal of a commutative Noetherian ring R and M an R-module. We
It is easy to see that H 0 I(M) = lim
H 0 I(M) = {m ∈ M : (∃i)I i m = 0}
−→
n≥0
HomR(R/I n , M). Since each Hom(R/I i , −) is left
exact and lim
−→ is exact, we see that H 0 I is an additive left exact functor from R-mod
to itself. The qthe right derived functor of H 0 I(−) applied on M is shown by H q
I (M)
and since the direct limit is exact, we have
H q
I
(M) = limExt
−→
n≥0
q
R (R/In , M)
So that H i I(M) is the ith cohomology module of the complex obtained by applying
H 0 I(−) to an injective resolution of M.
Our standard reference for local cohomology is the book of Brodmann and Sharp
[15] which gives a detailed and comprehensive account of this material. It covers
important applications and uses detailed examples designed to illustrate the geomet-
rical significance of aspects of local cohomology. In the following we recall some
preliminary facts.
Theorem 1.4.1. Let (R, m, k) be a Noetherian local ring or a homogeneous K-
algebra, and let M be a finitely generated R-module. Local cohomology detects depth
and dimension.
depth(M) = min{i | H i m(M) = 0},
dim(M) = max{i | H i m(M) = 0} Grothendieck’s non-anishing.