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

Chapter 1: Preliminaries 13
toll:
The following result which is known as the local duality theorem will be a handy
Theorem 1.4.2. Let (R, m, k) be a Cohen-Macaulay complete local ring of dimension
d with canonical module ωR. Then for all finitely generated R-modules M and all
integers i there exists natural isomorphisms
H i m(M) ∼ = HomR(Ext d−i
R (M, ωR), E(k)), and
Ext i R(M, ωR) ∼ = HomR(H d−i
m (M), E(k)).
In particular, Ext d−i
R (M, ωR) = 0 for all i < dim M.
(1.4)
Theorem 1.4.3. Let R, S be two Noetherian rings, I an ideal of R and ϕ : R −→
S be a ring map. Then H i I(M) ∼ = H i IS(M) for every S-module M. Consequently,
H i I(M) = 0 for i > dim(R/annR(M)).
1.5 Castelnuovo-Mumford regularity
Let K be a field and S = K[x1, · · · , xr] and let
F : · · · → Fi → Fi−1 → · · ·
be a graded complex of free S-modules, with Fi =
S(−ai,j). The Castelnuovo-
Mumford regularity, or simply regularity, of F is the supremum of the numbers ai,j −i.
The regularity of a finitely generated graded S-module M is the regularity of a minimal
graded free resolution of M. We will write reg(M) for this number. The regularity of
an ideal is an important measure of how complicated the ideal is. The above definition
of regularity shows how the regularity of a module governs the degrees appearing in
j