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

Chapter 5: Bass numbers of Local Cohomology modules 79
(b) We will again induct on s. For s = 0, the result is clear. Let s > 0 and s − 1
is settled. Assume that E and N are as in the proof of (i). For any 0 ≤ j < s − 1,
we have
Ext s−j
R (R/a, Hj
a(N)) ∼ = Ext s−j
R
Consider the following exact sequence
(R/a, Hj+1
a (M)).
Ext s R(R/a, M) → Ext s R(R/a, M/Γa(M)) → Ext s+1
R (R/a, Γa(M)).
Since Ext s R(R/a, M) is minimax (by assumption) and Ext s+1
R (R/a, Γa(M)) is minimax
(by hypothesis), we have Ext s R(R/a, M/Γa(M)) is minimax and hence Ext s−1
R (R/a, N)
is minimax. This shows that HomR(R/a, H s−1
a (N)) is minimax by induction hypoth-
esis. Now the assertion holds.
Finally we study the local cohomology of generalized reflexive modules intro-
duced by [8]. Suppose that E is the minimal injective cogenerator of the category
of R-modules. An R-module M is called reflexive with respect to E if the canonical
injection
M → HomR(HomR(M, E), E))
is an isomorphism. It is well-known that an R-module M is reflexive (with respect
to E) if and only if M is minimax and R/Ann (M) is a complete semilocal ring,
cf. [8, Theorem 2]. Recall that if N is an arbitrary submodule of a module M,
then M is reflexive if and only if both N and M/N are reflexive, cf. [8, Lemma 5].
Consequently, a finite direct sum of modules is reflexive if and only if each direct
summand is reflexive. In [12, Theorem 2.5] we proved the following:
Theorem 5.2.8. Let M be a finitely generated R-module such that M is reflex-
ive with respect to a minimal injective cogenerator E in the category of R-modules.