A Complex of Modules and Its Applications to Local Cohomology and Extension Functors
Let (R,m) be a commutative Noetherian complete local ring and let M be a non-zero Cohen-Macaulay R-module of dimension n. It is shown that, if projdim_R(M)<∞, then injdim_R(D(H^n_m (M)))<∞, and if injdim_R(M)<∞, then projdim_R(D(H^n_m (M)))<∞, where D(-):= Hom_R(-,E) denotes the Matlis dual functor and E := E_R(R/m) is the injective hull of the residue field R/m. Also, it is shown that if (R,m) is a Noetherian complete local ring, M is a non-zero finitely generated R-module and x_1,…,x_k, (k≥ 1), is an M-regular sequence, then \[ D(H^k_(x_1,…,x_k)(D(H^k_(x_1,…,x_k)(M))))M. \] In particular, Ann H^k_(x_1,…,x_k)(M)=Ann M. Moreover, it is shown that if R is a Noetherian ring, M is a finitely generated R-module and x_1,…,x_k is an M-regular sequence, then \[ Ext^k+1_R(R/(x_1,…,x_k),M)=0. \] , number=1, journal=MATHEMATICA SCANDINAVICA, author=Bahmanpour, Kamal, year=2015, month=Sep., pages=150–160