A COMPLEX OF MODULES AND ITS APPLICATIONS TO LOCAL COHOMOLOGY AND EXTENSION FUNCTORS

• Published in: Mathematica scandinavica 2015-01, Vol.117 (1), p.150-160
• Main Author: BAHMANPOUR, KAMAL
• Format: Article
• Language: English
• ISSN: 0025-5521; 1903-1807
• DOI: 10.7146/math.scand.a-22240

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, (i) if projdim R(M) < ∞, then injdim _R\left( {D\left( {H_m^n\left( m \right)} \right)} \right)$ < ∞ and (ii) if injdim R(M) < ∞, then projdim _R\left( {D\left( {H_m^n\left( m \right)} \right)} \right)$ < ∞, where D(–) := HomR(–, E) denotes the Mathis dual functor and E := ER(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 x₁, ..., xk, (k ≥ 1), is an M-regular sequence, then $D\left( {H_{\left( {{x_1},...{x_k}} \right)}^k\left( {D\left( {H_{\left( {{x_1},...{x_k}} \right)}^k\left( M \right)} \right)} \right)} \right) \simeq M.$ In particular, Ann ${H_{\left( {{x_1},...{x_k}} \right)}^k\left( M \right)}$ = Ann M. Moreover, it is shown that if R is a Noetherian ring, M is finitely generated R-module and x₁, ... , xk is an M-regular sequence, then $Ext_R^{k + 1}\left( {R/\left( {{x_1},...{x_k}} \right),M} \right) = 0$.