Finiteness properties of extension functors of cofinite modules
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Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T) ≤ 1. In this paper, for any finitely generated R-module N with support in V (I), we show that the R-modules Ext i R(T,N) are finitely generated for all integers i ≥ 0. This immediately implies that if I has dimension one (i.e., dimR/I = 1), then Exti R(Hj I (M),N) is finitely generated for all integers i, j≥ 0, and all finitely generated R-modules M and N, with Supp(N) ⊆ V (I).
Abstract
Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T) ≤ 1. In this paper, for any finitely generated R-module N with support in V (I), we show that the R-modules Ext i R(T,N) are finitely generated for all integers i ≥ 0. This immediately implies that if I has dimension one (i.e., dimR/I = 1), then Exti R(Hj I (M),N) is finitely generated for all integers i, j≥ 0, and all finitely generated R-modules M and N, with Supp(N) ⊆ V (I).
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