Abstract

Let n and t be non-negative integers, R a commutative Noetherian ring with $\dim (R)\le n+2,\mathfrak{a}$ an ideal of R, M and N finite R-modules, and X an arbitrary R-module. We prove that if ${\text{Ext}}_R^i(M/\mathfrak{a}M,X)$ is an ${\text{FD}}_{<n}$ R-module for all $i\le t,$ then ${\mathrm{H}}_{\mathfrak{a}}^i(M,X)$ is an $({\text{FD}}_{<n},\mathfrak{a})$-cofinite R-module for all i < t, ${\text{Hom}}_R(R/\mathfrak{a},{\mathrm{H}}_{\mathfrak{a}}^t(M,X))$ is an ${\text{FD}}_{<n}$ R-module, and $\{\mathfrak{p}\in {\text{Ass}}_R({\mathrm{H}}_{\mathfrak{a}}^i(M,X)):\dim (R/\mathfrak{p})\ge n\}$ is a finite set for all $i\le t.$ It shows that ${\mathrm{H}}_{\mathfrak{a}}^i(M,N)$ is $({\text{FD}}_{<n},\mathfrak{a})$-cofinite and $\{\mathfrak{p}\in {\text{Ass}}_R({\mathrm{H}}_{\mathfrak{a}}^i(M,N)):\dim (R/\mathfrak{p})\ge n\}$ is finite for all i. In particular, ${\mathrm{H}}_{\mathfrak{a}}^i(M,N)$ is $\mathfrak{a}$-cofinite for all i whenever $\dim (R)\le 2.$ Also, ${\text{Ass}}_R({\mathrm{H}}_{\mathfrak{a}}^i(M,N))$ is finite for all i when R is semi-local with $\dim (R)\le 3.$

摘要

令n和t为非负整数，R为可交换的诺特环$\mathrm{暗淡}(\mathrm{电阻})\le n+2,\mathfrak{一种}$R、M和N有限R模的理想，X是任意R模。我们证明如果${\text{分机}}_{\mathrm{电阻}}^{\mathrm{一世}}(米/\mathfrak{一种}米,X)$ 是一个 ${\text{FD}}_{<n}$ 所有人的R模块$\mathrm{一世}\le 吨,$ 然后 ${\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,X)$ 是一个 $({\text{FD}}_{<n},\mathfrak{一种})$-cofinite R -module 对于所有i  <  t，${\text{坎}}_{\mathrm{电阻}}(\mathrm{电阻}/\mathfrak{一种},{\mathrm{H}}_{\mathfrak{一种}}^{吨}(米,X))$ 是一个 ${\text{FD}}_{<n}$ R模块，和$\{\mathfrak{磷}\in {\text{屁股}}_{\mathrm{电阻}}({\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,X))：\mathrm{暗淡}(\mathrm{电阻}/\mathfrak{磷})\ge n\}$ 是所有的有限集 $\mathrm{一世}\le 吨.$ 它表明 ${\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,N)$ 是 $({\text{FD}}_{<n},\mathfrak{一种})$-cofinite 和 $\{\mathfrak{磷}\in {\text{屁股}}_{\mathrm{电阻}}({\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,N))：\mathrm{暗淡}(\mathrm{电阻}/\mathfrak{磷})\ge n\}$对于所有i是有限的。特别是，${\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,N)$ 是 $\mathfrak{一种}$- cofinite for all i每当$\mathrm{暗淡}(\mathrm{电阻})\le 2.$ 还， ${\text{屁股}}_{\mathrm{电阻}}({\mathrm{H}}_{\mathfrak{一种}}^{\mathrm{一世}}(米,N))$当R是半局部的时，对所有i都是有限的$\mathrm{暗淡}(\mathrm{电阻})\le 3.$