论文标题
模块辅助性的标准
A criterion for cofiniteness of modules
论文作者
论文摘要
让$ a $为可交换的noetherian戒指，$ \ frak a $是$ a $，$ m，n $的理想，是非负整数，让$ m $是$ a $ a $ module，以至于$ \ ext^i_a（a//\ frak a，m）$是$ i \ i \ leq m+n $ wity willy Genaline Geralisy Genaline Geralinaling Genaline Geralinaling Genaline Geralinaling。我们定义一个模块的类$ \ cs_n（\ frak a）$，我们假设$ h _ {\ frak a}^s（m）\ in \ cs_ {n}（\ frak a）$ s \ s \ leq m $。我们表明，如果$ n = 1 $ n = 1 $或$ n = 1 $或$ n \ geq 2 $和$ n \ geq 2 $和$ h_a^{i}（a//\ frak a，a/h _ _ _ _ _ {\ frak a}， $ 1 \ leq t \ leq n-1 $，$ i \ leq t-1 $和$ s \ leq m $。如果$ a $是\ cs_n（\ frak a）的尺寸$ d $和$ m \的环，则对于任何理想的$ \ frak a $ a $ dimension $ \ leq d-1 $，那么我们证明$ m \ in \ cs_n（\ frak a）$ a $ \ frak a $ \ frak a $ a $ a $ a $ a $ a $。
Let $A$ be a commutative noetherian ring, $\frak a$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $\Ext^i_A(A/\frak a,M)$ is finitely generated for all $i\leq m+n$. We define a class $\cS_n(\frak a)$ of modules and we assume that $H_{\frak a}^s(M)\in\cS_{n}(\frak a)$ for all $s\leq m$. We show that $H_{\frak a}^s(M)$ is $\frak a$-cofinite for all $s\leq m$ if either $n=1$ or $n\geq 2$ and $\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$ is finitely generated for all $1\leq t\leq n-1$, $i\leq t-1$ and $s\leq m$. If $A$ is a ring of dimension $d$ and $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of dimension $\leq d-1$, then we prove that $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of $A$.
