论文标题
相对顺序的Cohen-Macaulay和相对Cohen-Macaulay模块的一些表征
Some Characterizations of Relative Sequentially Cohen-Macaulay and Relative Cohen-Macaulay Modules
论文作者
论文摘要
令$ m $为$ r $ -module上的noetherian ring $ r $和$ \ mathfrak {a} $是$ r $的理想，$ c = {\ rm cd}（\ mathfrak {a}，m）$。首先，我们证明$ m $是有限的$ \ mathfrak {a} $ - 相对cohen-macaulay，并且仅当$ {\ rm h} _i（λ_ {\ mathfrak {\ mathfrak {a}}}}}}（\ rm h} {\ rm h} $ {\ rm H} _C（λ_{\ Mathfrak {a}}（{\ rm H} _ {\ Mathfrak {\ Mathfrak {a}}}^c（m））\ cong \ cong \ wideHat {m}接下来，在$ \ mathfrak {a} $ - 相对相对Cohen-Macaulay本地环$（r，\ Mathfrak {m}）$上，我们提供了$ \ mathfrak {a} $ - 相对顺序cohen-macaulay模块$ m $ $ \ mathfrak $ $ \ a a} $ - a} $ - a} $ a的表征$ {\ rm ext}^{d-i} _ {r}（m，{\ rm d} _ {\ mathfrak {a}}} $ for $ i \ geq 0 $，其中$ {\ rm d} _ {\ rm d} _ {\ rm d} h}^d _ {\ mathfrak {a}}（r），{\ rm e}（r/\ mathfrak {m}））$ and $ d = {\ rm cd}（\ rm cd}（\ mathfrak {a a}，r）$。最后，我们提供了$ \ mathfrak {a} $ - 相对顺序的cohen-macaulay模块$ m $的另一个特征，以消失本地同源模块$ {\ rm h} _j（λ_{λ_{\ mathfrak {\ mathfrak {a}}（\ rm rm） h} _ {\ mathfrak {a}}^i（m））= 0 $ for ALL $ 0 \ leq i \ leq c $，以及所有$ j \ neq i $。
Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be an ideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ is finite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rm H}_i(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M)))=0$ for all $i\neq c$ and ${\rm H}_c(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M))) \cong \widehat{M}^{\mathfrak{a}}$. Next, over an $\mathfrak{a}$-relative Cohen-Macaulay local ring $(R,\mathfrak{m})$, we provide a characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of $\mathfrak{a}$-relative Cohen-Macaulayness of the $R$-modules ${\rm Ext}^{d-i}_{R}(M,{\rm D}_{\mathfrak{a}})$ for all $i\geq 0$, where ${\rm D}_{\mathfrak{a}} = {\rm Hom}_R({\rm H}^d_{\mathfrak{a}}(R),{\rm E}(R/\mathfrak{m}))$ and $d={\rm cd}(\mathfrak{a},R)$. Finally, we provide another characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of vanishing of the local homology modules ${\rm H}_j(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^i(M)))=0$ for all $0\leq i\leq c$ and for all $j\neq i$.
