Let ( R , $$\mathfrak{m}$$ m ) be a commutative Noetherian local ring, $$\mathfrak{a}$$ a be an ideal of R and M a finitely generated R -module such that $$\mathfrak{a}M \ne M$$ a M ≠ M and cd( $$\mathfrak{a}$$ a , M ) — grade( $$\mathfrak{a}$$ a , M ) ⩽ 1, where cd( $$\mathfrak{a}$$ a , M ) is the cohomological dimension of M with respect to $$\mathfrak{a}$$ a and grade( $$\mathfrak{a}$$ a , M ) is the M -grade of $$\mathfrak{a}$$ a . Let D (−):= Hom R (−, E ) be the Matlis dual functor, where E := E ( R / $$\mathfrak{m}$$ m ) is the injective hull of the residue field R / $$\mathfrak{m}$$ m . We show that there exists the following long exact sequence $$\begin{array}{l}{0 \longrightarrow H_{\mathfrak{a}}^{n-2}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{\mathfrak{a}}^{n}(M))) \longrightarrow D(M)} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n-1}(D(H_{\mathfrak{a}}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n+1}(D(H_{\mathfrak{a}}^{n}(M)))} \\ {\quad \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(x_{1},\ldots, x_{n-1})}^{n-1}(M))) \longrightarrow H_{\mathfrak{a}}^{n}(D(H_{(}^{n-1} M))) \longrightarrow \cdots},\end{array}$$ 0 → H a n − 2 ( D ( H a n − 1 ( M ) ) ) → H a n ( D ( H a n ( M ) ) ) → D ( M ) → H a n − 1 ( D ( H a n − 1 ( M ) ) ) → H a n + 1 ( D ( H a n ( M ) ) ) → H a n ( D ( x ( x 1 , … , x n − 1 ) n − 1 ( M ) ) ) → H a n ( D ( H ( n − 1 M ) ) ) → ⋯ , where n := cd( $$\mathfrak{a}$$ a , M ) is a non-negative integer, x 1 ,…, x n −1 is a regular sequence in $$\mathfrak{a}$$ a on M and, for an R -module L , $$H_{\mathfrak{a}}^{n}(L)$$ H a n ( L ) is the i th local cohomology module of L with respect to $$\mathfrak{a}$$ a .

中文翻译：

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局部上同调模块的 Matlis 对偶

设 ( R , $$\mathfrak{m}$$ m ) 是一个可交换的 Noetherian 局部环，$$\mathfrak{a}$$ a 是 R 的理想，M 是一个有限生成的 R 模，使得 $$\ mathfrak{a}M \ne M$$ a M ≠ M and cd( $$\mathfrak{a}$$ a , M ) — Grade( $$\mathfrak{a}$$ a , M ) ⩽ 1, 其中cd( $$\mathfrak{a}$$ a , M ) 是 M 关于 $$\mathfrak{a}$$ a 和 grade( $$\mathfrak{a}$$ a , M ) 的上同调维数是 $$\mathfrak{a}$$ a 的 M 级。令 D (−):= Hom R (−, E ) 是 Matlis 对偶函子，其中 E := E ( R / $$\mathfrak{m}$$ m ) 是残差域 R / $ 的单射包$\mathfrak{m}$$ m 。