局部上同调的一个分裂定理及其应用
A Splitting Theorem for Local Cohomology and its Applications
摘要：
设$R$是交换诺特环，$M$是交换生成的$R$ -模.本文证明了，对于整数$t$，如果对于所有的$i<t$，关于理想$\frak a$的局部上同调模$H^{i}_\mathfrak{a}(M)$都是连续生成的，则对于所有的$\frak a$，$ $H^{i}_\mathfrak{a}(M/xM)\cong H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)$包含$\frak a$和所有的$i<t-1$的足够大的幂的正则元$x$.作为结果，我们得到推广，非常短的证明，M的主要结果。Brodmann和A.L. Faghani（A finiteness report for associated primes of local cohomology modules，Proc. Amer. Math. Soc.，128（2000），2851-2853）和H.L. Truong和第一作者（广义Cohen-Macaulay模中参数理想的渐近行为，J. Algebra，320（2008），158 -168）。
英文摘要：
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this paper that, for an integer $t$, if the local cohomology module $H^{i}_\mathfrak{a}(M)$ with respect to an ideal $\frak a$ is finitely generated for all $i<t$, then $$H^{i}_\mathfrak{a}(M/xM)\cong H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)$ for all $\frak a$-filter regular elements $x$ containing in a enough large power of $\frak a$ and all $i<t-1$. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani (A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(2000), 2851-2853) and of H.L. Truong and the first author (Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra, 320(2008),158-168).
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