$k$-regular sequences to extension functors and local cohomology modules

Abstract

In this paper we generalize the Zero Divisor Conjecture and Rigidity Theorem for $k$-regular sequence. For this purpose for any $k$-regular $M$-sequence ${x_1},...,{x_n}$ we prove that if $\dim{\rm Tor}_2^R({\frac{R}{{({{x_1},...,{x_n}} )}},M}) \le k$, then $\dim{\rm Tor}_i^R({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, for all $i \ge 1$. Also we show that if $\dim{\rm Ext}_R^{n + 2}({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, then $\dim{\rm Ext}_R^{i}({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, for all integers $i \ge 0$ $({i \ne n})$.