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 $i \ge 0$ $({i \ne n})$.
Acknowledgment
The authors wish to thank the referee for his/her comments which made many improvments.
Citation
Sajjad Arda. Seadat Ollah Faramarzi. Khadijeh Ahmadi Amoli. "$k$-regular sequences to extension functors and local cohomology modules." Kodai Math. J. 46 (3) 263 - 274, October 2023. https://doi.org/10.2996/kmj46302
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