Decent intersection and Tor-rigidity for modules over local hypersurfaces

Serre's condition for tensor products and $n$-Tor-rigidity of modules

In this paper, we study Serre's condition $(S_n)$ for tensor products of modules over a commutative noetherian local ring. The paper aims to show the following. Let $M$ and $N$ be finitely generated module over a commutative noetherian local ring $R$, either of which is $(n+1)$-Tor-rigid. If the tensor product $M \otimes_R N$ satisfies $(S_{n+1})$, then under some assumptions $\mathrm{Tor}_{i}^R(M, N) = 0$ for all $i \ge 1$. The key role is played by $(n+1)$-Tor-rigidity of modules. As applications, we will show that the result recovers several known results.

Rigidity of Ext and Tor with coefficients in residue fields of a commutative noetherian ring

Let p be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies Tor^R_n(k(p),M) = 0 for some n \geq dim R_p, where k(p) is the residue field at p, then Tor^R_i(k(p),M) = 0 holds for all i \geq n. Similar rigidity results concerning Ext_R^*(k(p),M) are proved, and applications to the theory of homological dimensions are explored.

多重托模块：刚性与梅耶-维蒂里斯谱序列

Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences

本文研究多重理想下的托模块性质扩展，提出新的复形并探讨其同调关系，揭示托独立性在多格下的支撑区域关联。

通过平坦-余挠理论看 Ext 与 Tor 的刚性

Rigidity of Ext and Tor via flat-cotorsion theory

本文利用 Nakamura 和 Thompson 建立的极小半平坦-余挠替换，将 Ext 刚性结果中 n 的下界降至 dim R，并改进 Tor-刚性结论。

Ulrich 模的复杂性与刚性及其一些应用

Complexity and rigidity of Ulrich modules, and some applications

本文研究 Ulrich 模作为测试模的可行性，证明 CM 局部环上的 Ulrich 模具有极大复杂性和曲率，给出局部环的新刻画，并探讨其 Tor 刚性性质。