优势Cohen-Macaulay环上的导出函子与希尔伯特多项式
Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings
- Indian Institute of Technology Bombay（印度理工学院孟买分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对优势Cohen-Macaulay局部环，探讨Tor函子与Ext函子相关多项式型函数的次数性质，得出特定条件下次数的极限上界结论。
AI中文摘要：
设$(A,\mathfrak{m})$是维数为$d$的Cohen-Macaulay局部环，剩余域为$k$，$I$是$\mathfrak{m}$-准素理想，假设$k$是无限的；设$N$是维数$t\geq1$的完美$A$-模，$M$是极大Cohen-Macaulay（MCM）$A$-模。函数$n\mapsto\ell(\text{Tor}^A_j(M, N/I^{n+1}N))$具有多项式型，记其次数为$r_{I, N}^j(M)$。一般地，对$j\geq d + 2$有$r_{I, N}^j(M) \leq r_{I,N}^2(\text{Syz}^d_A(k))$。若$A$还满足Takahashi意义下的优势条件且$M$非自由，则证明$\limsup_{j \rightarrow \infty} r_{I, N}^j(M) = r_{I,N}^2(\text{Syz}^d_A(k))$；当$V$是有限内射维数的Cohen-Macaulay模，且多项式型函数$n\mapsto\ell(\text{Ext}_A^j(M, V/I^{n+1}V))$时，也证明了类似结果。
英文摘要：
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of polynomial type and let $r_{I, N}^j(M)$ be its degree. In general we have $r_{I, N}^j(M) \leq r_{I,N}^2(\text{Syz}^d_A(k))$ for $j \geq d + 2$. If $A$ is also dominant in the sense of Takahashi and $M$ is non-free then we show $\limsup_{j \rightarrow \infty} r_{I, N}^j(M) = r_{I,N}^2(\text{Syz}^d_A(k))$. We prove an analogous result when $V$ is a Cohen-Macaulay module of finite injective dimension and the polynomial type function $n \rightarrow \ell(\text{Ext}_A^j(M, V/I^{n+1}V))$
