Ext functors, support varieties and Hilbert polynomials over complete intersection rings
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910841096372224 |
|---|---|
| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d \geq 1$ and codimension $c \geq 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i \geq 1$ let $ψ_i^I(M)$ be the degree of the polynomial type function $n \rightarrow \ell(Ext^i_A(M, A/I^n))$. We show that for $j = 0, 1$ and for all $i \gg 0$ we have $ψ_{2i +j}^I(M)$ is a constant and let $r_0^I(M)$ and $r_1^I(M)$ denote these constant values. Set $r^I(M) = \max\{ r_0^I(M), r_1^I(M) \}$. We show that $r^I(M)$ is an invariant of $I, A$ and the support variety of $M$. We set the degree of the zero polynomial to be $-\infty$. If $r^I(M) \leq 0$ then we show that $reg \ G_I(Ω^i(M))$ for $i \geq 0$ is bounded. We give an application of this result to syzgetic Artin-Rees property of $M$. We also give several examples which illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ext functors, support varieties and Hilbert polynomials over complete intersection rings Puthenpurakal, Tony J. Commutative Algebra Primary 13D07, 13D40, Secondary 13A30 Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d \geq 1$ and codimension $c \geq 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i \geq 1$ let $ψ_i^I(M)$ be the degree of the polynomial type function $n \rightarrow \ell(Ext^i_A(M, A/I^n))$. We show that for $j = 0, 1$ and for all $i \gg 0$ we have $ψ_{2i +j}^I(M)$ is a constant and let $r_0^I(M)$ and $r_1^I(M)$ denote these constant values. Set $r^I(M) = \max\{ r_0^I(M), r_1^I(M) \}$. We show that $r^I(M)$ is an invariant of $I, A$ and the support variety of $M$. We set the degree of the zero polynomial to be $-\infty$. If $r^I(M) \leq 0$ then we show that $reg \ G_I(Ω^i(M))$ for $i \geq 0$ is bounded. We give an application of this result to syzgetic Artin-Rees property of $M$. We also give several examples which illustrate our results. |
| title | Ext functors, support varieties and Hilbert polynomials over complete intersection rings |
| topic | Commutative Algebra Primary 13D07, 13D40, Secondary 13A30 |
| url | https://arxiv.org/abs/2502.16494 |