Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on cohomological vanishing theorems

Mohsen Asgharzadeh

Abstract

We study $cd(M,N):=\sup\{j:H^j_{m}(M,N)\neq0\}$, and we prove the following over $AB$-rings: $cd(M,N)<\infty$ iff $cd(M, N)\leq2 dim R$. For locally free over the punctured spectrum, we present the better bound, namely $cd(M, N)<\infty$ iff $cd(M, N)\leq dim R,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of $M$. We present some explicit examples to compute $cd(M, N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map $\varphi$. We show for some $n\gg 0$ that $cd(^{\varphi_n}R,M)<\infty$ iff $id_R(M)<\infty.$ This presents some criteria on regularity. Also, some vanishing results on $Ext^i_R(^{\varphi}R,-)$ are given, where $(-)\in\{R,^{\varphi}R\}$. We determine conditions under which the vanishing $Ext^i_R(^{\varphi}R,-)$ of restricted many $i$-th, implies the vanishing of all.

A note on cohomological vanishing theorems

Abstract

We study , and we prove the following over -rings: iff . For locally free over the punctured spectrum, we present the better bound, namely iff and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of . We present some explicit examples to compute . Now, suppose is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map . We show for some that iff This presents some criteria on regularity. Also, some vanishing results on are given, where . We determine conditions under which the vanishing of restricted many -th, implies the vanishing of all.
Paper Structure (3 sections, 18 theorems, 33 equations)

This paper contains 3 sections, 18 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. When is $H_m^+(M,N)=0$?
  3. Vanishing of $\operatorname{Ext}^\ast({{}^{\varphi_n}\!}R,-)$

Key Result

Proposition 2.1

Let $R$ be an $\operatorname{AB}$ ring, let $M$ and $N$ be finitely generated $R$-modules. If $\operatorname{cd}(M,N)<\infty$ then $\operatorname{cd}(M,N)\leq2\dim R$.

Theorems & Definitions (44)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Example 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • ...and 34 more