Homological subsets of Spec

TL;DR

This paper explores how to recover module data from Ext-families by analyzing homological subsets of the spectrum, extending classical calculations, and computing supports and associated primes in specific cases.

Contribution

It introduces new methods for studying homological subsets of Spec via Ext-families and extends Grothendieck's dimension calculations for Ext modules.

Findings

Extended Grothendieck's calculation of Ext module dimensions.

Computed support and associated primes for Ext-families in various cases.

Provided new insights into the structure of modules via homological subsets.

Abstract

We recover some data of a module $M$ from the Ext-family $\{\Ext^i_R(M,R)\}$. In this regard, we investigate homological subsets of $\Spec(R)$, defined by the help of Ext-family. We extend Grothendieck's calculation of $\dim(\Ext^g_R(M,R))$. Also, we compute support and the set of all associated prime ideals of the Ext-family in a serial of nontrivial cases.