# The (ir)regularity of Tor and Ext

## TL;DR

This paper studies the asymptotic behavior of Castelnuovo-Mumford regularity of Ext and Tor modules over complete intersection rings, revealing linearity in high degrees under certain conditions and complex behavior otherwise.

## Contribution

It establishes linearity results for the regularity of Ext and Tor modules in high degrees over complete intersection rings, with specific support conditions for Tor modules.

## Key findings

- Linearity of Ext regularity in high degrees
- Conditional linearity of Tor regularity
- Examples showing complex behavior without support restrictions

## Abstract

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.