# Equivariant isomorphisms of Ext and Tor modules

## TL;DR

This paper explores how Ext and Tor modules over different relative complete intersections are related through equivariant isomorphisms, using perturbation theory to connect DG modules when varying Koszul-regular sequences.

## Contribution

It establishes equivariant isomorphisms of Ext and Tor modules over varying complete intersections, employing perturbation techniques for DG modules.

## Key findings

- Isomorphisms of Ext and Tor modules are established for different complete intersections.
- Perturbation theory is used to relate DG modules in this context.
- The results provide a new understanding of module behavior under sequence variation.

## Abstract

In this article we establish equivariant isomorphisms of Ext and Tor modules over different relative complete intersections. More precisely, for a commutative ring $Q$, this paper investigates how $Ext_{Q/(\boldsymbol{f})}^*(M,N)$ and $Tor^{Q/(\boldsymbol{f})}_*(M,N)$ change when one varies $\boldsymbol{f}$ among all Koszul-regular sequences of a fixed length such that $\boldsymbol{f} M=0$ and $\boldsymbol{f} N=0$. Of notable interest is how the theory of perturbations is used to establish isomorphisms of certain DG modules.