Torsor-cotorsor duality of Ext groups

TL;DR

This paper explores a duality between torsors and cotorsors in the context of Ext groups, revealing a symmetric interpretation of Ext$^1(M, N)$ via module categories over rings.

Contribution

It introduces a dual perspective on Ext groups, interpreting them as classes of torsors and cotorsors, highlighting a torsor-cotorsor duality in module categories.

Findings

Ext$^1(M, N)$ corresponds to isomorphism classes of $N$-torsors.

Ext$^1(M, N)$ also corresponds to isomorphism classes of $M$-cotorsors.

The duality provides a symmetric framework for understanding extensions in module categories.

Abstract

Let $A$ be a ring, and let $M$ and $N$ be $A$-modules. Then $N$ can be viewed as a group object in the category $A$-Mod/$M$ of $A$-modules over $M$ and Ext$^1(M, N)$ can be interpreted as the set of isomorphism classes of $N$-torsors. Alternatively, $M$ can be viewed as a cogroup object in the category $N$/$A$-Mod of $A$-modules under $N$ and Ext$^1(M, N)$ can be interpreted as the set of isomorphism classes of $M$-cotorsors.