Yoneda extensions of abelian quotient categories

TL;DR

This paper studies the behavior of Ext functors under quotient categories of abelian categories, providing conditions for when natural maps between Ext groups are isomorphisms, with an application included.

Contribution

It characterizes when the natural maps between Ext groups in an abelian category and its quotient are invertible up to a certain degree, extending understanding of Yoneda extensions in quotient categories.

Findings

Conditions for invertibility of Ext maps in quotient categories

Extension of Yoneda extension theory to abelian quotients

Application demonstrating the main theoretical results

Abstract

Let $\mathcal{A}$ be a essentially small abelian category and $\mathcal{C}$ be a Serre subcategory of $\mathcal{A}$. Consider the quotient functor $q:\mathcal{A}\rightarrow \mathcal{A}/\mathcal{C}$. For an object $A\in \mathcal{A}$ and a non-negative integer $k$ we investigate when the natural map $q_{X,A}^i: \rm Ext^i_{\mathcal{A}}(X,A)\rightarrow Ext^i_{\mathcal{A}/\mathcal{C}}(q(X),q(A))$ is invertible for every $X\in \mathcal{A}$ and every $i\in\{0,1,\cdots,k\}$. In the end we give an application of the main theorem.