Yoneda extensions of abelian quotient categories

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.