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 $q:\mathcal{A}\to \mathcal{A}/\mathcal{C}$. For an object $A\in \mathcal{A}$ and a non-negative integer $k$ we investigate when the natural map $q_{X,A}^i:\mathrm{Ex}{\mathrm{t}}_{\mathcal{A}}^{\mathrm{i}}(\mathrm{X},\mathrm{A})\to \mathrm{Ex}{\mathrm{t}}_{\mathcal{A}/\mathcal{C}}^{\mathrm{i}}(\mathrm{q}(\mathrm{X}),\mathrm{q}(\mathrm{A}))$ 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.

arXiv commentRevised version following referee's report. Comments are welcome!