On equivalence of derived categories
Consider an abelian category \({\mathcal A}\) and a thick subcategory \({\mathcal B}\) of \({\mathcal A}\). Let \(D^b ({\mathcal B})\) denote the derived category of cohomologically bounded chain complexes of objects in \({\mathcal B}\) and \(D^b_{\mathcal B} ({\mathcal A})\) the derived category of cohomological bounded chain complexes of objects in \({\mathcal A}\) with cohomology in \({\mathcal B}\). The paper considers the question whether the embedding of \({\mathcal B}\) into \({\mathcal A}\) induces an equivalence of categories from \(D^b ({\mathcal B})\) to \(D^b_{\mathcal B} ({\mathcal A})\). Two positive answers to this question are given. First, let us denote by \(y \text{Ext}^n_{\mathcal A} (A,B)\) the Yoneda \(n\)-extension, i.e., the set of all \(n\)-extensions from \(A\) by \(B\), modulo the equivalence relation generated by morphisms of \(n\)-extensions. With this notation, the authors prove that the inclusion \(i\) of \({\mathcal A}\) into \({\mathcal B}\) induces an equivalence of derived categories between \(D^b ({\mathcal A})\) and \(D^b_{\mathcal B} ({\mathcal A})\) if and only if for any \(A,B\) in \({\mathcal B}\) and any positive integer \(n\), we have \[ y \text{Ext}^n_{\mathcal B} (A,B) = y \text{Ext}^n_{\mathcal A} (iA, iB), \] up to isomorphism. Examples when this condition is satisfied include the case when \({\mathcal B}\) has enough \({\mathcal A}\)-injectives and the geometric example, where \(X\) is a quasicompact and semi-separated scheme, in which case there is an equivalence between the categories \(D^b (\text{Qcoh} (X))\) and \(D^b_{qc} (X)\). If one works over a Grothendieck category \({\mathcal A}\), a second positive answer to the previous question is given by the following result. Consider a thick subcategory \({\mathcal T}\) of \({\mathcal A}\) which is closed under taking direct sums and denote by \(t\) the torsion theory (or the idempotent kernel functor) associated to \({\mathcal T}\). Then the embedding of \({\mathcal T}\) in \({\mathcal A}\) induces an equivalence between \(D^b ({\mathcal T})\) and \(D^b_{\mathcal T} ({\mathcal A})\) if and only if for any injective object \(E\) in \({\mathcal A}\), the quotient \(E/tE\) has infinite \(t\)-dominant dimension.
scientific article; zbMATH DE number 895839
scientific article