A note on thick subcategories of stable derived categories (Q2873892)

In this very clear paper, the authors propose some classification results regarding the subcategory of projective objects of an exact category \(A\). By a result of Verdier, these subcategories correspond to thick subcategories of the bounded derived category \(D^b(A)\) containing all the projective objects. Let \(A\) be an exact category having enough projective objects. The main result (Theorem 1 in Section 2) is devoted to describe a bijection between these subcategories of \(D^b(A)\) and thick subcategories of \(A\) containing all the projective objects. The maps that realize the bijection are the one sending a subcategory \(D\) to \(D \cap A\) and the inverse sending a thick subcategory \(C\) of \(A\) to the replete closure of \(D^b(C)\).NEWLINENEWLINEThe applications of this result are described in the next sections. Let \(A\) be a strict local complete intersection ring; there is a regular local ring \(B\) surjecting on \(A\) with kernel generated by the regular sequence \(\{b_1, \cdots, b_c\}\). Let \(Y\) be the hypersurface defined by the section \(\sum_{i=1}^c b_ix_i\) of \(\mathcal{O}_{\mathbb{P}^{c-1}_B}(1)\). There is an order-preserving bijection between thick subcategories of \(\operatorname {mod} A\) and specialization closed subsets of the singular locus of \(Y\).NEWLINENEWLINEMoreover, if \(G\) is a finite group, denote by \(H^*(G,k)\) the cohomology of \(G\) with coefficient in \(k\). There is an order-preserving bijection between the specialization closed subsets of \(\operatorname {Proj}H^*(G,k)\) and the thick tensor ideals of the category \(\operatorname {mod}kG\) containing \(kG\).NEWLINENEWLINEEventually let \(X\) a separated Noetherian scheme. The authors prove a characterization of the category of coherent \(O_X\)-modules as the smallest thick subcategory of \(\operatorname {Coh}(X)\) containing a particular set of coherent \(\mathcal{O}_X\)-modules \(S\) described by an ample family of line bundles on \(X\).