Algebraic stratification in representation categories
The paper begins by reviewing the categorical framework set up by \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne} [Astérisque 100 (1982; Zbl 0536.14011)] to construct glueings of derived categories: this involves several adjoint functors and compatibility conditions. For the case of an inclusion of module categories defined by a quotient \(B=A/J\), the conditions to set up this framework are reduced to ring-theoretic properties. In particular, if A has finite global dimension the conditions reduce to \(Ext^ n_ A(B,B)=0\) for all \(n>0\). If A is hereditary, it suffices if B is the largest quotient ring all whose composition factors (as A-module) belong to a preassigned set. If A is commutative, the conditions only hold if B is a direct factor. However, if \({\mathcal C}\) is the category of rational modules over a semisimple, simply-connected algebraic group G over an algebraically closed field, and \({\mathcal C}(\Gamma)\) the full subcategory of those all of whose composition factors have highest weight \(\leq\) some \(\lambda\in \Gamma\) (for preassigned finite \(\Gamma)\), then the \({\mathcal C}(\Gamma)\) are isomorphic to categories of modules and their inclusions satisfy the above mentioned conditions.
scientific article; zbMATH DE number 4077518
scientific article