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Representable functors, Serre functors, and mutations. (English. Russian original) Zbl 0703.14011
Math. USSR, Izv. 35, No. 3, 519-541 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1183-1205 (1989).
Let A be a triangulated category endowed with a filtration \(W_ 0A\subset W_ 1A\subset\dots\subset A\), where \(W_ iA\) is a thick subcategory of A such that the inclusion \(W_ iA\subset A\) admits an adjoint functor. For example, let A be the derived category \(D^ b_{coh}(P^ n)\) of all coherent sheaves on the projective space \(P^ n\). By a result of Beilinson, A is generated by the sheaves \({\mathcal O}(j),\quad j=0,1,\dots,n.\) Let \(W_ iA\) be the subcategory of A generated by \({\mathcal O}(j),\quad j=0,1,\dots,i.\) The aim of this paper is to study the data \((A,W_ 0A\subset W_ 1A\subset\dots\subset A).\)
The philosophy is that many properties of the quotient categories \(W_ iA/W_{i-1}A\) are inherited by the category A itself. This is useful if one wants to study the representability properties of the cohomological functors \(A\to Vect\). The latter properties are also related to some duality questions. Another application of the above philosophy concerns the construction of a t-structure on A. This amounts to give on each quotient category \(W_ iA/W_{i-1}A\) a t-structure (via the theory of perverse sheaves).
Reviewer: L.Bădescu

14F25 Classical real and complex (co)homology in algebraic geometry
18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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