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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

##### MSC:
 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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