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Relative integral functors for singular fibrations and singular partners. (English) Zbl 1221.18010

Summary: We study relative integral functors for singular schemes and characterise those which preserve boundedness and those which have integral right adjoints. We prove that a relative integral functor is an equivalence if and only if its restriction to every fibre is an equivalence. This allows us to construct a non-trivial auto-equivalence of the derived category of an arbitrary genus one fibration with no conditions on either the base or the total space and getting rid of the usual assumption of irreducibility of the fibres. We also extend to Cohen-Macaulay schemes the criterion of Bondal and Orlov for an integral functor to be fully faithful in characteristic zero and give a different criterion which is valid in arbitrary characteristic. Finally, we prove that for projective schemes both the Cohen-Macaulay and the Gorenstein conditions are invariant under Fourier-Mukai functors.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14E30 Minimal model program (Mori theory, extremal rays)
13D22 Homological conjectures (intersection theorems) in commutative ring theory
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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References:

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