Fernando Sancho de Salas

Relative integral functors for singular fibrations and singular partners

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.

Number of singularities of a foliation on P^n

Let D be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of D are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let m be the dimension of the singular locus of D. We give an upper bound of the number of singularities of dimension m, counted with multiplicity and degree, that D may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.

A direct proof of the theorem on formal functions

We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion. This proof also works in some non-noetherian cases.

Homotopy of finite ringed spaces

A finite ringed space is a ringed space whose underlying topological space is finite. The category of finite ringed spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a finite ringed space. We study the homotopy of finite ringed spaces, extending Stong’s homotopy classification of finite topological spaces to finite ringed spaces. We also prove that the category of quasi-coherent modules on a finite ringed space is a homotopy invariant.

Finite Spaces and Schemes

Abstract A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a ringed finite space. We introduce the notions of schematic finite space and schematic morphism, showing that they behave, with respect to quasi-coherence, like schemes and morphisms of schemes do. Finally, we construct a fully faithful and essentially surjective functor from a localization of a full subcategory of the category of schematic finite spaces and schematic morphisms to the category of quasi-compact and quasi-separated schemes.

Euler sequence and Koszul complex of a module

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential p

The linear dual of the derived category of a scheme

p

Desingularization and integralityof the automorphism scheme of a finite field extension

-forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.

Fourier¿Mukai transforms for Gorenstein schemes

Let X → S be a projective morphism of schemes. We study the category D(X/S) * of S-linear exact functors D(X) → D(S), and we study the Fourier transform D(X) → D(X/S) * .

Principal bundles, quasi-abelian varieties and structure of algebraic groups☆

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