arXiv subjects
Publications and source records attributed to F. Sancho de Salas.
We prove that a topological space is aspherical if and only if it satisfies Bökstedt-Neeman Theorem, i.e., the derived category of complexes of locally constant sheaves is equivalent to the derived category of complexes of sheaves with locally constant cohomology.
We construct affine spaces, projective spaces and grassmannians in the ca\-te\-gory of ringed spaces. We show how finite posets and sheaves of rings on them appear in a natural way.
We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth algebraic group and $\mathcal L$ is a $G$-invariant invertible sheaf on a $G$-variety $X$, then the action of $G$ on $X$ extends to a projective action on the complete linear ${\mathbb P}(H^0(X,{\mathcal L})$. As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups.
We study relative integral functors for singular schemes and characterise those which preserve boundness 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.
We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also contemplate a criterion for equivalence. We offer a proof that is new even if we restrict to the smooth case. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre also is it. These results imply the invertibility of the usual relative Fourier-Mukai transform for an elliptic fibration as a direct corollary.