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Fernando Sancho de Salas

Publications and source records attributed to Fernando Sancho de Salas.

16 recordsLinked to original sources
arXiv2024

Hochster's type formulae

We give an elementary proof and generalize some Hochsters's type formulae on local cohomology and Ext's of squarefree modules

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arXiv2024

On Koszul complex of a supermodule

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting computing the cohomology of twisted differential p-forms sheaves on the projective superspaces is given.

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arXiv2023

Indecomposability of derived categories for arbitrary schemes

We extend the criterion of Kawatani and Okawa for indecomposability of the derived category of a smooth projective variety to arbitrary schemes. For relative schemes, we also give a criterion for the nonexistence of semiorthogonal decompositions that are linear over the base. These criteria are based on the base loci of the global or relative dualizing complexes.

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arXiv2021

Homology of sheaves via Brown representability

We give an elementary construction of homology of sheaves from Brown representability for the dual and see how its main properties are derived easily from the construction. Comparison with Poincaré-Verdier duality and with homology of groups are also developed.

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arXiv2019

Derived category of Finite Spaces and Grothendieck Duality

We obtain some fundamental results, as Bokstedt-Neeman Theorem and Grothendieck duality, about the derived category of modules on a finite ringed space. Then we see how these results are transfered to schemes in a simple way and generalized to other ringed spaces.

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arXiv2016

Finite Spaces and Schemes

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.

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arXiv2015

Homotopy of ringed finite spaces

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 study the homotopy of ringed finite spaces, extending Stong's homotopy classification of finite topological spaces to ringed finite spaces. We also prove that the category of quasi-coherent modules on a ringed finite space is a homotopy invariant.

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arXiv2015

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 $p$-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.

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arXiv2014

Ringed Finite Spaces

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 make a study of the homotopy of ringed finite spaces (that extends the homotopy of finite topological spaces) and a study of the quasi-coherent sheaves on a ringed finite space. This leads to set and develop basic notions on ringed finite spaces and morphisms, as being affine, schematic, semi-separated, etc, focusing on its cohomological properties. Finally, we see how to embed the category of quasi-compact and quasi-separated schemes in a localization of the category of finite ringed spaces.

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arXiv2011

Non-exact Integral Functors

We give a natural notion of (non-exact) integral functor in the context of k-linear and graded categories. In this broader sense, we prove that every k-linear and graded functor is integral.

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arXiv2008

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

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.

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arXiv2007

Koszul complexes and fully faithful integral functors

We characterise those objects in the derived category of a scheme which are a sheaf supported on a closed subscheme in terms of Koszul complexes. This is applied to generalize to arbitrary schemes the fully faithfullness criteria of an integral functor.

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