Quasicoherent sheaves should be thought of as the scheme-theoretic analogs of vector bundles. This analogy might lead one to think that vector bundles over a scheme X should be locally free OX-modules, but this turns out not to be an abelian category, and the category of quasicoherent sheaves contains the category of locally free OX-modules. There are two different different definitions we coul...

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P. The generic noncommutative plane corresponds to the Sklyanin algebra S = Skl(E,σ) constr...

The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S,H), over the field k = k̄ of zero characteristic, is constructed in previous papers of the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the surfaces which are certain modifications of S....

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometric...

This paper is motivated by questions such as P vs. NP and other questions in Boolean complexity theory. We describe an approach to attacking such questions with cohomology, and we show that using Grothendieck topologies and other ideas from the Grothendieck school gives new hope for such an attack. We focus on circuit depth complexity, and consider only finite topological spaces or Grothendieck...

We use a complex version of the celebrated Atiyah-Hitchin-DrinfeldManin matrix equations to construct admissible torsion-free sheaves on P3 and complex quantum instantons over our quantum Minkowski space-time. We identify the moduli spaces of various subclasses of sheaves on P3, and prove their smoothness. We also define the Laplace equation in the quantum Minkowski space-time, study its soluti...

These notes explain the equivalence between certain topological and coherent data on complex-analytic manifolds, and also discuss the phenomenon of “regular singularities” connections in the 1-dimensional case. It is assumed that the reader is familiar with the equivalence of categories between the category of locally constant sheaves of sets on a topological space X and the category of coverin...