Local cohomology and quasicoherent sheaves

Quasicoherent Sheaves on Complex Noncommutative Two-tori

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T . We define the rank of a quasicoherent sheaf that can take arbitrary nonnegative real values. We study the category Qcoh(ηT ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank ze...

Sheaves and More Cohomology

of a Laurent series. If Op is the local ring of holomorphic functions around p. Take Mp the field of meromorphic functions around p, a principle part is just an element of the quotient groupMp/Op. The Mittag-Leffler question is, given a discrete set {pn} of points in S and a principle part at pn for each n, does there exist a meromorphic function f on S, holomorphic outside {pn}, whose principl...

Cohomology of Sheaves

The Cohomology of Sheaves

A fast introduction to the the construction of the cohomology of sheaves pioneered by A. Grothendieck and J.L. Verdier. The approach is from the point of view of derived categories, though this concept is never mentioned. 1. Sheaves Let R commutative ring with 1. We denote by RMod the category of R-modules. For any topological space X we denote by OpenpXq the collection of open subsets. It can ...

Cohomology of Sheaves