Zoran Skoda talk Tallinn 25

PSSL 110 May 3-4, 2025 programme

Noncommutative schemes and torsors via categories of generalized quasicoherent sheaves.

Grothendieck Abelian categories have many features analogous to toposes. Categories of quasicoherent sheaves of O-modules on schemes supply an important class of examples. Any quasicompact quasiseparated scheme can be reconstructed up to an isomorphism from its category of quasicoherent sheaves (Gabriel-Rosenberg theorem). For an affine scheme, by the affine Serre’s theorem, the quasicoherent sheaves correspond to the modules over the coordinate ring (global sections of the structure ring). Noncommutative algebraic geometry may be formulated as studying ‘spaces’ represented by Abelian categories which are glued from local models for which one usually takes the categories of modules over (noncommutative) ring(oid)s. I will sketch two developments from this point of view. Firstly, how to introduce noncommutative analogues of torsors in wide generality, using properties of certain adjunctions. Secondly, how to describe a 2-category of noncommutative spaces by defining such spaces by gluing representable 2-presheaves (of categories) on an appropriate (analogue of a) site of noncommutative affine schemes.