nLab sheaf on a noncommutative space

In noncommutative algebraic geometry the analogues of collection of open sets, and more generally, the analogues of sites usually fail to be stable under pullback: a pullback of an open cover is not an open cover in general. This requires an extension of the usual concepts of Grothendieck topology and Grothendieck pretopology. Consequently the notion of a sheaf and of a stack need to be adapted to this formalism.

The point of view that localizations are analogues of Zariski open sets, and the appropriate notion of descent for quasicoherent sheaves for covers by noncommutative localizations is implied already in Gabriel’s thesis Des Categories Abeliennes and later explicitly studied in a number of works, including

• Freddy M. J. Van Oystaeyen, Alain H. M. J. Verschoren, Noncommutative algebraic geometry. An introduction, Lec. Notes in Math. 887, Springer 1981. vi+404 pp.

• A. L. Rosenberg, Non-commutative affine semischemes and schemes, Seminar on supermanifolds 26, Dept. Math., U. Stockholm (1988)

• Fred Van Oystaeyen, Luc Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras, J. Pure Appl. Algebra 104 (1995), no. 1, 109–122, MR97a:16086, doi

• F. van Oystaeyen, Algebraic geometry for associative algebras, Marcel Dekker 2000. vi+287 pp.

• M. Kontsevich, A. L. Rosenberg, Noncommutative spaces and flat descent, MPI-2004-36 dvi,ps,

More general point of view closer to the formalism of topologies/sieves than to Grothendieck pretopologies is also a notion of Q-category due Rosenberg and his work on sheaves and later work with Kontsevich on stacks on Q-categories. For example, noncommutative analogues of smooth, fppf and fpqc topologies can be formalized in this framework.

• A. L. Rosenberg, Almost quotient categories, sheaves and localizations, 181 p. Seminar on supermanifolds 25, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)

• M. Kontsevich, A. Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (dvi,ps), Noncommutative stacks, MPI-2004-37 dvi,ps

category: noncommutative geometry