Editing Noncommutative geometry (section)

==Noncommutative affine and projective schemes==
In analogy to the [[Duality (mathematics)|duality]] between [[affine scheme]]s and [[commutative ring]]s, we define a category of '''noncommutative affine schemes''' as the dual of the category of associative unital rings. There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.

There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a theorem of [[Jean-Pierre Serre|Serre]] on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition of '''noncommutative projective geometry''' by [[Michael Artin]] and J. J. Zhang,<ref>{{cite journal | last1=Artin | first1=M. | last2=Zhang | first2=J.J. | title=Noncommutative Projective Schemes | journal=[[Advances in Mathematics]] | volume=109 | issue=2 | year=1994 | issn=0001-8708 | doi=10.1006/aima.1994.1087 | pages=228–287| doi-access=free }}</ref> who add also some general ring-theoretic conditions (e.g. Artin–Schelter regularity).

Many properties of projective schemes extend to this context. For example, there exists an analog of the celebrated [[Serre duality]] for noncommutative projective schemes of Artin and Zhang.<ref>{{cite journal | last1=Yekutieli | first1=Amnon | last2=Zhang | first2=James J. |title=Serre duality for noncommutative projective schemes| journal=[[Proceedings of the American Mathematical Society]] | publisher=American Mathematical Society (AMS) | volume=125 | issue=3 | date=1997-03-01 | issn=0002-9939 | doi=10.1090/s0002-9939-97-03782-9 | pages=697–708|doi-access=free}}</ref>

A. L. Rosenberg has created a rather general relative concept of '''noncommutative quasicompact scheme''' (over a base category), abstracting Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.<ref>A. L. Rosenberg, Noncommutative schemes, Compositio Mathematica 112 (1998) 93--125, [https://dx.doi.org/10.1023/A:1000479824211 doi]; Underlying spaces of noncommutative schemes, preprint MPIM2003-111, [http://www.mpim-bonn.mpg.de/preprints/send?bid=1947 dvi], [http://www.mpim-bonn.mpg.de/preprints/send?bid=1948 ps]; [[Mathematical Sciences Research Institute|MSRI]] lecture ''Noncommutative schemes and spaces'' (Feb 2000): [http://www.msri.org/publications/ln/msri/2000/interact/rosenberg/1/index.html video]</ref> There is also another interesting approach via localization theory, due to [[Fred Van Oystaeyen]], Luc Willaert and Alain Verschoren, where the main concept is that of a '''schematic algebra'''.<ref>Freddy van Oystaeyen, Algebraic geometry for associative algebras, {{isbn|0-8247-0424-X}} - New York: Dekker, 2000.- 287 p. - (Monographs and textbooks in pure and applied mathematics, 232)</ref><ref>{{cite journal | last1=Van Oystaeyen | first1=Fred | last2=Willaert | first2=Luc | title=Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras | journal=[[Journal of Pure and Applied Algebra]] | publisher=Elsevier BV | volume=104 | issue=1 | year=1995 | issn=0022-4049 | doi=10.1016/0022-4049(94)00118-3 | pages=109–122| hdl=10067/124190151162165141 | url=https://repository.uantwerpen.be/docman/irua/3d00aa/5163.pdf | hdl-access=free }}</ref>