Noncommutative algebraic geometry

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Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations, taking noncommutative stack quotients etc.). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry multiply by points; as the values of these functions commute, the functions also commute: a times b equals b times a. It is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far reaching geometric intuition, though it formally looks like a fallacy.

Much of motivations for noncommutative geometry, and in particular for the noncommutative algebraic geometry is from physics; especially from quantum physics, where the algebras of observables are indeed viewed as noncommutative analogues of functions, hence why not looking at their geometric aspects.

One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.

The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogues in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.

Modern viewpoint via categories of sheaves

In modern times, one accepts a paradigm implicit in Pierre Gabriel's thesis and partly justified by Gabriel–Rosenberg reconstruction theorem (after Pierre Gabriel and Alexander Rosenberg) that a commutative scheme can be reconstructed, up to isomorphism of schemes, solely from the abelian category of quasicoherent sheaves on the scheme. Alexander Grothendieck taught us that to do a geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra via Yuri Manin. There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry.

Non-commutative deformations of commutative rings

As a motivating example, consider the one-dimensional Weyl algebra over the complex numbers C. This is the quotient of the free ring C{x, y} by the relation

xy - yx = 1.

This ring represents the polynomial differential operators in a single variable x; y stands in for the differential operator ∂_x. This ring fits into a one-parameter family given by the relations xy - yx = α. When α is not zero, then this relation determines a ring isomorphic to the Weyl algebra. When α is zero, however, the relation is the commutativity relation for x and y, and the resulting quotient ring is the polynomial ring in two variables, C[x, y]. Geometrically, the polynomial ring in two variables represents the two dimensional affine space A², so the existence of this one-parameter family says that affine space admits non-commutative deformations to the space determined by the Weyl algebra. In fact, this deformation is related to the symbol of a differential operator and the fact that A² is the cotangent bundle of the affine line.

Studying the Weyl algebra can lead to information about affine space: The Dixmier conjecture about the Weyl algebra is equivalent to the Jacobian conjecture about affine space.

Non-commutative localization

Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally, scheme) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Furthermore, the theory of non-commutative localization and descent theory is much more difficult in the non-commutative setting than in the commutative setting. While it works sometimes, there are rings that cannot be localized in the required fashion. Nevertheless, it is possible to prove some theorems in this setting.

Proj of a noncommutative ring

One of the basic constructions in commutative algebraic geometry is the Proj of a graded commutative ring. This construction builds a projective algebraic variety together with a very ample line bundle whose homogeneous coordinate ring is the original ring. Building the underlying topological space of the variety requires localizing the ring, but building sheaves on that space does not. By a theorem of Jean-Pierre Serre, quasi-coherent sheaves on Proj of a graded ring are the same as graded modules over the ring up to finite dimensional factors. The philosophy of topos theory promoted by Alexander Grothendieck says that the category of sheaves on a space can serve as the space itself. Consequently, in non-commutative algebraic geometry one often defines Proj in the following fashion: Let R be a graded C-algebra, and let Mod-R denote the category of graded right R-modules. Let F denote the subcategory of Mod-R consisting of all modules of finite length. Proj R is defined to be the quotient of the abelian category Mod-R by F. Equivalently, it is a localization of Mod-R in which two modules become isomorphic if, after taking their direct sums with appropriately chosen objects of F, they are isomorphic in Mod-R.

This approach leads to a theory of non-commutative projective geometry. A non-commutative smooth projective curve turns out to be a smooth commutative curve, but for singular curves or smooth higher-dimensional spaces, the non-commutative setting allows new objects.

References

M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228--287, doi.
Yuri I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.
Yuri I Manin, Topics in noncommutative geometry, 176 pp. Princeton 1991.
A. Bondal, M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Math J 2003
A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (2001), 327–344 doi
O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509--580; euclid.
Fred Van Oystaeyen, Alain Verschoren, Non-commutative algebraic geometry, Springer Lect. Notes in Math. 887, 1981.
Fred van Oystaeyen, Algebraic geometry for associative algebras, Marcel Dekker 2000. vi+287 pp.
A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN 0-7923-3575-9
M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996--1999, 85--108, Gelfand Math. Sem., Birkhäuser, Boston 2000; arXiv:math/9812158
A. L. Rosenberg, Noncommutative schemes, Compositio Mathematica 112 (1998) 93--125, doi; Underlying spaces of noncommutative schemes, preprint MPIM2003-111, dvi, ps; MSRI lecture Noncommutative schemes and spaces (Feb 2000): video
Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France 90 (1962), p. 323-448, numdam
Zoran Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, arXiv:0811.4770.
Dmitri Orlov, Quasi-coherent sheaves in commutative and non-commutative geometry, Izv. RAN. Ser. Mat., 2003, vol. 67, issue 3, 119–138 (MPI preprint version dvi, ps)
M. Kapranov, Noncommutative geometry based on commutator expansions, J. reine und angew. Math. 505 (1998), 73-118, math.AG/9802041.

External links

MathOverflow, Theories of Noncommutative Geometry
noncommutative algebraic geometry in nLab
equivariant noncommutative algebraic geometry in nLab
noncommutative scheme in nLab
Kapranov's noncommutative geometry in nLab

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