Noncommutative algebraic geometry

Noncommutative algebraic geometry is a branch of

stack quotients

).

For example, noncommutative algebraic geometry is supposed to extend a notion of an

pointwise multiplication; 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.^{[citation needed}

]

Much of the motivation 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 having the ability to observe their geometric aspects is desirable.

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

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.

History

Classical approach: the issue of non-commutative localization

Commutative algebraic geometry begins by constructing the

irreducible representations, or at least primitive ideals

, can be thought of as “non-commutative points”.

Modern viewpoint using categories of sheaves

As it turned out, starting from, say, primitive spectra, it was not easy to develop a workable

sheaf theory

. One might imagine this difficulty is because of a sort of quantum phenomenon: points in a space can influence points far away (and in fact, it is not appropriate to treat points individually and view a space as a mere collection of the points).

Due to the above, one accepts a paradigm implicit in

quasicoherent sheaves on the scheme. Alexander Grothendieck taught that to do 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 by Yuri Manin

. There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just below).

Derived algebraic geometry

Perhaps the most recent approach is through the

deformation theory, placing non-commutative algebraic geometry in the realm of derived algebraic geometry

.

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

symbol of a differential operator and 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.)

In this line of the approach, the notion of operad, a set or space of operations, becomes prominent: in the introduction to (Francis 2008), Francis writes:

We begin the study of certain less commutative algebraic geometries. … algebraic geometry over ${\mathcal {E}}_{n}$ -rings can be thought of as interpolating between some derived theories of noncommutative and commutative algebraic geometries. As n increases, these ${\mathcal {E}}_{n}$ -algebras converge to the derived algebraic geometry of Toën-Vezzosi and Lurie.

Proj of a noncommutative ring

One of the basic constructions in commutative algebraic geometry is the

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.

Notes

^ M. Artin, noncommutative rings

References

M. Artin, J. J. Zhang, Noncommutative projective schemes, Advances in Mathematics 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 Mathematical Journal 3 (2003), no. 1, 1–36.
A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (2001), 327–344 doi
John Francis, Derived Algebraic Geometry Over ${\mathcal {E}}_{n}$ -Rings
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,
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,
Journal für die reine und angewandte Mathematik 505 (1998), 73-118, math.AG/9802041
.

External links

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

[1] M. Artin, noncommutative rings