mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutativealgebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
For example, noncommutative algebraic geometry is supposed to extend a notion of an
algebraic scheme
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry ...
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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
, 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 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 group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:-
* Alfred Brauer (1894–1985), German-American mathematician, brother of Richard
* Andreas Brauer (born 1973), German film producer
* Arik ...
s.
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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
and especially the study of
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
s. These do not have a ring-theoretic analogue 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.
History
Classical approach: the issue of non-commutative localization
Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally,
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
) 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 In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
. Therefore, one can for instance attempt to replace a prime spectrum by a primitive spectrum: there are also the theory of non-commutative localization as well as descent theory. This works to some extent: for instance, Dixmier's ''enveloping algebras'' may be thought of as working out non-commutative algebraic geometry for the primitive spectrum of an enveloping algebra of a Lie algebra. Another work in a similar spirit is
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.noncommutative rings /ref> which in part is an attempt to study representation theory from a non-commutative-geometry point of view. The key insight to both approaches is that 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 Pierre Gabriel's thesis and partly justified by the Gabriel–Rosenberg reconstruction theorem (after Pierre Gabriel and Alexander L. 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 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 Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutativ ...
.
As a motivating example, consider the one-dimensional Weyl algebra over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s 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 . 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.'' This deformation is related to the symbol of a differential operator and that A2 is the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
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
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero c ...
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 writes:
Proj of a noncommutative ring
One of the basic constructions in commutative algebraic geometry is the Proj construction of a
graded commutative ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the s ...
. 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.
* 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), 327–34 doi * John Francis Derived Algebraic Geometry Over -Rings * O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509--580 euclid
*
Fred Van Oystaeyen Fred Van Oystaeyen (born 1947), also ''Freddy van Oystaeyen'', is a mathematician and emeritus professor of mathematics at the University of Antwerp. He has pioneered work on noncommutative geometry, in particular noncommutative algebraic geometry.
...
, 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.
* 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 dvips 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 versio dvips
* M. Kapranov, Noncommutative geometry based on commutator expansions, Journal für die reine und angewandte Mathematik 505 (1998), 73-118 math.AG/9802041
Further reading
* A. Bondal, D. Orlov, Semi-orthogonal decomposition for algebraic varieties_, PreprintMPI/95–15 alg-geom/9506006 * Tomasz Maszczyk, Noncommutative geometry through monoidal categories math.QA/0611806 * S. Mahanta, On some approaches towards non-commutative algebraic geometry math.QA/0501166 * Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev, Hodge theoretic aspects of mirror symmetry arxiv/0806.0107 * Dmitri Kaledin, Tokyo lectures "Homological methods in non-commutative geometry" pdfTeX and (similar but different Seoul lectures