Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...

, 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 or taking noncommutative stack quotients). 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 have a product defined by

pointwise multiplication In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be mult ...

; 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 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. Promi ...

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 algebraic ...

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 Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...

and

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geom ...

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 In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...

. 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 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 simple ...

. 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 In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vect ...

. 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 Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

from a non-commutative-geometry point of view. The key insight to both approaches is that irreducible representations, or at least

primitive ideal In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals ...

s, 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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

. 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 Alexander Lvovich Rosenberg (russian: Александр Львович Розенберг, 1946–2012) was a Russian-American mathematician who worked on functional analysis, representation theory and noncommutative algebraic geometry. He gra ...

) that a commutative scheme can be reconstructed, up to isomorphism of schemes, solely from the

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...

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 Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical lo ...

. 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 In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...

, 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 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 for ...

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 In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

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. This ...

of the affine line. (Studying the Weyl algebra can lead to information about affine space: The

Dixmier conjecture In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixm ...

about the Weyl algebra is equivalent to the Jacobian conjecture about affine space.) In this line of the approach, the notion of ''

operad In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one define ...

'', 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 In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not fun ...

of a graded commutative ring. This construction builds a

projective algebraic variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

together with a

very ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...

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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...

, 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 In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...

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 In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. Examples *The quantum plane, the most basic example, is the quotient ring of the free ring ...

. 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

References

* M. Artin, J. J. Zhang, Noncommutative projective schemes,

Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...

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 ''Compositio Mathematica'' is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London ...

125 (2001), 327–34
doi
* John Francis
Derived Algebraic Geometry Over

\mathcal_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. * 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,

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 ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics''). History The journal wa ...

505 (1998), 73-118
math.AG/9802041

External links

* MathOverflow
Theories of Noncommutative Geometry
* * * * {{nlab, id=Kapranov%27s%20noncommutative%20geometry, title=Kapranov's noncommutative geometry Algebraic geometry Noncommutative geometry