Noncommutative algebraic geometry
Encyclopedia
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 ring
s 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 falacy.
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 group
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 and especially the study of local ring
s. 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.
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.
s C. This is the quotient of the free ring C{x, y} by the relation
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.
. 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and more specifically a direction in noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
that studies the geometric properties of formal duals of non-commutative algebraic objects such as ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
s 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 falacy.
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 group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
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 and especially the study of local ring
Local ring
In abstract algebra, more particularly in 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 number fields examined at a particular place, or...
s. 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
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
and K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
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 GabrielPierre Gabriel
Pierre Gabriel is a mathematician at Universität Zürich who works on category theory, algebraic groups, and representation theory of algebras. He was elected a correspondent member of the French Academy of Sciences in November 1986.-See also:...
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
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
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 numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 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
Symbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...
and the fact that A2 is the cotangent bundle
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...
of the affine line.
Studying the Weyl algebra can lead to information about affine space: The Dixmier conjecture
Dixmier conjecture
In algebra the Dixmier conjecture, asked by , is the conjecture that any endomorphism of a Weyl algebra is an automorphism. showed that the Dixmier conjecture is equivalent to the Jacobian conjecture....
about the Weyl algebra is equivalent to the Jacobian conjecture
Jacobian conjecture
In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller...
about affine space.
Non-commutative localization
Commutative algebraic geometry begins by constructing the spectrum of a ringSpectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
. The points of the algebraic variety (or more generally, scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
) 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
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
. 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 ringHomogeneous coordinate ring
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient 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
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, 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
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
theory promoted by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
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.
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 KontsevichMaxim KontsevichMaxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
, Tony Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107 - Dmitri Kaledin, Tokyo lectures "Homological methods in non-commutative geometry", pdf, TeX; and (similar but different) Seoul lectures