Noncommutative geometry
Encyclopedia
Noncommutative geometry (NCG) 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 (possibly in some generalized sense). A noncommutative algebra is an associative algebra
in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure
in which one of the principal binary operation
s is not commutative; one also allows additional structures, e.g. topology
or norm to be possibly carried by the noncommutative algebra of functions. The leading direction in noncommutative geometry has been laid by French mathematician Alain Connes
since his involvement from about 1979.
on them. In general, such functions will form a commutative ring
. For instance, one may take the ring C(X) of continuous
complex
-valued functions on a topological space
X. In many important cases (e.g., if X is a compact
Hausdorff space
), we can recover X from C(X), and therefore it makes some sense to say that X has commutative geometry.
More specifically, in topology, compact Hausdorff
topological spaces can be reconstructed from the Banach algebra
of functions on the space (Gel'fand-Neimark). In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and schemes can be reconstructed from the categories of quasicoherent sheaves of modules on them (P. Gabriel-A. Rosenberg). For Grothendieck topologies, the cohomological properties of a site are invariant of the corresponding category of sheaves of sets viewed abstractly as a topos (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—some category of sheaves on that space.
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
The dream of noncommutative geometry is to generalize this duality to the duality between
and interact between the algebraic and geometric description of those via this duality.
Regarding that the commutative rings correspond to usual affine schemes, and commutative C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of topological space
s, as "non-commutative spaces". For this reason, some talk about non-commutative topology, though the term has also other meanings.
are described on the entries Noncommutative standard model
and Noncommutative quantum field theory
. Sudden rise in interest in noncommutative geometry in physics, follows after the speculations of its role in M-theory
made in 1997.
to handle noncommutative geometry at a technical level has roots in older attempts, in particular in ergodic theory
. The proposal of George Mackey
to create a virtual subgroup theory, with respect to which ergodic group action
s would become homogeneous space
s of an extended kind, has by now been subsumed.
, which shows that commutative C*-algebras are dual
to locally compact Hausdorff space
s. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.
For the duality
between σ-finite measure spaces and commutative von Neumann algebra
s, noncommutative von Neumann algebra
s are called non-commutative measure spaces.
M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple
. It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L2(M,E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L2(M,E) with compact resolvent (e.g. the signature operator
), such that the commutators [D,f] are bounded whenever f is smooth. A recent deep theorem states that M as a Riemannian manifold can be recovered from this data.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple
(A,H,D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that [D,a] is bounded for all a in some dense subalgebra of A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
between affine schemes and commutative ring
s, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings. There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.
There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J. J. Zhang, who add also some general ring-theoretic conditions (e.g. Artin-Schelter regularity).
Many properties of projective schemes extend to this context. For example, there exist an analog of the celebrated Serre duality
for noncommutative projective schemes of Artin and Zhang.
A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme (over a base category), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors. There is also another interesting approach via localization theory, due to Fred Van Oystaeyen
, Luc Willaert and Alain Verschoren, where the main concept is that of a schematic algebra.
' direction in noncommutative geometry is his spectacular discovery (and independently by Boris Tsygan) of a very important new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology
and its relations to the algebraic K-theory (primarily via Connes-Chern character map).
The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator K-theory
and cyclic cohomology. Several generalizations of now classical index theorems allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the JLO cocycle, generalizes the classical Chern character.
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...
concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
in which one of the principal binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s is not commutative; one also allows additional structures, e.g. topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
or norm to be possibly carried by the noncommutative algebra of functions. The leading direction in noncommutative geometry has been laid by French mathematician Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
since his involvement from about 1979.
Motivation
The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functionsFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on them. In general, such functions will form a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. For instance, one may take the ring C(X) of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
complex
Complex 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...
-valued functions on a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X. In many important cases (e.g., if X is a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
), we can recover X from C(X), and therefore it makes some sense to say that X has commutative geometry.
More specifically, in topology, compact Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
topological spaces can be reconstructed from the Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
of functions on the space (Gel'fand-Neimark). In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and schemes can be reconstructed from the categories of quasicoherent sheaves of modules on them (P. Gabriel-A. Rosenberg). For Grothendieck topologies, the cohomological properties of a site are invariant of the corresponding category of sheaves of sets viewed abstractly as a topos (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—some category of sheaves on that space.
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
The dream of noncommutative geometry is to generalize this duality to the duality between
- noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures
- and geometric entities of certain kind,
and interact between the algebraic and geometric description of those via this duality.
Regarding that the commutative rings correspond to usual affine schemes, and commutative C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, as "non-commutative spaces". For this reason, some talk about non-commutative topology, though the term has also other meanings.
Applications in mathematical physics
Some applications in particle physicsParticle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
are described on the entries Noncommutative standard model
Noncommutative standard model
In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on the...
and Noncommutative quantum field theory
Noncommutative quantum field theory
In mathematical physics, noncommutative quantum field theory is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative...
. Sudden rise in interest in noncommutative geometry in physics, follows after the speculations of its role in M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
made in 1997.
Motivation from ergodic theory
Some of the theory developed by Alain ConnesAlain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
to handle noncommutative geometry at a technical level has roots in older attempts, in particular in ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
. The proposal of George Mackey
George Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
to create a virtual subgroup theory, with respect to which ergodic group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s would become homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s of an extended kind, has by now been subsumed.
Non-commutative C*-algebras, von Neumann algebras
(The formal duals of) non-commutative C*-algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representationGelfand representation
In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
, which shows that commutative C*-algebras are dual
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
to locally compact Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.
For the duality
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
between σ-finite measure spaces and commutative von Neumann algebra
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s, noncommutative von Neumann algebra
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s are called non-commutative measure spaces.
Non-commutative differentiable manifolds
A smooth Riemannian manifoldRiemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple
Spectral triple
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint...
. It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L2(M,E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L2(M,E) with compact resolvent (e.g. the signature operator
Signature operator
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the...
), such that the commutators [D,f] are bounded whenever f is smooth. A recent deep theorem states that M as a Riemannian manifold can be recovered from this data.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple
Spectral triple
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint...
(A,H,D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that [D,a] is bounded for all a in some dense subalgebra of A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
Non-commutative affine and projective schemes
In analogy to the dualityDuality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
between affine schemes and commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings. There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.
There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J. J. Zhang, who add also some general ring-theoretic conditions (e.g. Artin-Schelter regularity).
Many properties of projective schemes extend to this context. For example, there exist an analog of the celebrated Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
for noncommutative projective schemes of Artin and Zhang.
A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme (over a base category), abstracting the Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors. There is also another interesting approach via localization theory, due to Fred Van Oystaeyen
Fred Van Oystaeyen
Fred Van Oystaeyen , also Freddy van Oystaeyen, is a mathematician and professor of mathematics at the University of Antwerp.He has pioneered work on noncommutative geometry, in particular noncommutative algebraic geometry....
, Luc Willaert and Alain Verschoren, where the main concept is that of a schematic algebra.
Invariants for noncommutative spaces
Some of the motivating questions of the theory are concerned with extending known topological invariants to formal duals of noncommutative (operator) algebras and other replacements and candidates for noncommutative spaces. One of the main starting points of the Alain ConnesAlain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
' direction in noncommutative geometry is his spectacular discovery (and independently by Boris Tsygan) of a very important new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology
Cyclic homology
In homological algebra, cyclic homology and cyclic cohomology are homology theories for associative algebras introduced by Alain Connes around 1980, which play an important role in his noncommutative geometry...
and its relations to the algebraic K-theory (primarily via Connes-Chern character map).
The theory of characteristic classes of smooth manifolds has been extended to spectral triples, employing the tools of operator 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...
and cyclic cohomology. Several generalizations of now classical index theorems allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the JLO cocycle, generalizes the classical Chern character.
Examples of non-commutative spaces
- In Weyl quantizationWeyl quantizationIn mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
, the symplecticSymplectic manifoldIn mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
phase spacePhase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of classical mechanicsHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
is deformed into a non-commutative phase space generated by the position and momentum operators.
- The standard modelStandard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics is another example of a noncommutative geometry, cf noncommutative standard modelNoncommutative standard modelIn theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on the...
.
- The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.
- Snyder space
- Noncommutative algebras arising from foliationFoliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s.
- Examples related to dynamical systems arising from number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, such as the Gauss shift on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.
See also
- CommutativityCommutativityIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
- Moyal productMoyal productIn mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
- Fuzzy sphereFuzzy sphereIn mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is...
- Noncommutative algebraic geometryNoncommutative algebraic geometryNoncommutative 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...
External links
- Introduction to Quantum Geometry by Micho Đurđevich
- Lectures on Noncommutative Geometry by Victor Ginzburg
- Very Basic Noncommutative Geometry by Masoud Khalkhali
- Lectures on Arithmetic Noncommutative Geometry by Matilde Marcolli
- Noncommutative Geometry for Pedestrians by J. Madore
- An informal introduction to the ideas and concepts of noncommutative geometry by Thierry Masson (an easier introduction that is still rather technical)
- Noncommutative geometry on arxiv.org
- MathOverflow, Theories of Noncommutative Geometry
- S. Mahanta, On some approaches towards non-commutative algebraic geometry, math.QA/0501166