Noncommutative algebraic geometry is the study of ‘spaces’ represented or defined in terms of algebras, or categories. Commutative algebraic geometry, restricts attention to spaces whose local description is via commutative rings and algebras, while noncommutative algebraic geometry allows for more general local (or affine) models. The categories are viewed as categories of quasicoherent modules on noncommutative locally affine space, and by affine one can think of many algebraic models, e.g. -algebras; the algebra and its category of modules are in the two descriptions viewed as representing the same space (Morita equivalence should not change the space). Categories are needed as the global “algebra of functions” is usually an incomplete global description while it may be good enough locally. The categories are typically abelian, triangulated, dg-categories or A-infinity-catgories.
Noncommutative algebraic geometry extends the algebraic geometric methods (local rings, intersection theory etc.) to study noncommutative algebras, and conversely, uses noncommutative algebras in the study of commutative algebraic varieties (Brauer group?s, noncommutative desingularizations, stacks etc.). But some features, phenomena and methods do not have commutative analogues.
Noncommutative algebraic geometry may be considered a subfield of general noncommutative geometry. Among prominent other subfields, the most influential is the direction lead by Alain Connes. In Connes’ noncommutative geometry the algebras in question are operator algebras viewed as algebras of continuous, smooth or measurable functions or more general continuous global sections. There it is customary and sufficient to describe a space globally by a single algebra: smooth and continuous context allow for partition of unity so there is no need for sheaves and gluing local descriptions, to define the very spaces.
The subject is older than mainstream noncommutative geometry and it is very much implicit in Pierre Gabriel’s thesis “Des catégories Abéliennes” (Bull. Soc. Math. France 1962). The Gabriel spectrum (consisting of injective indecomposables) amounts to a reconstruction of a quasiseparated? quasicompact scheme from its category of quasicoherent? -modules (now generalized to all schemes, as the Gabriel-Rosenberg theorem). He looked at localization functors, and corresponding subcategories (reflective and coreflective), and proved some theorems on localization of quasicoherent sheaves. At the same time, Grothendieck taught us that to study a space it is enough to have a category of sheaves – a Grothendieck topos – on this would-be space.
Together with two Auslander-Goldman papers ” Maximal orders ” and ” The Brauer group of a commutative ring ” – both in Trans. AMS 1960 – Gabriel’s paper is a precursor of a more articulated development starting with Mike Artin’s ” On Azumaya algebras and finte dimensional representations of rings ” (J. Alg. 1969). Artin’s 1969 paper characterizes Azumaya algebras via their identities and paved the way to generalize algebraic geometry to rings satisfying polynomial identities, via their representations and invariant theory. Subsequent major contributions were made by Claudio Procesi with his book ” Rings with polynomial identities ” (Marcel Dekker 1973) and his paper ” The invariant theory of matrices ” (Adv. Math. 1976) and by Mike Artin and Bill Schelter’s ” A version of Zariski’s main theorem for PI rings ” (Amer. J. Math. 1979). The link with representation theory, using seminal work by Paul Cohn and George Bergman, was clarified by Claus Ringel ” The simple Artinian spectrum of a finite dimensional algebra ” (in Dekker Lect. Notes 51, 1979) and culminated in Aidan Schofield’s book ” Representations of rings over skew fields ” (1985). These developments have been influential for recent work in ‘formal’ noncommutative geometry, such as Maxim Kontsevich’s ” Formal (non)commutative symplectic geometry ” (1993), Joachim Cuntz and Daniel Quillen’s “Algebra extensions and nonsingularity” (JAMS 1995), “Noncommutative smooth spaces” by M. Kontsevich and A. L. Rosenberg (Gelfand seminar and math.AG/9812158) (see quasi-free algebra).
In the 70ties, localization theory was popular among ring theorists, and, several people tried to generalize structure sheaves to noncommutative algebras. Gabriel-like localizations via alternative formalism of “kernel functors” (Goldman 1969) led to the books by Jonathan Golan ” Localization of noncommutative rings ” (Marcel Dekker 1975) and Fred Van Oystaeyen ” Prime spectra in noncommutative algebra ” (Springer LNM 444, 1975), and making the link with the Artin-Procesi approach, the book
Localizations were viewed as analogues of open sets. However, the lattice of torsion theories? is not distributive, what presented major difficulties at the time. Around 1995, a major descent theorem was proved by van Oystaeyen and Willaert and independently in a work of A. Rosenberg in Stockholm (1988). It is sort of generalized sheaf condition for quasicoherent modules, with respect to covers by Gabriel localizations.
Cohn’s universal localizations were used by Paul Cohn in ” The affine scheme of a general ring ” (in Springer LNM 753, 1979) and later by Claus Ringel and Aidan Schofield as mentioned before. In England, Goldie’s localization theory led to the theory of ‘cliques’ and ‘clans’ of prime ideals (Jategaonkar, Hajarnavis and others) which may in retrospect be viewed as influential to present local description via quivers and A-infinity structures.
Around 1980, the Leningrad school of integrable systems (Sklyanin, Fadeev, Takhtajan) created examples of quantum groups, and soon after that were joined by Drinfel’d, Novikov, Jimbo, and independently, Manin and Woronowicz; these in turn give good examples in affine noncommutative geometry (spaces represented by a single nongraded ring), moreover the actions of group-like objects are considered starting the subject of equivariant noncommutative algebraic geometry.
In the early 80ties maximal orders were revisited as noncommutative versions of normal varieties. In particular their local structure was investigated. Extending prior work by Mark Ramras, Mike Artin succeeded in describing the etale local structure of maximal orders over surfaces. In trying to generalize this to higher dimensions, one was quickly led to impose strong homological properties on the orders (and later, more general algebras). This led to the investigation of homological homogeneous rings (work by Ken Brown and others), Artin-Schelter regular algebras and, finally, Auslander-regular rings.
In this vein, around 1989, noncommutative projective geometry was developed by Artin, Zhang and collaborators, where Serre’s theorem on of a graded ring is taken as a definition in the noncommutative case: quasicoherent sheaves over a of a graded ring is the category of graded modules modulo the modules of finite length. They assume strong regularity properties on graded rings, limiting to the geometry which is very close to commutative projective geometry. Using Ore localization, F. van Oystaeyen has defined another version of noncommutative projective geometry, based on his notion of schematic algebra.
A huge project started, trying to classify Auslander-regular algebras, graded-connected and generated in degree one (that is, noncommutative analogons of projective n-space) for small dimensions n. In dimension 3, the initial work was done by Artin and Schelter, culminating in the influential papers ATV1 and ATV2 (ATV for Mike Artin, John Tate and Michel Van den Bergh, ATV1 in Grothendieck Festschrift 1990, ATV2 in Invent. Math 1991). In this classification a class of algebras re-appeared which were initially discovered by Odeskii and Feigin, the so called Sklyanin algebras. Ring-theoretic investigations of the Sklyanin algebras were done by Tate and Van den Bergh (Noetherianity for all dimensions n) and in dimension 4 a geometric study via ‘point’ and ‘line’ modules was carried out by Thierry Levasseur, Toby Stafford and Paul Smith. Further classifications in dimension 3 (dropping the degree one generator condition) and 4 were done by these people and their students. In the ungraded case, maximal orders with excellent homological properties were recently used in connection with quotient-singularities and their desingularizations as in the noncommutative crepant resolution by Michel Van den Bergh (Abel symposium 2002).
Lieven Le Bruyn to the Lab-elves : what follows is part of the original text by Zoran Skoda based on his association with Alex Rosenberg. I’ve tried to include the major developments above with precise references. I don’t understand the Van Oystaeyen-Willaert vs. Rosenberg part nor the Le Bruyn (that is,me) vs. Laudal part. But, feel free to edit in any way you feel like.
Zoran Škoda: my (very unfinished and criticised before any level of completion beyond quick initial start) entry is not at all based on any association, but rather on much more general viewpoint on noncommutative geometry including not only ring-theoretic aspect but also the categorical aspect on the line GABRIEL-Manin-Rosenberg in abelian and Kapranov-Bondal-Orlov-van den Bergh-Kontsevich in triangulated, dg and A-infinity setup. My emphasis was more on concepts, and categorical viewpoint, than on listing all references (history entry should be separately and welcome). For example, in the addition above casually is mixed graded and nongraded setup: nlab regulars should be warned about points of entry and motivation of affine and proijective geometries. On the other hand, I am not competent enough to write extensively on the very developed, but for category theorists rather special case of slightly noncommutative projective geometry of line Artin-Schelter-Tate-Zhang-Smith though intend to do part of this in specialized entries (one of the reasons why I listed ALL papers in Grothendieck Festschrift was that interest in Artin’s work; last week I went to integrable systems conference partly because of interest in elliptic algebras which belong to the class from this paper; Le Bruyn should notice that I was long time ago passworded member of his web chat group on his style of ncg which to my regret was not very active). I placed some of the stuff from below into the text above to make it all-inclusive without disposing any material except the repetitions of facts. Much more is missing here, for example algebraic K-theory for nc rings and cyclic homology etc.
When one is serious about the point of view that every space is represented by some abelian category and that every abelian category represents a (noncommutative) space this way, one wants to find the spectrum of an abelian category in the way that one forms the spectrum (geometry) of a ring. In particular since in the noncommutative case the classical spectra from ring theory (like primitive spectrum and Gabriel’s spectrum are usually too small to represent noncommutative spaces)
Motivated by this A. L. Rosenberg has developed many spectral constructions extending from the left spectrum of a ring to various spectra of abelian categories, triangulated categories and right exact categories (in his sense) and created a categorical version of a notion of (quasicompact relative) noncommutative scheme over a base category (viewed as category of sheaves on ).
Homological mirror symmetry by Maxim Kontsevich is a source of lots of examples of spaces of derived algebraic geometry, which are represented by triangulated categories or their dg- or -enhancements; this has been enhanced by the insight in the structure of derived categories of coherent sheaves and their deformations like Landau-Ginzburg models, by Beilinson(1977/8), Kapranov (1985-), Orlov, Bondal, van den Bergh, Polishchuk, Bridgeland and others. Another big source of example is deformation quantization and related topics involving usage of and -algebras. Understanding of the role of quivers in noncommutative geometry has also being stimulated by independent and original works of Lieven le Bruyn and Arfinn Laudal, where the latter has also being motivated by deformation theory.
Derived noncommutative geometry is a subject related to the commutative derived algebraic geometry of C. Simpson’s school. Cyclic and Hochschild homology play a major role.
This approach represents spaces by categories enriched in cochain complexes (dg-category approach) or enriched in spectra. A-infinity categories have some advantages over working with dg-categories.
Notice that such categories are models for stable (∞,1)-categories. Moreover, every stable (∞,1)-category may automatically be regarded as a stable (∞,1)-topos. Therefore the method of thinking of a generalized space in terms of a triangulated category is in line with the way topos theory and in particular higher topos theory characterizes generalized spaces by toposes.
The most nontrivial result seem to be a noncommutative version of the degeneration of Hodge-to-de Rham spectral sequence, conjectured by Kontsevich and in one version proved by D. Kaledin.
M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287 (doi).
A. Bondal, M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow MJ 2003
O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509–580; (euclid).
the n-geometry cafe, the NAG canon (blog)
F. van Oystaeyen, Algebraic geometry for associative algebras, Marcel Dekker 2000. vi+287 pp.
S. Mahanta, On some approaches towards non-commutative algebraic geometry, math.QA/0501166
L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107
D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, pdf, TeX
MathOverflow, Theories of Noncommutative Geometry