Contents

Idea

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. $A_\infty$-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-categories.

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 groups, noncommutative desingularizations, stacks etc.). But some features, phenomena and methods do not have commutative analogues.

Relation to other kinds of geometry

Relation to ordinary algebraic geometry

The direct “naive” generalization of Grothendieck-style algebraic geometry via sheaves on a site (Zariski site, etale site etc.) of commutative rings-op to non-commutative rings does not work, for reasons discussed in some detail in (Reyes 12). This is the reason why non-commutative algebraic geometry is phrased in other terms, mostly in terms of monoidal categories “of (quasicoherent) abelian sheaves” (“2-rings”). See at 2-algebraic geometry for more.

Relation to noncommutative geometry a la Connes

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.

Early history

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 isomorphism classes of injective indecomposables) amounts to a reconstruction of a noetherian quasiseparated quasicompact scheme from its category of quasicoherent $\mathcal{O}$-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 $n\times n$ 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) and Kontsevich-Rosenberg’s ”Noncommutative smooth spaces“ (cf. bibl. and 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

• Fred Van Oystaeyen, Alain Verschoren, Non-commutative algebraic geometry, Springer LNM 887, 1981.

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. This fact is now seen almost obvious; namely it is the comonadic descent for a comonad induced by a conservative family of flat localizationshaving right adjoint, which in the Gabriel localization clearly satisfy the conditions for Beck’s comonadicity theorem.

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 A. Brown and others), Artin-Schelter regular algebras and, finally, Auslander-regular rings.

In this vein, around 1989, the noncommutative projective geometry was developed by Artin, Zhang and collaborators, where Serre’s theorem on $Proj$ of a graded ring is taken as a definition in the noncommutative case: quasicoherent sheaves over a $Proj$ 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 a 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), (ATV2) by Mike Artin, John Tate and Michel Van den Bergh (see bibl.). 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).

Approaches

Spectrum of an abelian category

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 $X$ over a base category $C$ (viewed as category of sheaves on $Spec k$). Many other variants of spectra of abelian categories can be constructed; there is a general pattern though, cf. spectral cookbook.

Sheaves of sets on a $Q$-category

A space in the sense of a Grothendieck school is a sheaf $F$ of sets on the category of affine schemes $Aff$ equipped with some subcanonical Grothendieck topology and which is locally affine, i.e. there is a cover of $F$ (in the sense of induced Grothendieck topology on the category of presheaves) by representables. The noncommutative analogue of $Aff$ is the opposite to the category of noncommutative unital rings $NAff = Ring^{op}$. The problem is that the natural candidates for covers in $NAff$ are not making classes which satisfy the axioms for Grothendieck topology, namely, the crucial stability under pullback fails. However $A$ can be equipped with a structure of a Q-category, and then one can consider the sheaves of sets on a Q-category and define in an analogous way when they are locally affine. Similar approach works for noncommutative stacks.

Derived noncommutative geometry

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.

Here one thinks of dg-categories, A-infinity categories, Spectra-enriched categories and ultimately of stable (∞,1)-categories as formal duals to non-commutative derived spaces:

Any locally presentable stable (∞,1)-category may be regarded as a stable (∞,1)-topos (the “stable Giraud theorem”, see there). Therefore, thinking of a non-commutative space as the formal dual to a triangulated category or rather to a stable (∞,1)-category is directly analogous to the way topos theory and in particular higher topos theory characterizes generalized spaces as formal duals of toposes.

More in detail, every stable (∞,1)-category with a set of generators is equivalently the $\infty$-category of (∞,1)-modules over an A-∞ algebra (or rather $A_\infty$-algebroid, in general, due to Schwede-Shipley 01) and in this sense manifestly the formal dual to a non-commutative derived variety (see there for more).

The notion of noncommutative motive as localizations of stable $\infty$-categories (see there), due to Blumberg-Gepner-Tabuada 10, directly ties into this.

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.

Homological mirror symmetry

Homological mirror symmetry by Maxim Kontsevich is a source of many examples of spaces of derived algebraic geometry, which are represented by triangulated categories or their dg- or $A_\infty$-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), Mikhail Kapranov (1985-), Dmitri Orlov, Alexei Bondal, Michel van den Bergh, Alexander Polishchuk, Tom Bridgeland and others.

Other motivations

Another big source of examples is deformation quantization and related topics involving usage of $A_\infty$ and $L_\infty$-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?. Laudal was also motivated by deformation theory.

Related concepts

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References

• the entries on the subject include: noncommutative scheme, sheaf on a noncommutative space, noncommutative projective geometry, localization, schematic algebra, Q-category, quasicoherent sheaf, spectrum of an abelian category, spectral cookbook, quantum group, noncommutative principal bundle, quantum homogeneous space, equivariant noncommutative geometry?

• list of pages with the tag noncommutative geometry

• related subjects include: noncommutative geometry, Hopf algebra, gebra, bialgebroid, quantization, deformation theory

• 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

• M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Birkhäuser 2000; arXiv:math/9812158

• M. Kontsevich, A. Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (dvi,ps), Noncommutative spaces and flat descent, MPI-2004-36 dvi,ps, Noncommutative stacks, MPI-2004-37 dvi,ps

• A. L. Rosenberg, Noncommutative schemes, Compos. Math. 112 (1998) 93–125, MR99h:14002, doi

• O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509–580; (euclid).

• (ATV2) M. Artin, J. Tate, M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388, MR93e:16055, doi

• (ATV1) Michael Artin, John Tate, Michel Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, 33–85, Progr. Math. 86, Birkhäuser 1990.

• 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

• Zoran Skoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770; preprint MPIM2009-3.

• MathOverflow, Theories of Noncommutative Geometry

• Reyes, Sheaves that fail to represent matrix rings (arXiv:1211.4005)

Discussion of noncommutative formal geometry of infinitesimal neighborhood of commutative schemes within noncommutative ambient schemes is in

• Mikhail Kapranov, Noncommutative geometry based on commutator expansions (arXiv:9802041)

For more on this see at Kapranov's noncommutative geometry. See also the references at derived noncommutative geometry