May 21, 2010

Historical observations of noncommutative algebraic geometry III

Posted in Uncategorized at 9:56 pm by noncommutativeag

2. “Spaces” of noncommutative algebraic geometry

One of the benefits of the pseudo-geometric viewpoint in noncommutative algebraic geometry  is a considerable increase of its range. Roughly, the picture is as follows.

2.1. Spaces and algebras. The duality between compact topological spaces and commutative unital C*- algebra is a fundamental fact of functional analysis discovered by I.M.Gelfand in the late thirties. A.Connes extended formally this duality to the noncommutative setting identifying “noncommutative space” with noncommutative C*- algebras. This eventually led to the creation of noncommutative differential geometry [C1],[C2]. Following Connes’s example. V.Drinfeld [Dr] defined the category of noncommutative affine schemes(he called them “quantum space”) in a similar way, as the category dual to the category of unital associative algebras,forcing to the noncommutative case the duality

[algebras <—>affine schemes]

of commutative algebraic geometry.

2.2 Noncommutative Proj. Noncommutative projective spaces were introduced(by Manin’s suggestion)via a formal extension of the Serre’s description of the category of quasi-coherent sheaves on a projective variety [S]: the category of quasi coherent sheaves on the projective spectrum of an associative graded ring R is the quotient category of the category of graded R-modules by the subcategory of locally finite ones(this approach was further developed in [V1],[V2],[A2],[AZ],[OW], and in a number of other works)

Thus, a noncommutaive projective space X is represented by a category C_{X}, which is regarded as category of quasi coherent sheaves on X. This point of view is well adapted to the affine case: for any associative ring R, the category of quasi coherent sheaves on the corresponding affine schemes is identified with the category R-mod of left R-modules

2.3. “Spaces” represented by abelian categories. From the perspective of the above mentioned developments, a point of view which looked plausible at the end of the eighties(and was later, after appearance of [R1] and [R], adopted by most mathematicians working in the area)is that “spaces” of noncommutative algebraic geometry are represented by abelian categories(thought as their category of quasi coherent or coherent sheaves). If X and Y are “spaces” represented by abelian categories, respectively C_{X} and C_{Y}, then morphism from X to Y are isomorphism classes of additive functors C_{Y}\rightarrow C_{X} called inverse image functors of the morphism they represent.

2.4 “Spaces” represented by triangulated categories. Another viewpoint motivated in the first place by representation theory of reductive groups, and later (around 1993) by problems of mathematical physics(-homological mirror symmetry) is to consider “spaces” represented by(enhanced)triangulated categories, which sometimes can be thought as derived categories of quasi coherent sheaves on thses “space”

2.5. “Spaces”represented by A-infinity categories. At the end of nineties, working on deformation theory, M.Kontsevich expanded geometric flavor by considering “spaces” represented y A-infinity categories.

Historical observations of noncommutative algebraic geometry II

Posted in Uncategorized at 9:20 pm by noncommutativeag

1.5 Cohn’s spectrum. There was another approach to noncommutative local algebra, due to P.Cohn, which is based on the notion of universal localization. Technically, the main difference between Cohn’s approach and the other approaches mentioned above is the instead of dealing with abelian categories of modules over a ring. Cohn’s theory operates with the exact category of projective modules of finite type(Cohn’s original formulations use only matrix ring over a given associative unital ring)

It is worth mentioning that Cohn’s philosophy serves as a base for works of Gelfand and Retakh and their collaboratos on birational noncommutative algebras. Recently, Cohn’s universal localizations found applications in topology.

1.6. Imposing naive geometric spaces. The above mentioned approaches to noncommutative algebraic geometry insisted on a naive generalization of the standard pattern of commutative scheme theory: noncommutative version of schemes were sought as geometric spaces, and the latter were understood as topological spaces endowed with a structure sheaf of associative rings. This holds for D-scheme of Beilinson-Bernstein and for much more recent Kapranov’s version of formal Noncommutative geometry[Ka], because, by nature, D-schemes as well as Kapronov’s NC-schemes, are quasi coherent sheaves of associative algebras on commutative schemes. But, an arbitrary left noetherian associative algebra is not isomorphic to the algebra of global sections of the corresponding structure sheaf on Gabriel’s or Cohn’s (or any other)spectrum. It is therefore not surprising that imposing ringed space as the frame work for noncommutative algebraic geometry and trying to literally mimic the pattern of commutative algebra and algebraic geometry, led to considerable difficulties already on a very basic level.

1.7. Pseudo-geometry verus geometry.

The discovery of quantum groups triggered a flow of new examples supplied mostly by mathematical physics and attributed to noncommutative geometry, reviving some stagnating areas(e.g. Hopf algebra) and involving a big number of mathemaicians and theoretical physicists fascinated by the geometric flavor of this suddenly wide open field of research. This rise of the interest in noncommutative algebraic geometry was marked by the transition from attempts to build the foundations relying on naive generalizations of geometric spaces to the opposite extreme-viewing noncommutative algebraic geometry as pseudo-geometry, that is geometry in which spaces are replaced by something else. The transition was greatly influenced by Connes’ s approach to noncommutative differential geometry. On a more advanced stage, its root can be found in the pseudo-geometric development of Grothendieck’s algebraic geometry between the end of fifties and the beginning of the seventies-going from the category of geometric spaces(that is locally ringed)to the category Esp of spaces which are sheaves of sets on the fpqc presite of affine schemes, then expanding to topoes, algebraic spaces and stacks. Note that in commutative algebraic geometry, all these notions and points of view coexisted and complemented each other.

1.8. Points from commutative algebraic geometry. The abandon of the geometric point of view was due not so much to the limitations of Gabriel’s injective spectrum and shortcomings in the attempts of using it, but, mostly due to the fact the Gabriel’s spectrum was known and appreciated by a few algebraist,while the dominating paradigm of a point came from commutative algebraic geometry: points of commutative schemes are equivalence classes of geometric points, i.e. the morphism from spectra of fields. A naive noncommutative generalization of this notion is obtained by replacing fields by skew fields. Thus the naive points of an affine ‘space’ corresponding to an associative untial ring R are morphisms from R to skew fields, and the equivalence classes of morphisms from R to skew fields are in natural bijective correspondence with complete prime two-sided ideals of the ring R. Noncommutative rings usually have very few completely prime two-sided ideals. One consequence of this other transplantation of a commutative paradigm into noncommutative setting, was a widely adopted opinion that noncommutative algebraic geometry is essentially a geometry without points. Such a viewpoint reduces noncommutative algebraic geometry to the condition of a poor relative of its commutative predecessor: one can not count on a noncommutative version of local algebra, in particular, one can not count on a local study of spaces and morphisms of a spaces, which constitute at least a half of the content of commutative algebraic geometry. Fortunately, this opinion is wrong.

Historical observations on noncommutative algebraic geometry I

Posted in Uncategorized at 8:25 pm by noncommutativeag

Copied from Rosenberg&Kontsevich’s books.

1.1 Serre’s Proj and Gabriel’s spectrum.
The most important early sources of noncommutative algebraic geometry are the description by Serre of the category of coherent sheaves on a projective variety[S] and the introduction by Gabriel of the injective spectrum of a locally noetherian Grothendieck category. Gabriel assigned, in a canonical way, to every locally noetherian Grothendieck category a locally ringed space,whose underlying topological spaces is the injective spectrum-the set of isomorphism classes of indecomposable injectives endowed with Zariski topology. He proved that this assignment reconstructs any noetherian scheme uniquely up to isomorphism[Gab,Chapter 6]
Note that the work of Serre appeared several years prior to scheme theory and the Gabriel’s work around the time as the first two volumes of EGA
1.2.First attempt to define noncommutative scheme.. There were attempts(which started around the end of the sixties and continued to be visible for more than a decade)to initiate noncommutative scheme theory based on a rather straight forward extension of the Gabriel spectrum to the category of left modules over an arbitrary associative untial ring R (its point are those isomorphism classes of indecomposable injectives  (E) for which the quotient category by the left orhogonal to E has simple object)endowed with Zariski topology and a structure sheaf of associative rings detemined by the ring R. Schemes were defined as ringed spaces which are locally affine(See [Go1],[Go2] and references therein)
If R is commutative ring, then there is a natural embedding of the prime spectrum of R into the above defined spectrum of category of R-modules, which is an isomorphism if the R is noethrian(the case considered by Gabriel),but not in general. So, the restriction of this concept of the noncommutative scheme to the commutative case recovers only locally noetherian schemes, which is already an indication of a certain inadequacy of the spectrum used here. Nevetheless, even under noetherian hypothesis, this theory did go beyond the above mentioned definition of a scheme(given in the last section of [Go2]). The declared goal—the creation of a noncommutative version of local algebra was never achieved.
Other movements towards noncommutative algebraic geometry(initiated around the mid-seventies) were based on the prime spectrum of rings endowed with Zariski topology and a structure sheaf of associative rings whose construction required noethrian hypothesis. This produced a version of an affine noetherian scheme. General noetherian schemes were defined as locally affined ringed spaces[VOV].One can show that this version of noncommutative schemes can be obtained from the previous one by considering only left noetherian rings and taking a much coarser version of Zariski topology than the one used in [Go2]. Note that the prime spectrum of most of noncommutative algebra of interesr is rather poor(e.g. It is trivial in the case of Weyl algebra over fields of zero characteristic)
1.3 Supporting motivations. There was a certain outside interest in the quest of noncommutative algebraic geometry already at that time. i.e. In the middle of seventies(see the introduction to [Dix]),which was mostly due to algebraization of representation theory(initiated by works of Kirillov, Gelfand and Kirillov, Dixmer, and his school)and a promise of new insights and possible applications to representation theory of algebraic groups. Enveloping algebra of Lie algebra,and some other algebra of interest
1.4 D-modules and D-schemes. Then, starting from 1980, Beilinson and Bernstein developed a compromise type noncommutative algebraic geometry-the theory of D-schemes(which are usual commutative scheme equipped with a subsheaf of the sheaf of (twisted)differential operators)in order to study representation theory of reductive algebraic groups. This important development led to a break through in representation theory and distracted the curiosity of most working mathematcians from attempts to construct noncommutative scheme theory based on Gabriel’s spectrum, or on the prime spectrum of associative rings.