May 21, 2010
Historical observations of noncommutative algebraic geometry II
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.