May 21, 2010
Historical observations on noncommutative algebraic geometry I
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 (its point are those isomorphism classes of indecomposable injectives 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 . 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 -modules, which is an isomorphism if the 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.