May 21, 2010
Historical observations of noncommutative algebraic geometry III
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 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 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 is represented by a category , which is regarded as category of quasi coherent sheaves on . 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 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 and , then morphism from X to Y are isomorphism classes of additive functors 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.