June 17, 2010
Historical observation of noncommutative algebraic geometry IV
Copied from A.Rosenberg’s book
Geometry of noncommutative “spaces” and schemes
2.6 “Spaces” defined by presheaves of sets on the category of noncommutative affine schemes.
The category of affine noncommutative k-schemes is the category opposite to the category of associative unital k-algebras. Some of the important examples of noncommutative “space”, such as noncommutative Grassmannians, flag varieties and many others [KR1],[KR2],[KR3], are defined in two steps. The first step is a construction of a presheaf of sets on (i.e. a functor from category of unital associative k-algebras to the category of sets). In commutative algebraic geometry, the second step is taking the associated sheaf with respect to an appropriate(fpqc or Zariski)topology on . In noncommutative geometry, we assign, instead, to every presheaf of sets on a fibered category whose fibers are categories of modules over k-algebras and define the category of quasi-coherent sheaves on this presheaf as the category opposite to the category of cartesian sections of this fibered category [KR4]. The category of quasi coherent presheaves represents the “space” corresponding to the presheaf of sets.
2.7. Commutative “spaces” ,which “live” in symmetric monoidal categories.
After the formalism of Tannakian categories appeared at the end of the sixties-beginning of the seventies [Sa],[DeM], and super-mathematics approximately at the same time, the idea of mathematics(or at least algebra and geometry),which uses general symmetric monoidal categories, instead of the symmetric monoidal category of vector spaces, became familiar. In [De], Deligne presented a sketch of a fragment of commutative projective geometry in symmetric monoidal k-linear abelian categories as a part of his proof of the characterization of rigid monoidal abelian categories having a fiber functor.
Manin defined the(category of coherent sheaves on the)Proj of a commutative -graded algebra in a symmetric monoidal abelian category endowed with a fiber functor [M1]using, once again, the Serre’s description of the category of coherent sheaves on a projective variety as its definition.
2.8. Quantized enveloping algebras and algebraic geometry in braided monoidal categories.
While working(in 1995) on a quantum analog of Beilinson-Bernstein localization construction, it was discovered that “spaces” of noncommutative algebraic geometry could be something different from just abelian or Grothendieck categories. In this particular situation, the natural action of the quantized enveloping algebra if a semisimple Lie algebra on its quantum base affine space becomes differential only if the whole picture is put into the monoidal category of -graded modules endowed with a braiding determined by the Cartan matrix of the Lie algebra (see [LR2],[LR3],[LR4]) . This list(which is far from being complete)shows that the range of objects-spaces and morphisms of spaces, of noncommutative algebraic geometry is considerably larger than the range of objects of commutative algebraic geometry.