June 17, 2010

Pseudo Geometry I

Posted in Uncategorized at 2:51 pm by noncommutativeag

Copied from A.Rosenberg’s book

Geometry of noncommutative “spaces” and schemes

3. Pseudo-geometric start.

The pseudo-geometric noncommutative landscape sketched above is a natural point of departure, by simple reason that it includes most examples of interest. Instead of trying to impose, from the very beginning, general notions of spaces and morphism of spaces, which absorb all the known case, we approach these notions by studying algebraic geometry in certain key pseudo-geometric settings, which are simple enough to not to get lost and, at the same time, sufficient to obtain a rich theory and to see what one should expect or look for in more sophisticated pseudo-geometries.

3.1. “Spaces” represented by categories. In the very first, in a sense the simplest, setting of this kind, “spaces”are represented by svetle(equivalent to small)categories and morphisms of “spaces” X\rightarrow Y are isomorphism classes of(inverse image)functors C_Y\rightarrow C_X between the corresponding categories. This defines the category |Cat|^o of “spaces”. A morphism of “space” is called continuous if its inverse image functor has a right adjoint (called a direct image functor), and it is called flat, if ,in addition, the inverse image functor is left exact(i.e. preserves finite limits). A continuous morphism is called affine if its direct image functor is conservative(i.e. reflects isomorphism) and has a right adjoint. These notions(introduced in [R])unveil unexpectedly rich algebraic geometry,more precisely, geometries, living inside of |Cat|^o. They appear as follows.

3.2. Continuous monads. Fix a “space” S such that the category C_S has cokernels of paris of arrows. We consider End(C_S) of continuous endofunctors of C_S. It is a monoidal category with repsect to the composition of functors whose unit object is the identical functor. The monads in this category are called continuous monads on C_S. In other words, continuous monads on C_S are pairs (F,u),where F is continuous functor C_S\rightarrow C_S and u is a functor morphism F^2\rightarrow F such that uFu=uuF,uF\eta=id_F=u\eta F ¬†for a unique morphism \eta :Id_{C_S}\rightarrow F called the unit of the monad (F,u). A monad morphism is given in a natural way. This defines the category Mon_c(S) of continuous monads on C_S.

If C_S=Z-mod, then the category Mon_c(S) is naturally equivalent to the category Rings of associative unital rings. If C_S is the category of quasi coherent sheaves on a scheme (X,O_X), then Mon_c(S) is equivalent to the category of quasi coherent sheaves A of rings on (X,O_X) endowed with a morphism O_X\rightarrow A of sheaves of rings. In particular, the sheaf of rings of (twisted)differential operators can be regarded as a monad on C_S. If C_S is the category of sets, then the category Mon_c(S) is equivalent to the category of monoids in the usual sense.

3.3. Relative affine “space”. Give a space S, we define the category Aff_S of affine S-space as the full subcategory of |Cat|^o/S whose objects are pairs (X,f:X\rightarrow S),where f is an affine morphism.

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