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.