June 17, 2010

Pseudo Geometry II

Posted in Uncategorized at 5:20 pm by noncommutativeag

Copied from A.Rosenberg’s book

Geometry of noncommutative “spaces” and schemes

3.4 Theorem

The category Aff_S is equivalent to the category Ass_S whose objects are continuous monads on the category C_S and morphisms are conjugacy classes of monads morphisms.

If C_S=Z-mod, then the category Ass_S is equivalent to the category whose objects are associative unital rings and morphisms are conjugacy classes of ring morphisms. If C_S=Sets, then Ass_S is equivalent to the category whose objects are monoids and morphisms are conjugacy classes of monoids morphism. This shows that the choice of base “space” S influences drastically the rest of the story.

3.5. Locally affine relative “space”. Locally affine S-“space” are defined in an obvious way, once a notion of a cover(a quasi-pretopology)is fixed. We introduce several canonical quasi-pretopologies on the category |Cat|^o. Their common feature is the following: if a set of morphisms to X is a cover, then the set of their inverse image functors is conservative and all inverse image functors are exact in a certain mild way. If, in addition, morphisms of covers are continuous, X has a finite affine cover, and the category C_S has finite limits, then this requirement suffices to recover the object X from the covering date uniquely up to isomorphism(i.e the category C_X is reovered uniquely up to equivalence)via “flat descent”

3.6 “Spaces” determined by presheaves of sets on Aff_k.

By definition, the category Aff_k of noncommutative affine k-schemes is the category opposite to the category Alg_k of associative unital k-algebras; so that presheaves of sets on Aff_k are functors from Alg_k to Sets. The presheaves of sets on Aff_k appeared in our work with Maxim Kontsevich, for the first time in order to introduce noncommutative projective spaces. It was an attempt to imitate the standard commutative approach realizing schemes(and more general spaces) as sheaves of sets on the category of affine schemes endowed with an appropriate Grothendieck  pretopology. It turned out that it is not clear a priori what an appropriate pretopology in the noncommutative case is: Zariski pretopology is irrelevant, because the noncommutative projective space is not a scheme- it does not have an affine Zariski cover. Flat affine covers seemed to be a as a natural choice, but, they do not form a pretoplogy-invariance under base change fails. Similar story with Grassmannians and other analogs of commutative constructions. The elucidation of this problem is as follows. Consider the fibered category \widetilde{Aff_k}  with the base Aff_k whose fibers are categories of left modules over corresponding algebras. For every presheaf of sets X on Aff_k, we have the fibered category \widetilde{Aff_k}/X induced by \widetilde{Aff_k} along the forgetful functor \widetilde{Aff_k}/X\rightarrow Aff_k. The category Qcoh(X) of quasi coherent sheaves on the presheaf X is defined as the category opposite to the category of cartesian sections of the fibered category \widetilde{Aff_k}. For a pretopology \tau on Aff_k/X, we define the subcategory Qcoh(X,\tau) of quasi-coherent sheaves on (Aff_k/X,\tau).

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