June 17, 2010

From Pseudo-geometry to geometry

Posted in Uncategorized at 10:01 pm by noncommutativeag

The general idea is that pseudo-geometric “space” have canonical spectral theories, and a choice of a spectral theory implies a geometric realization of “spaces”, which associates with every “space” a stack whose base is a topological space(the spectrum of the “space”) and fibers at points are local(in an appropriate sense)”space”. Thus, if “spaces” are represented by categories of a certain type(say abelian, exact or triangulated), then their stalks at points are local categories of the same type.

An important realization is the spectral theory of “spaces” represented by svetle abelian categories [R,Ch3],which was started in the middle of the eighties. Below follows a brief outline of some of its basic notions and facts.

4.1. Topologizing subcategories and the spectrum $Spec(-)$.

A full subcategory of an abelian category $C_X$ is called topologizing if it is closed under finite coproducts and subquotients. For an object $M$, we denote by $[M]$ the smallest topologizing subcategory of $C_X$ containing $M$. One can show that objects of $[M]$ are subquotient of finite coproducts of copies of $M$. The spectrum $Spec(X)$ of the “space” $X$ consists of all non zero $M$ such that $[L]=[M]$ for any nonzero suboject $L$ of $M$. We endow $Spec(X)$ with the preorder $\supseteq$ which is called the specialization preorder.

If $M$ is a simple object, then the objects of $[M]$ are isomorphic to finite direct sums of copies of $M$ and $[M]$ is a minimal element of $(Spec(X), \supseteq)$, if $C_X$ is the category of modules over a commutative unital ring $R$, then the map: $p|\rightarrow [R/p]$ is an isomorphism between the prime spectrum of $R$ with specialization preorder and $(Spec(X),\supseteq)$.

4.2. Local “space”

An abelian category $C_Y$ is called local if it has the smallest nonzero topologizing suncategory. It follows that this subcategory coincides with