Re: Grothendieck's relative point of view

[zoran skoda <zskoda@gmail.com>]

Dear colleagues,

the flexibility of definitions with respect to the change of base is in my
understanding not the main aspect in Grothendieck's relative point of view.

The idea is that the general *study of properties* of objects should be
better replaced by the more fundamental study of properties of morphisms.
This is different from just looking how the properties of *objects* change
under base change. Thus the notions like affine scheme, quasicompact scheme
and so on, are replaced by the study of affine, flat, quasicompact,
quasi-separated, proper etc. morphisms over an arbitrary base. The
definition of many local properties can be extended say from schemes to
algebraic stacks by pulling back to a chart by a scheme (or algebraic space)
and checking the property there. Many properties of morphisms can be
expressed in terms of properties of functors among the corresponding
categories of quasicoherent sheaves, what enables nowdays further
"noncommutative" generalizations in terms of generalizations of such
modules.

Thus while affine scheme corresponds to (the spectrum of a) ring, affine
morphism generalizes a ring morphism in the sense that the direct inverse
image functor is faithful and has both left and right adjoint; therefore it
corresponds to a monad having a right adjoint. Thus being represented by an
algebra on the object level gets generalized to being represented by a
sufficiently good monad at the morphism level.

Now, if one emphasises the change of base aspect then this means that you
look just at properties of morphisms measured with respect to the codomain
fibration; if we emphsise on the behaviour with respect to quasicoherent
modules, then we measure the properties of morphism with respect to to the
stack of quasicoherent sheaves. It is often the case that the properties of
morphisms in geometry are measured by action on/behaviour with respect to
specific kinds of objects. For example, in algebraic geometry there are two
main kinds of global finiteness: quasicompact morphism and quasi-separated
morphism; both derived from a notion of quasi-compact object.

Zoran Škoda


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]