Grothendieck's relative point of view

Grothendieck's relative point of view

[Graham White]

I'm looking for some history of what seems to be called "Grothendieck's
relative point of view": the idea that what ostensibly seems to be an
isolated mathematical object may well be better studied by
systematically looking at families of such objects. Now Grothendieck
certainly uses it in his proof of, and generalisation of, Riemann-Roch.
And there are historical essays in various Wikis that say so. But:
i) how does he come to this idea?
ii) are there any places where he talks about it and about what it
means? What generality is it meant to apply in?
iii) does anybody before him talk about it? (In particular, there is
some explicit parametrisation involved in Weil's approach to algebraic
number theory, and the number field/function field analogy: does Weil
ever talk explicitly about this?)

Thanks

Graham



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Re: Grothendieck's relative point of view

[Eduardo J. Dubuc]

Graham White wrote:
> I'm looking for some history of what seems to be called "Grothendieck's
> relative point of view":

What about "Grothendieck's change of base", so much used in SGA4, and
extensively advocated and used by Joyal and others by considering the
category of Grothendieck topoi over a given Grothendieck topos (not Sets) ?.

Is not this Grothendieck's relative point of view ?

I am just asking.   e.d.


   the idea that what ostensibly seems to be an
> isolated mathematical object may well be better studied by
> systematically looking at families of such objects. Now Grothendieck
> certainly uses it in his proof of, and generalisation of, Riemann-Roch.
> And there are historical essays in various Wikis that say so. But:
> i) how does he come to this idea?
> ii) are there any places where he talks about it and about what it
> means? What generality is it meant to apply in?
> iii) does anybody before him talk about it? (In particular, there is
> some explicit parametrisation involved in Weil's approach to algebraic
> number theory, and the number field/function field analogy: does Weil
> ever talk explicitly about this?)
>
> Thanks
>
> Graham


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Re: Grothendieck's relative point of view

[Colin McLarty]

2010/7/7 Eduardo J. Dubuc <edubuc@dm.uba.ar>:
> Graham White wrote:
>>
>> I'm looking for some history of what seems to be called "Grothendieck's
>> relative point of view":
>
> What about "Grothendieck's change of base", so much used in SGA4, and
> extensively advocated and used by Joyal and others by considering the
> category of Grothendieck topoi over a given Grothendieck topos (not Sets) ?.

It is an interesting question exactly how Grothendieck came to this.
The Grothendieck-Riemann-Roch probably was the first thing that made
the idea prominent.  But Grothendieck would also have been encouraged
to think this way by his interest in sheaves (of sets) and the
more-or-less well known fact (in the 1950s) that the idea of a "sheaf
X over a sheaf S on space T" has two equivalent definitions: as either
an arrow in the category of sheaves on T, or a local homeomorphism to
the espace etale of  the sheaf S.  This is just the special case of
what Eduardo remarks, a slice topos over a spatial topos is again a
spatial topos.

These things arose in the 1950s.

But in itself, the relative point of view is an extension to geometry
of the usual practice in algebraic number theory which goes back
before the 1850s.  An algebraic number field is not just a field but a
field extension.  We constantly view it as an extension of the
rational field.  But more than that, number theorists have long
focused on questions about, say, all cyclic extensions of any fixed
algebraic number field, or which groups can occur as Galois groups of
one algebraic number field over another.

And those are just the natural outgrowths of a much older question,
clarified by Abel and Galois but asked long before (e.g. it is the
subject of Euclid Book X, stated in terms of constructions rather than
polynomials):  If a given polynomial has no rational roots, does it
have roots expressible in terms of square roots of rationals, or n-th
roots?

best, Colin


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Re: Grothendieck's relative point of view

[zoran skoda]

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


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