Grothendieck's relative point of view
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
participants (4)
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Colin McLarty -
Eduardo J. Dubuc -
Graham White -
zoran skoda