Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later works of the Grothendieck school, such as Berthelot's Cohomologie Cristalline des Schemas de Caracteristique $p>0$ or Illusie's Complexe Cotangent et Déformations I et II.

Question I. What are some instances in which working in the full generality of a ringed topos gives one more powerful tools than just working with $S$-schemes?

One example I'm aware of is in Illusie's Complexe Cotangent books. As remarked by Jonathan Wise in this MO question, working with ringed topoi in this setting enables one to study more interesting deformations.

I heard that a modern example might be the Falting topos, which appears in Abbes–Gros–Tsuji's book The p-adic Simpson Correspondence. (I don't really understand this example, however.)

  • $\begingroup$ I do not know enough algebraic geometry to comment on use of the notion. But from the point of view of topos theory I would say there is not enough difference between a site and a topos to make question II interesting. They are justs two different way to talk about the exact same thing, and you will often need to jump back and forth between the two to get things done. $\endgroup$ – Simon Henry Mar 6 '20 at 8:16
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    $\begingroup$ I don't think it's a question about more powerful vs. less powerful, but just that there are certain naturally arising situations which you'd like a nice precise formalism to be able to discuss. Basically, families of varieties where the parameter (e.g. coefficients of a defining equation) varies not algebraically but analytically, or maybe just continuously, or whatever. $\endgroup$ – Dustin Clausen Mar 6 '20 at 13:36
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    $\begingroup$ @DustinClausen Do you know concrete(-ish) examples of such situations? For instance, what kind of problems would "require" working with schemes over (say) formal schemes, adic spaces, or other kinds of spaces appearing in algebraic or arithmetic geometry? $\endgroup$ – Sofia Mar 6 '20 at 20:28
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    $\begingroup$ @sofia : The main drawback of the site formalism compared to toposes is the treatment of geometric morphisms (A "site morphisms" corresponds to a very special kind of geometric morphisms, essentially one that preserves the basis). So one generally prefer a definition in terms of toposes as it makes it easier to understand the action of geometric morphisms on the object you are defining. But at some point you also want to have a site definition for computation in concrete example. The question of whether you start from the topos definition and translate it as a site definition, or... $\endgroup$ – Simon Henry Mar 7 '20 at 10:41
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    $\begingroup$ ...you start from a site definition and show invariance under equivalence of site is really only a matter of choice of exposition. Maybe I should also note that in the case of Hakim definition of relative scheme the Topos/site translation is very simple: She defines scheme over a ringed topos (T,R) as the stackification of the pre-stack "X -> R(X)-Scheme" for X an object of a topos. But it is enough to consider X in a site of definition for this definition to apply. So here this discussion is really irrelevant. $\endgroup$ – Simon Henry Mar 7 '20 at 10:46

The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question stood at the more basic level of the relevance of relative schemes over something else than a scheme.

In Grothendieck's philosophy, a relative scheme $f\colon X\to S$ functions as a family of schemes $X_s$ (for $s\in S$) parameterized by the base $S$, which is a scheme. Both $X$ and $S$ are schemes, and $f$ is a morphism of schemes (possibly with additional properties such as being flat, proper, finite, étale, smooth, etc.)

There are instances where one wants to consider families of schemes which are parameterized by something else, such as a (complex, Berkovich, Huber…) analytic space. For example for formulating GAGA-type theorems: what do complex analytic families of complex varieties in a given projective space look like, when the parameter set is an open disk, say.

In most important cases, those spaces are essentially characterized by rings (eg, local rings, or rings of functions over a compact, affinoid subspace) and sometimes the above study can be reduced to the study of relative schemes over these rings. Such a technique is systematically used in nonarchimedean geometry, for instance.

Nevertheless, it might be interesting to have under disposal a full-fledged theory of relative schemes over bases. Being over a ringed topos, Monique Hakim's theory can encompass all of the above situations.

In any case, such a theory won't prove the basic (but difficult) results from commutative algebra that are probably needed. In nonarchimedean geometry, some work is needed, for example, to compare the scheme $\mathop{\rm Spec}(A)$ and the affinoid space $\mathscr M(A)$, when $A$ is an affinoid algebra, and similarly for a relative family $X\to \mathop{\rm Spec}(A)$ and its analytification $X^{\mathrm {an}}\to \mathscr M(A)$.

  • $\begingroup$ Thanks for your explanation! From what I understand, it seems to me that being able to formulate GAGA-type theorems over general bases is one of the biggest selling points of Hakim's theory. Is this correct? $\endgroup$ – Sofia Mar 15 '20 at 23:08
  • $\begingroup$ Also, do you know about situations outside the setting of GAGA-type theorems in which it is useful to (be able to) work over such a general base? $\endgroup$ – Sofia Mar 15 '20 at 23:10

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