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It is often said that one of the most influential concepts Grothendieck introduced through the scheme theory is the emphasis on the "relative perspective," that is, properties should be interpreted as a property of morphisms instead of one of the objects. However, I don't know exactly what was the main idea which made Grothendieck think about this point of view. My guess is that it was the consciousness of the fact that all rings can be seen as a space (i.e. affine scheme): in the pre-Grothendieck era, one thought that a variety is an absolute object that existed by itself (here, the base field $k$ was not seen as a "space"). After scheme theory, however, variety was defined as a scheme over an affine scheme (which comes from some field $k$) $\mathrm{Spec}\ k$ with some nice properties, that is, a variety is a morphism between spaces $V \to \mathrm{Spec}\ k$. I don't know whether this is the case. I don't have any evidence to support my conjecture. My question is:

  1. Is my guess correct?
  2. If it is wrong, what is the answer?

Sorry for asking a somewhat vague question, but I will appreciate your answers.

Takuei ASAI
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  • I've always thought it came from homological algebra, where some properties about modules (like projectiveness, flatness or injectiveness) are essentially relative properties: they tell something about morphisms of those objects. – Compacto Mar 12 '22 at 14:04
  • Also, many concepts can be explained using morphisms, so the relative picture is very flexible and covers many cases at once. A subvariety is understood as the inclusion map. A submersion is also a map. A family of varieties are preimages by a map. Generalized points are also maps. – Compacto Mar 12 '22 at 14:05
  • Also I want to say people were starting to dare to think that giving rigorous constructions of serious moduli spaces (i.e. other than toy examples) was a real possibility, but as it was Weil had had to move mountains "just" to construct the Jacobian in characteristic $p$. So given the central role that the relative perspective plays in moduli theory, that might have been part of it. – Tabes Bridges Mar 13 '22 at 17:41

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