Grothendieck’s theory of schemes and the algebra–geometry duality

Abstract

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \(A \rightarrow B\) into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and (what we shall call) the structure-semiotics duality (of which the syntax-semantics duality for propositional and predicate logic are particular cases). Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general (commutative) rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics (e.g. Stone duality, Gelfand duality, Pontryagin duality, Galois-Grothendieck duality, etc.) can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” (Friedman) that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding (co)domains.