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 → B into other similar structures B. This vantage point will allow us to analyze the relationship between the algebrageometry 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.1 According to J.-P. Serre, "[t] his first approximation to the theory of affine schemes was 'in the air ' in 1954-1955, at least with some restrictions on the rings in question. One of Grothendieck's contributions was to remove these restrictions: the best category of commutative rings is the category of all commutative rings! " [51, 256]. For a discussion of the genesis of the notion of scheme see Refs. [36,35,92].

SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics

“…The prime spectrum of a ring is the set of prime ideals of the ring R and denoted by Spec(R). The sheaf of rings O is the relevant algebraic geometrical notion introduced by Grothendieck to develop this field [2,32]. We touched this important concept of schemes while investigating non-zero dimensional sets of singularities in surfaces belonging to the SL(2, C) character variety of an infinite group.…”

SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics

“…The prime spectrum of a ring is the set of prime ideals of the ring R and denoted by Spec(R). The sheaf of rings O is the relevant algebraic geometrical notion introduced by Grothendieck to develop this field [8,41]. We touched this important concept of schemes while investigating non-zero dimensional sets of singularities in surfaces belonging to the SL(2, C) character variety of an infinite group.…”