Loading…
Grothendieck’s theory of schemes and the algebra–geometry duality
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 math...
Saved in:
Published in: | Synthese (Dordrecht) 2022-06, Vol.200 (3), p.234, Article 234 |
---|---|
Main Authors: | , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Summary: | 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
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. |
---|---|
ISSN: | 1573-0964 0039-7857 1573-0964 |
DOI: | 10.1007/s11229-022-03675-1 |