Affine C-theoretic schemes

Abstract

This thesis provides a way of generalizing the construction of affine schemes.
Starting from objects in an arbitrary complete category C, the method will
produce objects which resemble affine schemes. These objects will be called
affine C-theoretic schemes. To each affine C-theoretic scheme a topological
space and presheaf is produced similar to the theory of affine schemes. The
study of affine C-theoretic schemes will eventually motivate the definitions of
mathematical objects which will be called categorical nets and connections. The
general theory is then applied to the special case when considering the category
of abelian groups. Finally the theory is used to define an analog notion of a
scheme in this context.

Popular Abstract

Algebraic geometry is a broad field with various results and applications. A
famous theoretical result is the proof of Fermat’s Last Theorem which states
that the equation x
n + y
n = z
n, n ≥ 3, xyz ̸= 0 does not have any integer
solutions. There are also a wide range of real world applications. One such is
elliptic curve cryptography, where properties of elliptic curves are used in encryption algorithms.
The subject originated from the study of solutions of systems of polynomial
equations. The geometric properties of these solution sets gave rise to the notion of algebraic sets and affine varieties. The original study mainly regarded
solution sets over algebraically closed fields such as for instance the field of
complex numbers. The question arose how to deal with the more general case
in the study of non-algebraically closed fields and even rings. The key insight
that led the study forward was the realization that the algebraic sets could be
constructed as the prime ideals of the so called coordinate ring, which is a ring
that is closely related to the starting polynomials. Not only the set, but also
the topology and structure sheaf could be constructed directly from this ring.
This led to the construction of affine schemes as generalizations of algebraic
sets in the sense that “construction procedure” could be applied to arbitrary
commutative rings with unity. The affine schemes were then used to define
the notion of a scheme. There are several ways to think about schemes. One
could see these objects as generalizations of rings, roughly in the same way that
manifolds are generalizations of the n-dimensional real space. One might call
a manifold a “global n-dimensional space”. Motivated by this comparison one
might call a scheme a “global ring”. A question that one might ask is the following. If a scheme is a “global ring” what would be a “global object” for some
object in some category? For instance how could one define the notion of a
“global group”? An observation to be made is that the “global” objects in the
study of algebraic geometry (schemes) are constructed by first turning rings into
“local geometric objects” (affine schemes) which are then glued. This leads to
the idea of constructing “local geometric objects” from objects in some general
category. This is the focus of this thesis. Using manifolds as a metaphor this
would correspond to the process of constructing the “local” n-dimensional real
space R
n. Thus the thesis is not going to focus on constructing “global” (i.e.
scheme-like) objects but rather the “local objects” (i.e. affine scheme-like) from
objects in a category C. These objects are going to be called affine C-theoretic
schemes. This thesis thereby hopes to enable a “global” study. A global study
of objects in a category C could enable one to use arguments and tools similar
to the ones used for schemes and rings but in a wider context. Thus the thesis
should be seen as a first stepping stone towards this over arching goal.

• author: Lindholm, Oskar
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Engineering)
• course: FMAM05 20241
• year: 2024
• type: H1 - Master's Degree (One Year)
• subject: Mathematics and Statistics
• keywords: Scheme, Affine scheme, Category theory, Algebraic geometry
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMA-3555-2024
• ISSN: 1404-6342
• other publication id: 2024:E64
• language: English
• id: 9172143
• date added to LUP: 2025-03-21 10:06:10
• date last changed: 2025-03-21 10:06:10