La geometr\'ia de los Monoides

TL;DR

This paper develops the foundations of monoid theory from a categorical perspective, exploring their relation to affine schemes, toric varieties, and logarithmic geometry in arithmetic geometry.

Contribution

It introduces a categorical framework for monoid theory, generalizes affine schemes to monoidal schemes, and links these concepts to toric varieties and logarithmic geometry.

Findings

Establishes the categorical foundations of monoid theory.

Generalizes affine schemes to affine monoidal schemes.

Connects monoidal schemes with toric varieties and logarithmic geometry.

Abstract

In this article we present the basis of Monoid Theory from a categorical point of view, emphasizing the analogies and differences between the theory of modules over commutative rings. We present the generalization of afine schemes and afine monoidal schemes. We study the relationship between the former two with opic varieties and we set up the basis for the study of Fontaine-Kato-Illusie's Logaritmic Geometry, widely used in Arithmetic Geometry. En este art\'iculo presentamos las bases de la teor\'ia de monoides desde el punto de vista categ\'orico, haciendo \'enfasis en las analog\'ias y diferencias entre la teor\'ia de m\'odulos sobre anillos conmutativos. Se presenta la generalizaci\'on de esquema af\'in a esquema af\'in monoidal; estudiamos la relaci\'on que \'estos tienen con las variedades t\'oricas y sentamos las bases para el estudio de la Geometr\'ia Logar\'itmica de Fontaine-Kato-Illusie~\cite{kato1989logarithmic}, ampliamente usada en Geometr\'ia Aritm\'etica.