On the Relative Projective Space

TL;DR

This paper develops a framework for non-associative relative algebraic geometry by defining projective spaces in symmetric monoidal categories, with the octonionic projective space as a key example.

Contribution

It introduces a new construction of projective spaces within symmetric monoidal categories, extending algebraic geometry to non-associative contexts.

Findings

Defined a presheaf $ ext{P}^{n}_{ ext{C}}$ on commutative algebras in $ ext{C}$

Proved this presheaf is a $ ext{C}$-scheme in the sense of Toen and Vaquié

Provided the octonionic projective space as a primary example

Abstract

Let $(\mathcal C,\otimes,\mathbb 1)$ be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf $\mathbb P^{n}_{\mathcal C}$ on the category of commutative algebras in $\mathcal C$ and we prove that this functor is a $\mathcal C$-scheme in the sense of Toen and Vaqui\'e. This construction gives us a context of non-associative relative algebraic geometry. The most important example of the construction is the octonionic projective space.