# On the relative projective space

## Matias Data and Juliana Osorio

Let $(C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf $Proj{C}$ on the category of commutative algebras in $C$ and we prove that this functor is a $C$-scheme in the sense of B. Toen and M. Vaquie. We give another definition and prove that they give isomorphic $C$-schemes. This construction gives us a context of non-associative relative algebraic geometry. The most important example of the construction is the octonionic projective space.

Keywords: symmetric monoidal category, algebra object, line object, relative scheme

2010 MSC: 14A22, 18F99

Theory and Applications of Categories, Vol. 34, 2019, No. 3, pp 58-79.

Published 2019-02-22.

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