On the relative projective space

Abstract

Let (C, ⊗, 1) be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf P n 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. Vaquié. 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