Derived schemes and the field with one element

Abstract

The purpose of this paper is to define derived schemes over F1, the “field with one element”. More generally, we define our derived algebraic geometry over a symmetric monoidal category (C,⊗,1) by adapting the constructions of Lurie. We also suggest that the construction of derived schemes over F1 has an analogue in the “homotopy theory of algebraic varieties” over an algebraically closed field k, which we refer to as “derived schemes over T1(k)”. This is motivated by the fact that the S-modules in the category of schemes over k are analogous to Z-modules.