Cech cohomology over $\mathbb{F}_{1^2}$

Abstract

In this text, we generalize Cech cohomology to sheaves F with values in blue B-modules where B is a blueprint with −1. If X is an object of the underlying site, then the cohomology sets H l (X,F) turn out to be blue B-modules. For locally free O X -module F on a monoidal scheme X, we prove that H l (X,F) + = H l (X + ,F + ) where X + is the scheme associated with X and F + is the locally free O X + -module associated with F.In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over F 1 .