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

Abstract

In this text, we generalize Cech cohomology to sheaves $\mathcal F$ 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 $H^l(X,\mathcal F)$ turn out to be blue $B$-modules. For locally free $\mathcal O_X$-module $\mathcal F$ on a monoidal scheme $X$, we prove that $H^l(X,\mathcal F)^+=H^l(X^+,\mathcal F^+)$ where $X^+$ is the scheme associated with $X$ and $\mathcal F^+$ is the locally free $\mathcal O_{X^+}$-module associated with $\mathcal F$. In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over $\mathbb F_1$.