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

TL;DR

This paper extends Cech cohomology to sheaves valued in blue modules over blueprints with -1, establishing new properties and demonstrating infinite-dimensional issues over the projective line in the context of $un$-geometry.

Contribution

It generalizes Cech cohomology to blue $B$-modules with $-1$, linking it to scheme cohomology and highlighting limitations of naive derived functor approaches.

Findings

Cech cohomology sets are blue $B$-modules.

Isomorphism between cohomology of $X$ and $X^+$ for locally free modules.

Naive derived functor cohomology is infinite-dimensional over $un$.

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 $\mathbb F_1$.