  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
$\mathbb F_1$.
