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 ${\mathbb{F}}_1$.

14 pages

References in corpus (1)

• Cech Cohomology of Semiring Schemes