Abstract. We study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.
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 .
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