Cartier theory with coefficients

Abstract

We develop a Cartier theory to describe formal groups with an action of a number ring A. Such formal groups are called formal A-modules. An important example of a formal A-module is the formal group of A-typical Witt vectors that generalize the big Witt vectors: instead of indexing over the non-zero natural numbers, we index over a subset of non-zero ideals of A. Using a variant of the A-typical Witt vectors, we define the A-typical Cartier ring $$\mathbb {E}_{A}$$ and prove that the category of formal A-modules is equivalent to the category of $$\mathbb {E}_{A}(R)$$ -modules under the assumption that the tangent space is finitely generated and projective.