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.
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Verhoek, H. Cartier theory with coefficients. manuscripta math. 153, 455–499 (2017). https://doi.org/10.1007/s00229-016-0892-5
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DOI: https://doi.org/10.1007/s00229-016-0892-5