# Cartier theory with coefficients

Research paper by **Hendrik Verhoek**

Indexed on: **10 Nov '16**Published on: **19 Oct '16**Published in: **Manuscripta Mathematica**

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#### Abstract

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.AbstractWe 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.AAAAAAA
\(\mathbb {E}_{A}\)
\(\mathbb {E}_{A}\)A
\(\mathbb {E}_{A}(R)\)
\(\mathbb {E}_{A}(R)\)