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Witt Vector Rings and the Relative de Rham Witt Complex | | Joachim Cuntz
; Christopher Deninger with an appendix by Umberto Zannier
; | | Date: |
20 Oct 2014 | | Abstract: | In this paper we develop a novel approach to Witt vector rings and to the
(relative) de Rham Witt complex. We do this in the generality of arbitrary
commutative algebras and arbitrary truncation sets. In our construction of Witt
vector rings the ring structure is obvious and there is no need for universal
polynomials. Moreover a natural generalization of the construction easily leads
to the relative de Rham Witt complex.
Our approach is based on the use of free or at least torsion free
presentations of a given commutative ring $R$ and it is an important fact that
the resulting objects are independent of all choices. The approach via
presentations also sheds new light on our previous description of the ring of
$p$-typical Witt vectors of a perfect $mathbb{F}_p$-algebra as a completion of
a semigroup algebra. We develop this description in different directions. For
example, we show that the semigroup algebra can be replaced by any free
presentation of $R$ equipped with a linear lift of the Frobenius automorphism.
Using the result in the appendix by Umberto Zannier we also extend the
description of the Witt vector ring as a completion to all
$ar{mathbb{F}}_p$-algebras with injective Frobenius map. | | Source: | arXiv, 1410.5249 | | Services: | Forum | Review | PDF | Favorites | |
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