Verschiebung maps among K-groups of truncated polynomial algebras
Introduction
There is a natural map from algebraic K-theory to topological cyclic homology called the cyclotomic trace map. By the famous theorem of Dundas-Goodwillie-McCarthy [1, Theorem 7.0.0.2] about this map, relative topological cyclic homology and relative algebraic K-theory coincide for the algebras we are studying. In [4, Theorem 4.2.10], Hesselholt and Madsen constructed the following exact sequence and gave an explicit presentation [4, Theorem A] of the algebraic K-theory of a truncated polynomial algebra over a perfect field k of positive characteristic by the Witt vectors: where TC denotes topological cyclic homology and denotes the big Witt vectors of length l. Furthermore they also show that such TC-groups in even degrees are trivial.
Using such stable homotopy theoretical results, in this paper, we consider the map of relative algebraic K-groups induced by the ring homomorphism We also consider the colimit over these 's indexed by the filtered category of natural numbers under multiplication. One of our main results is the following (Theorem 4.1). The notation in the diagram below will be introduced in Section 4.
Theorem 1.1 Let A be a ring in which p is nilpotent. There is a map of long exact sequences where is the integer part of t, V's denote the maps induced by Verschiebung maps, the maps in the limits are restriction maps, and the maps are induced by .
One of the remarkable facts about topological Hochschild homology is that its homotopy groups correspond to the groups of de Rham-Witt forms. Via this correspondence, stable homotopy theory and p-adic Hodge theory have been getting closer to each other. Considering this, we also produce a diagram similar to the one above for regular algebras over in terms of de Rham-Witt groups via the translation between topological Hochschild homology and the de Rham-Witt complex given by [2, §1, §2, §5]. See also [3].
Theorem 1.2 Let A be a regular -algebra. There is a map of long exact sequences where the subscript denotes the truncation set and denotes the big de Rham-Witt complex over A.
As an application of the above theorems, we calculate some relative K-groups. Let k be a perfect field of characteristic p. We define with the maximal ideal, and with the maximal ideal. Let K and L denote the quotient fields and respectively. Note that these K and L are perfectoid fields.
With these notations, we have where the colimits are indexed by the category of natural numbers under addition and denotes the big Witt vectors. Moreover, the relative K-groups in even degrees are zero.
Section snippets
The cyclic bar-construction
We recall some notations from [2] or [4]. For a positive integer k, we let denote the finite commutative pointed monoid defined by , , and . We let denote the circle group and , where as linear spaces and it has the -action defined by , for .
We let denote the cyclic set which is the cyclic bar construction of and denote its geometric realization. For , we also let
The geometric Verschiebung map
In order to study the map , we use two pointed commutative monoids and and their geometric realizations of cyclic bar constructions, and respectively, and construct a map between corresponding cofibration sequences.
In [5, 7.2], Hesselholt and Madsen defined an isomorphism between the geometric realization of the standard n-th cyclic set and the product topological space of the circle and the standard n-simplex as
Proofs of theorems
For a ring A, the topological Hochschild homology is a cyclotomic spectrum. See, for example, [7, III.4, III.5]. For a finite dimensional orthogonal -representation λ, we let denote its one point compactification. We define to be the abelian group of maps in the -stable homotopy category ([2, p. 92–93]).
As explained at page 93 of [2], for positive integers s, there are maps called Frobenius, called Verschiebung and called restriction
Acknowledgements
I am deeply grateful to Lars Hesselholt for suggesting the topic presented here and for helpful conversations. I also wish to express my gratitude to Martin Speirs for valuable comments, to the DNRF Niels Bohr Professorship of Lars Hesselholt for the support and to anonymous referee who gave me useful comments on an earlier version of this paper.
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