Verschiebung maps among K-groups of truncated polynomial algebras

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Abstract

Let p be a prime number, and let A be a ring in which p is nilpotent. In this paper, we consider the mapsKq+1(A[x]/(xm),(x))Kq+1(A[x]/(xmn),(x)), induced by the ring homomorphism A[x]/(xm)A[x]/(xmn), xxn. We evaluate these maps, up to extension, for general A in terms of topological Hochschild homology, and for regular Fp-algebras A, in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative K-group of OK/pOK for certain perfectoid fields K.

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 Wl 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-groupsvn:Kq+1(A[x]/(xm),(x))Kq+1(A[x]/(xmn),(x)) induced by the ring homomorphismA[x]/(xm)A[x]/(xnm),xxn. We also consider the colimit over these vn's indexed by the filtered category (N,) 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 t 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 vn are induced by xxn.

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 Fp 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 Fp-algebra. There is a map of long exact sequences where the subscript m(l+1) denotes the truncation set {1,2,...,m(l+1)} and W()ΩA 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 defineOK:=(colimnW(k)[p1/pn]) with mKOK the maximal ideal, andOL:=(colimnW(k)[ζpn]) with mLOL the maximal ideal. Let K and L denote the quotient fields OK[1/p] and OL[1/p] 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 W 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 Πk:={0,1,x1,...,xk1} denote the finite commutative pointed monoid defined by xnxm=xm+n, 0xn=0, x0=1 and xk=0. We let T denote the circle group and λd:=C(d)...C(1), where C(j)=C as linear spaces and it has the T-action defined by T×C(j)C(j),(z,w)zjw, for 1jd.

We let Ncy(Πk)[] denote the cyclic set which is the cyclic bar construction of Πk and Ncy(Πk) denote its geometric realization. For iN, we also let Ncy(Πk,i)[

The geometric Verschiebung map

In order to study the map K(A[x]/(xk),(x))K(A[x]/(xnk),(x)), we use two pointed commutative monoids Πk and Πnk and their geometric realizations of cyclic bar constructions, Ncy(Πk) and Ncy(Πnk) respectively, and construct a map between corresponding cofibration sequences.

In [5, 7.2], Hesselholt and Madsen defined an isomorphism between the geometric realization |Λ[n][]| of the standard n-th cyclic set and the product topological space T×Δn of the circle R/Z and the standard n-simplex as

Proofs of theorems

For a ring A, the topological Hochschild homology THH(A) is a cyclotomic spectrum. See, for example, [7, III.4, III.5]. For a finite dimensional orthogonal T-representation λ, we let Sλ denote its one point compactification. We defineTRqλn(A):=[Sq(T/Cn)+,SλTHH(A)]T to be the abelian group of maps in the T-stable homotopy category ([2, p. 92–93]).

As explained at page 93 of [2], for positive integers s, there are maps Fs called Frobenius, Vs called Verschiebung and Rs 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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