The big de Rham–Witt complex

Abstract

The big de Rham-Witt complex was introduced by the author and Madsen in [15] with the purpose of giving an algebraic description of the equivariant homotopy groups in low degrees of Bökstedt's topological Hochschild spectrum of a commutative ring. This functorial algebraic description, in turn, is essential for understading algebraic K-theory by means of the cyclotomic trace map of Bökstedt-Hsiang-Madsen [4]; compare [16,14,10]. The original construction, which relied on the adjoint functor theorem, was very indirect and a direct construction has been lacking. In this paper, we give a new and explicit construction of the big de Rham-Witt complex and we also correct the 2-torsion which was not quite correct in the original construction.The new construction is based on a theory, which is developed first, of modules and derivations over a λ -ring. The main result of this first part of the paper is that the universal derivation of a λ -ring is given by the universal derivation of the underlying ring together with an additional structure that depends directly on the λ -ring structure in question. In the case of the universal λ -ring, which is given by the ring of big Witt vectors, this additional structure consists in divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new direct construction of the big de Rham-Witt complex possible. This is carried out in the second part of the paper, where we also show that the big de Rham-Witt complex behaves well with respect to étale morphisms. Finally, we explicitly evaluate the big de Rham-Witt complex of the ring of integers.In more detail, let A be a ring, which we always assume to be commutative and unital. The ring W(A) of big Witt vectors in A is equipped with a natural action through ring homomorphisms by the multiplicative monoid N of positive integers, Generous assistance from DNRF Niels Bohr Professorship, JSPS Grant-in-Aid 23340016, and CMI Senior Scholarship is gratefully acknowledged

Algebraic Topology: Applications and New Directions

“…The big de Rham-Witt complex W S Ω * k is defined, up to unique isomorphism, to be an initial object in the category of Witt complexes over k. An explicit construction is given in [16,Section 4]. It is proved in loc.…”

“…We recall that for every subset S ⊂ N stable under division and every positive integer e, the big de Rham-Witt groups W S Ω q k and the Verschiebung maps V e : W S/e Ω q k → W S Ω q k are defined; see [16]. Theorem A.…”

Witt vector rings and the relative de Rham Witt complex

“…Except for the results comparing the objects that we obtain to the conventional ones, it can be read without knowing the classical theories in the title. For these we refer to [16,10,15,9,17,18,14,8,1] and the appendix of [13] as far as Witt vector rings are concerned. For the de Rham Witt theory we mention the works [3,11,8,12] and the references given there.…”

“…For these we refer to [16,10,15,9,17,18,14,8,1] and the appendix of [13] as far as Witt vector rings are concerned. For the de Rham Witt theory we mention the works [3,11,8,12] and the references given there. More advanced topics are discussed in [4][5][6] for example.…”

The norm map of Witt vectors

“…For a summary of the basic properties of Witt vectors, see, e.g.,[7, §3] or Hesselholt's article[6]. Here we give a very short summary of some of the properties of Witt vectors.Given a truncation set S and a commutative ring k, the ring W S (k) of Witt vectors is defined to be k S as a set, and addition and multiplication are defined by the requirement that the ghost map w : W S (k) → k S defined byw(a) s = d|s da s/d d is a ring homomorphism, functorially in k. We will write a Witt vector as a = (a s ) and the image of a Witt vector under the ghost map as x = x s .…”