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Introduction 1. Topological Hochschild homology and localization 2. The homotopy groups of T (A|K) 3. The de Rham-Witt complex and TR • * (A|K; p) 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T (A|K) 6. The pro-system TR • * (A|K; p, Z/p v ) Appendix A. Truncated polynomial algebras References * The first named author was supported in part by NSF Grant and the Alfred P. Sloan Foundation. The second named author was supported in part by The American Institute of Mathematics. i ! −→ TC(C b z (P A ); p) j −→ TC(C b q (P A ); p) ∂ −→ Σ TC(C b z (P A ) q ; p), *

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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

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