Lars Hesselholt

On the K-theory of finite algebras over witt vectors of perfect fields

Introduction i 1. The topological Hochschild spectrum 1 2. Witt vectors 12 3. Topological cyclic homology 18 4. Topological cyclic homology of perfect fields 22 5. Topological cyclic homology of finite W (k)-algebras 32 6. Pointed monoids 36 7. A formula for TC(L[ ]) 40 8. Topological cyclic homology of k[ ] 45 Appendix A: Spectra and prespectra 49 Appendix B: Continuity properties for K-theory 50 References 52

On the K-theory of local fields

In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. The fieldsK we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p > 2. When K contains the pth roots of unity, the relationship between the K-theory with Z/p-coefficients and the de Rham-Witt complex can be described by a sequence

On the p-typical curves in Quillen's K-theory

Twenty years ago Bloch, [Bl], introduced the complex C∗(A; p) of p-typical curves inK-theory and outlined its connection to the crystalline cohomology of Berthelot-Grothendieck. However, to prove this connection Bloch restricted his attention to the symbolic part of K-theory, since only this admitted a detailed study at the time. In this paper we evaluate C∗(A; p) in terms of Bökstedt’s topological Hochschild homology. Using this we show that for any smooth algebra A over a perfect field k of positive characteristic, C∗(A; p) is isomorphic to the de Rham-Witt complex of Bloch-Deligne-Illusie. This confirms the outlined relationship between p-typical curves in K-theory and crystalline cohomology in the smooth case. In the singular case, however, we get something new. Indeed, we calculate C∗(A; p) for the ring k[t]/(t ) of dual numbers over k and show that in contrast to crystalline cohomology, its cohomology groups are finitely generated modules over the Witt ring W (k). Let A be a ring, by which we shall always mean a commutative ring, and let K(A) denote the algebraic K-theory spectrum of A. Here we use the term spectrum in the sense of topology. More generally, if I ⊂ A is an ideal, K(A, I) denotes the relative algebraic K-theory, that is, the homotopy theoretical fiber of the map K(A)→ K(A/I). We define the curves on K(A) to be the homotopy limit of spectra

The big de Rham–Witt complex

This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a λ-ring which is developed first. The main result in 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 depending directly on the λ-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to étale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

Bi-relative algebraic K-theory and topological cyclic homology

It is well-known that algebraic K-theory preserves products of rings. However, in general, algebraic K-theory does not preserve fiber products of rings, and one defines bi-relative algebraic K-theory so as to measure the deviation. It was proved recently by Cortinas [6] that, rationally, bi-relative algebraic K-theory and bi-relative negative cyclic homology agree. In this paper, we show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application, we show that for a possibly singular curve over a perfect field k of positive characteristic p, the cyclotomic trace map induces an isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic homology groups in non-negative degrees. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology. Let F be a functor that to a unital associative ring A associates a symmetric spectrum F (A). If I ⊂ A is a two-sided ideal, the relative term F (A, I) is defined to be the homotopy fiber of the map F (A) → F (A/I) induced from the canonical projection. If further f : A → B is a ring homomorphism such that f : I → f(I) is an isomorphism onto a two-sided ideal of B, there is an induced map F (A, I)→ F (B, f(I)), and the bi-relative term F (A,B, I) is defined to be the homotopy fiber of this map. Then there is a distinguished triangle in the stable homotopy category

Witt vectors of non-commutative rings and topological cyclic homology

Classically, one has for every commutative ring A the associated ring of p-typical Wit t vectors W(A). In this paper we extend the classical construction to a functor which associates to any associative ring A an abelian group W(A). The extended functor comes equipped with additive Frobenius and Verschiebung operators. We also define groups Wn(A) of Witt vectors of length n in A. These are related by restriction maps R: Wn(A)-+Wn-I(A) and W(A) is the inverse limit. In particular, WI(A) is defined to be the quotient of A by the additive subgroup [A, A] generated by elements of the form xy-yx, x, yEA. There are natural exact sequences

On the

We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic (0,p), the p-adic etale K-theory and p-adic topological cyclic homology agree.

K

We determine the structure modulo p of the de Rham-Witt complex of a smooth scheme X over a discrete valuation ring of mixed characteristic with log-poles along the special fiber Y and show that the sub-sheaf fixed by the Frobenius is isomorphic to the sheaf of p-adic vanishing cycles. We use this together with results of the second author and Madsen to evaluate the

-theory and topological cyclic homology of smooth schemes over a discrete valuation ring

K

The de Rham-Witt complex and -adic vanishing cycles

-theory with finite coefficients of the quotient field K of the henselian local ring of X at a generic point of Y. The result affirms the Lichtenbaum-Quillen conjecture for the field K.