Search

Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell$ different from $p$ and integers $j \geq 1$ , there exists a "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. The purpose of this paper is to prove the analogous result for the $p$-adic $K$-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,|\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,|\,K,\mathbb{Z}_p)$ between two $p$-complete cyclotomic spectra associated with $D$ and $K$, respectively. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.

We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.

We consider unital associative rings in which a fixed prime number p p is nilpotent. It was proved long ago by Weibel that for such rings, the relative K K -groups associated with a nilpotent extension and the bi-relative K K -groups associated with a pull-back square are p p -primary torsion groups. However, the question of whether these groups can contain a p p -divisible torsion subgroup has remained an open and intractable problem. In this paper, we answer this question in the negative. In effect, we prove the stronger statement that the groups in question are always p p -primary torsion groups of bounded exponent.

Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel., Math. Ann. (to appear)

Let A be a noetherian -algebra that is finitely generated as a module over the subring Ap of pth powers. We give an explicit formula for the de Rham?Witt complex of the power series ring At in terms of that of the ring A. We use this formula to show that, for every complete regular local -algebra whose residue field is a finite extension of the subfield of pth powers, the canonical map from the algebraic K-theory with -coefficients to the topological K-theory with -coefficients is an isomorphism.

The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper with Ib Madsen, we used this comparison and an approximate evaluation of the structure of the pro-spectrum TR to evaluate the p-adic K-groups of a local field K of mixed characteristic (0,p) with perfect residue field. In this paper we completly determine the structure of the pro-spectrum TR for an algebrac closure of the local field K. This leads us to formulate a conjecture for the structure of the pro-spectrum TR for the field K. We also determine the structure of the absolute de Rham-Witt complex of the valuation ring R in the algebraic closure of K. The group in degree one is a p-divisible group whose Tate module is free module of rank one over the ring of Witt vectors in R. We give an explicit generator of this Tate module.

We compare the absolute and relative de Rham-Witt complexes considered by the author and Madsen [5, 4] and by Langer and Zink [10], which both generalize the classical de Rham-Witt complex of Bloch, Deligne, and Illusie [7] from Fp-schemes to Z(p)-schemes. From this comparison, we derive a Gauss-Manin connection on the crystalline cohomology of X/Wn(S) for a smooth family X/S.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$ . Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$ -module. But the ring of integers $\mathcal{O}_L$ is a projective $\mathcal{O}_K[G]$ -module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$ -module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$ , the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

The purpose of this paper is twofold. Firstly, it gives a thorough treatment of the de Rham–Witt complex for Z (p) -algebras, a construction we first considered in [L. Hesselholt, I. Madsen, Ann. of Math. 158 (2003) 1–113]. This complex is the natural generalization to Z (p) -algebras of the de Rham–Witt complex for F p -algebras of Bloch–Deligne–Illusie [L. Illusie, Ann. Sci. Ecole Norm. Sup. 12 (4) (1979) 501–661] (for p odd). We also give an explicit formula for the de Rham–Witt complex of a polynomial ring in terms of that of the coefficient ring. Secondly, we generalize the main Theorem C of [L. Hesselholt, I. Madsen, Ann. of Math. 158 (2003) 1–113] to smooth algebras over a discrete valuation ring of mixed characteristic (0,p) with perfect residue field and p odd.

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the groups of de Rham-Witt forms of the ring k. At present, the validity of the formula depends on a conjecture that concerns the combinatorial structure of a new family of polytopes that we call stunted regular cyclic polytopes. The polytopes in question appear as the intersections of regular cyclic polytopes with (certain) linear subspaces. We verify low-dimensional cases of the conjecture. This leads to unconditional new results on K_2 and K_3 which extend earlier results by Krusemeyer for K_0 and K_1.

This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×

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

We use the methods of topological Hochschild homology to shed new light on the groups satisfying the Bass trace conjecture. We show that the factorization of the Hattori-Stallings rank map through the Bokstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G with the property that every subgroup that is isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G., Comment: 16 pages

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 the author and Madsen 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 lambda-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a lambda-ring is given by the universal derivation of the underlying ring together with an additional structure depending on the lambda-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 K\"ahler 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 \'etale maps, and finally, the big de Rham-Witt complex of the ring of integers is explicitly evaluated. The latter complex may be interpreted as the complex of differentials along the leaves of a foliation of Spec Z., Comment: 63 pages

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type and suppose that R is K_{d+1}-regular. Then R is a regular ring.

The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss andWilliams shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed Riemannian manifold of negative sectional curvature can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead spectrum of the circle. The cyclotomic trace of Bokstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an expression for these homotopy groups in terms of the equivariant homotopy groups of the homotopy fiber of the map from the topological Hochschild T-spectrum of the sphere spectrum to that of the ring of integers induced by the Hurewicz map. We evaluate the latter homotopy groups, and hence, the homotopy groups of the topologicalWhitehead spectrum of the circle in low degrees. The result extends earlier work by Anderson and Hsiang and by Igusa and complements recent work by Grunewald, Klein, and Macko.

We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively., Comment: Journal of Topology (to appear)

Let A a regular F_p-algebra. The relative K-groups K_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluated earlier by the author and Madsen in terms of the big de Rham-Witt groups of the ring A. In this paper, we evaluate the maps of relative K-groups and Nil-groups induced by the canonical projection from A[x]/(x^m) to A[x]/(x^n). The result depends strongly on the prime p. It generalizes work of Stienstra on the groups in degrees 2 and 3.

Let k a regular noetherian p-algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I = (x,y) be the ideal that defines the intersection point. We evaluate the relative K-groups Kq(A, I) completely in terms of the big de Rham-Witt groups of k. This generalizes a formula for K1(A, I) and K2(A, I) by Dennis and Krusemeyer.

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 bi-relative algebraic K-theory measures the deviation. It was proved by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative 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 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.

We determine the structure of the reduction modulo p p of the absolute de Rham-Witt complex of a smooth scheme over a discrete valuation ring of mixed characteristic ( 0 , p ) (0,p) with log-poles along the special fiber and show that the sub-sheaf fixed by the Frobenius map is isomorphic to the sheaf of p p -adic vanishing cycles. We use this result together with the main results of op. cit. to evaluate the algebraic K K -theory with finite coefficients of the quotient field of the henselian local ring at a generic point of the special fiber. The result affirms the Lichtenbaum-Quillen conjecture for this field.

The authors 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. They consider fields K that are complete discrete valuation fields of characteristic zero with perfect residue fields k of characteristic p > 2. They evaluate the K-theory with Z/p^v-coefficients of K, and verify the Lichtenbaum-Quillen conjecture for K., added in migration; 113 pages, published version