Algebraic modules and the Auslander-Reiten quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander-Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander-Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.     

The homotopy limit problem for two-primary algebraic K-theory

We solve the homotopy limit problem for two-primary algebraic K-theory of fields, that is, the Quillen-Lichtenbaum conjecture at the prime 2.     

Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-nite Gorenstein algebras.     

Higher algebraic K-groups and {\mathcal D}-split sequences

In this paper, we use ${\mathcal D}$ -split sequences and derived equivalences to provide formulas for calculation of higher algebraic K-groups (or mod-p K-groups) of certain matrix subrings which occur both in commutative algebra as the endomorphism rings of direct sums of Prüfer modules or of chains of Glaz–Vasconcelos ideals and in noncommutative geometry as an essential ingredient of the study of singularities of orders over surfaces. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic K-theory literature to a more general context.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

Hermitian <Emphasis Type="Italic">K</Emphasis>-theory and 2-regularity for totally real number fields

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.     

Coefficients for the Farrell-Jones Conjecture

We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture.     

Δ-graphs of polytopes in Bruns and Gubeladze K-theory

W. Bruns and J. Gubeladze introduced a new version of algebraic K-theory where K-groups are additionally parameterized by polytopes of some type. In this paper we propose a concept of stable E-equivalence which can be used to calculate K-groups for high-dimensional polytopes. Polytopes which are stable E-equivalent have similar inner structures and isomorphic K-groups. In addition, for each polytope we define a Δ-graph which is an oriented graph being invariant under a stable E-equivalence.     

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.     

The Higher K-Theory of Complex Varieties

Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the K-groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-finite. We also describe the torsion in the K-groups of F and X.     

Sur l'indépendance l-adique de nombres algébriques

Let l a prime number and K a Galois extension over the field of rational numbers, with Galois group G. A conjecture is put forward on l-adic independence of algebraic numbers, which generalizes the classical ones of Leopoldt and Gross, and asserts that the l-adic rank of a G submodule of K^x depends only on the character of its Galois representation. When G is abelian and in some other cases, a proof is given of this conjecture by using l-adic transcendence results.     

The p-chain Spectral Sequence

We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah–Hirzebruch-type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum and Connes and Farrell and Jones to compute algebraic K- and L-groups of group rings and topological K-groups of reduced group C*-algebras.     

On Complete Functions in Jucys-Murphy Elements

The problem of computing the class expansion of some symmetric functions evaluated in Jucys-Murphy elements appears in different contexts, for instance, in the computation of matrix integrals. Recently, Lassalle gave a unified algebraic method to obtain some induction relations on the coefficients in this kind of expansion. In this paper, we give a simple purely combinatorial proof of his result. Besides, using the same type of argument, we obtain new simpler formulas. We also prove an analogous formula in the Hecke algebra of (S _2n , H _n ) and use it to solve a conjecture of Matsumoto on the subleading term of orthogonal Weingarten function. Finally, we propose a conjecture for a continuous interpolation between both problems.     

Primitive roots in quadratic fields, II

We consider an analogue of Artin's primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo (p) is p²−1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least for infinitely many inert primes p.     

Weak Leopoldt's conjecture for Hecke characters of imaginary quadratic fields

We give a proof of the weak Leopoldt's conjecture à la Perrin-Riou, under some technical condition, for the p-adic realizations of the motive associated to Hecke characters over an imaginary quadratic field K of class number 1, where p is a prime >3 and where the CM elliptic curve associated to the Hecke character has good reduction at the primes above p in K. This proof makes use of the 2-variable Iwasawa main conjecture proved by Rubin. Thus we prove the Jannsen conjecture for the above p-adic realizations for almost all Tate twists.     

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

We settle two conjectures for computing higher Grothendieck–Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of K  -theory under the natural Z/2$\mathbb{Z}/2$-action is a 2-adic equivalence. We also prove that the mod 2^ν$2^{\nu }$ comparison map between the Hermitian K-theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen–Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck–Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.     

Clean unital ℓ-groups

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ?-groups. We mention the relationship of clean unital ?-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ?-groups with distinguished weak order unit.