We investigate categorical and amalgamation properties of the functor $\operatorname{Id_c}$ assigning to every partially ordered abelian group $G$ its $\langle \vee, 0 \rangle$-semilattice of compact ideals $\operatorname{Id_c} G$. Our main result is the following.

Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the$\operatorname{Id_c}$functor, by a diagram of pseudo-simplicial vector spaces.

Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result.

Theorem 2. For any countable distributive$\langle \vee, 0 \rangle$-semilattices$S$and$T$and any field$K$, any$\langle \vee, 0 \rangle$-homomorphism$f \colon S \to T$can be lifted, with respect to the$\operatorname{Id_c}$functor on rings, by a homomorphism $f \colon A \to B$of$K$-algebras, for countably dimensional locally matricial algebras$A$and$B$over$K$.

We also state a lattice-theoretical analogue of Theorem 2 (with respect to the $\operatorname{Con_c}$ functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.

This is a contribution to the study of line-transitive groups of automorphisms of finite linear spaces. Groups which are almost simple are of particular importance. In this paper almost simple line-transitive groups whose socle is an alternating group are classified. It is proved that the only alternating groups to occur are those of degrees 7 and 8, and that only one linear space occurs, namely a well-known space with 15 points and 35 lines. Although much of the proof exploits special properties of alternating groups, some general theory of groups acting line-transitively on finite linear spaces is developed.

Suppose $\mathcal{O}$ is either the ring of integers of a number field, the ring of integers of a $p$-adic local field, or a field of characteristic $0$. Let $\mathcal{X}$ be a regular projective scheme which is flat and equidimensional over $\mathcal{O}$ of relative dimension $d$. Suppose $G$ is a finite group acting tamely on $\mathcal{X}$. Define ${\rm HCl}(\mathcal{O} G)$ to be the Hermitian class group of $\mathcal{O} G$. Using the duality pairings on the de Rham cohomology groups $H^*(X, \Omega^\bullet_{X / F})$ of the fiber $X$ of $\mathcal{X}$ over $F = {\rm Frac}(\mathcal{O})$, we define a canonical invariant $\chi_H(\mathcal{X}, G)$ in ${\rm HCl}(\mathcal{O} G)$ . When $d = 1$ and $\mathcal{O}$ is either $\mathbb{Z}$, $\mathbb{Z}_p$ or $\mathbb{R}$, we determine the image of $\chi_H(\mathcal{X}, G)$ in the adelic Hermitian classgroup ${\rm Ad\,HCl}(\mathbb{Z} G)$ by means of $\epsilon$-constants. We also show that in this case, the image in ${\rm Ad\,HCl}(\mathbb{Z} G)$ of a closely related Hermitian Euler characteristic $\chi_{H}(\mathcal{X}, G)(0)$ both determines and is determined by the $\epsilon_0$-constants of the symplectic representations of $G$.

Using results of Browkin and Schinzel one can easily determine quadratic number fields with trivial 2-primary Hilbert kernels. In the present paper we completely determine all bi-quadratic number fields which have trivial 2-primary Hilbert kernels. To obtain our results, we use several different tools, amongst which is the genus formula for the Hilbert kernel of an arbitrary relative quadratic extension, which is of independent interest. For some cases of real bi-quadratic fields there is an ambiguity in the genus formula, so in this situation we use instead Brauer relations between the Dedekind zeta-funtions and the Birch–Tate conjecture.

We consider the Zariski closure of Thom–Boardman strata $\Sigma^{i, j}$ in the jet space for map-germs $(\mathbf{C}^n, 0)\to(\mathbf{C}^p, 0)$. We apply the geometric technique of calculating syzygies to the desingularization of $\Sigma^{i, j}$ constructed by Ronga. We investigate when the normalization of the coordinate ring of $\Sigma^{i, j}$ is rational in the cases $i = n - p + 1$ or $i = j = 1$. We also consider when these coordinate rings are normal with rational singularities.

Let $(S, 0)$ be a normal surface germ and Let $f$ a non-constant regular function on Let $(S, 0)$ with Let $f(0) = 0$. Using any additive invariant on complex algebraic varieties one can associate a zeta function to these data, where the topological and motivic zeta functions are the roughest and the finest zeta functions, respectively. In this paper we are interested in a geometric determination of the poles of these functions. The second author has already provided such a determination for the topological zeta function in the case of non-singular surfaces. Here we give a complete answer for all normal surfaces, at least on the motivic level. The topological zeta function however seems to be too rough for this purpose, although for negative poles, which are the only ones in the non-singular case, we are able to prove exactly the same result as for non-singular surfaces.

We also give and verify a (natural) definition for when a rational number is a pole of the motivic zeta function.

In this paper we introduce a generalization of Picard groups to derived categories of algebras. First we study general properties of this group. Then we consider easy particular algebras such as commutative algebras, where we reduce to the classical case. Finally, we define and study a homomorphism of the braid group to the Picard group of the derived category of a Brauer tree algebra. In the smallest case we show that this homomorphism is injective and that its image is of finite index.

In this note we consider a stability property of p-harmonic functions in general metric measure spaces. Despite the fact that there may not be a corresponding differential equation, it is shown here that a family of uniformly convergent p-harmonic functions converges to a p-harmonic function; that is, the collection of p-harmonic functions forms a closed subspace in a certain Ho¨lder class. As a consequence, it is shown that a family of p-harmonic functions bounded in a Sobolev-type space (called the Newtonian space) has a sequence that converges locally uniformly in the domain of harmonicity to a p-harmonic function. This result is used to construct p-harmonic functions on unbounded domains. We also use this convergence result to prove a characterization of a parabolicity property of metric measure spaces. This characterization has been given for Riemannian manifolds by Holopainen, and the result here is a generalization of Holopainen's result.

In this paper we give lower bounds for the Mislin genus of the symplectic groups $\mathrm{Sp}(m)$. This result appears to be the exact analogue of Zabrodsky's theorem concerning the special unitary groups $\mathrm{SU}(n)$ . It is achieved by the determination of the stable genus of the quasi-projective quaternionic spaces $Q\mathbb{H}(m)$ , following the approach of McGibbon. It leads to a symplectic version of Zabrodsky's conjecture, saying that these lower bounds are in fact the exact cardinality of the genus sets. The genus of $\mathrm{Sp}(2)$ is well known to contain exactly two elements. We show that the genus of $\mathrm{Sp}(3)$ has exactly 32 elements and see that the conjecture is true in these two cases.

Independently, we also show that any homotopy type in the genus of $\mathrm{Sp}(m)$ fibers over the sphere $S^{4m-1}$ with fiber in the genus of $\mathrm{Sp}(m-1)$ , and that any homotopy type in the genus of $\mathrm{SU}(n)$ fibers over the sphere $S^{2n-1}$ with fiber in the genus of $\mathrm{SU}(n-1)$ . Moreover, these fibrations are principal with respect to some appropriate loop structures on the fibers. These constructions permit us to produce particular spaces realizing the lower bounds obtained.