On the Rank of Picard Groups of Modular Varieties Attached to Orthogonal Groups

We derive lower bounds for the rank of Picard groups of modular varieties associated with natural congruence subgroups of the orthogonal group of an even lattice of signature (2, l). As an example we consider the Siegel modular group of genus 2. The analytic part of this paper also leads to certain class number identities.     

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D ^b (X), where X is the blow up of ? ^n?r ×? ^r along a multilinear subspace ? ^n?r?1×? ^r?1 of codimension 2 of ? ^n?r ×? ^r . As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Bimodule Deformations, Picard Groups and Contravariant Connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K ₀-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.     

UMAP Classes of Groups

A class $ \mathcal{G} $ of maximally almost periodic (MAP) topological Abelian groups is called a UMAP-class if it has the following property: if G is an Abelian group and τ ₁ and τ ₂ are distinct group topologies in G such that (G, τ ₁) and (G, τ ₂) are in $ \mathcal{G} $ , then (G, τ ₁)^∧ ≠ (G, τ ₂)^∧. Several examples of UMAP-classes are discussed. In particular, it is shown that the class PMAP of all Polish MAP-groups is a UMAP-class.     

Elementary Classes of Groups

Let be a class of groups. The elementary class with base is defined as the minimal class of groups containing and closed with respect to taking subgroups, quotient groups, group extensions, and direct limits. Properties of such classes are studied. Some applications to the theory of elementary amenable groups and a relation to the Kurosh--Chernikov classes of generalized solvable groups are considered.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Picard Groups of Irrational Rotation C*-Algebras

Let be an irrational number in [0, 1] and A the correspondingirrational rotation C*-algebra. Let Aut (A) be the group ofall automorphisms of A and Int (A) the normal subgroup of Aut(A) of all inner automorphisms of A. Let Pic (A) be the Picardgroup of A. In the present note we shall show that if is notquadratic, then Pic (A)Aut (A)/Int (A) and that if is quadratic,then Pic (A) is isomorphic to a semidirect product of Aut (A)/Int(A) with Z. Furthermore, in the last section we shall discussPicard groups of certain Cuntz algebras.     

Varieties and Torsion Classes of m-Groups

We prove that every variety of m-groups is a torsion class; find basis of identities for a product variety of m-groups; and show that the product of every finitely based variety of m-groups and a variety of Abelian m-groups is a finitely based variety.     

Logarithmic Picard groups,chip firing,and the combinatorial rank

Mathematische Zeitschrift - Illusie has suggested that one should think of the classifying group of $$M_X^{gp}$$ -torsors on a logarithmically smooth curve X over a standard logarithmic point as a...     

Arithmetic Graphs and Classes of Finite Groups

Siberian Mathematical Journal - An arithmetic graph function is a mapping associating to a finite group G the graph whose vertices are the divisors of |G|. We formulate and study the problem of...     

Free Groups of Outer Commutator Varieties of Groups

If F is a free group, 1 < i j 2i and i k i + j + 1 thenF/[_j(F), _i(F), _k(F)] is residually nilpotent and torsion-free.This result is extended to 1 < i j 2i and i k 2i + 2j.It is proved that the analogous Lie rings, L/[L^j, Lⁱ, L^k] whereL is a free Lie ring, are torsion-free. Candidates are foundfor torsion in L/[L^j, Lⁱ, L^k] whenever k is the least of {i,j, k}, and the existence of torsion in L/[L^j, Lⁱ, L^k] is provedwhen i, j, k 5 and k is the least of {i, j, k}.     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .