One Dimensional Tilting Modules are of Finite Type

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one. Presented by Claus Michael Ringel.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

L—fuzzy模范畴的张量积与张量函子

本文在Ｌ—ｆｕｚｚｙ模范畴中，建立了相应的张量积，给出了它的结构性、存在性与唯一性定理，并讨论了张量函子与Ｈｏｍ函子的伴随性。所得结果为通常张量积的“良好推广”（ｇｏｏｄｅｘｔｅｎｓｉｏｎ）。     

Relative Projectivity of Modules and Cohomology Theory of Finite Groups

In the modular representation theory of finite groups the theory of projectivity relative to subgroups is of fundamental importance. To generalize this notion the theory of projectivity relative to modules was introduced by the first author. Our aim is to show some aspects of cohomology theory of finite groups concerning projectivity of modules relative to both subgroups and modules. We shall give some applications to the cohomology theory; especially we shall calculate the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups.     

Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products II

In this note we refine the methods of Etingof and Montarani (Represent. Theory, 9: 457–467, 2005) in order to extend the main result of that article to a wider class of finite dimensional representations of wreath product symplectic reflection algebras. Presented by Claus M. Ringel.     

Free Subproducts and Free Scaled Products of II1-Factors

The constructions of free subproducts of von Neumann algebras and free scaled products are introduced, and results about them are proved, including rescaling results and results about free trade in free scaled products.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

The Identities and Central Polynomials of the Finite Dimensional Unitary Grassmann Algebras Over a Finite Field

We describe the T-ideal of identities and the T-space of central polynomials for the unitary finite dimensional Grassmann algebra over a finite field.     

Tensor Products of Hilbert Modules Over Locally <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

In this paper the tensor products of Hilbert modules over locally C ^*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C ^*-modules are also valid in the context of Hilbert modules over locally C ^*-algebras.     

Finite Free Resolutions and 1-Skeletons of Simplicial Complexes

A technique of minimal free resolutions of Stanley—Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d–1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d–1 is d-connected.     

The Identities and the Central Polynomials of the Infinite Dimensional Unitary Grassmann Algebra Over a Finite Field

We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field.     

Finite Projective Geometries and Classification of the Weight Hierarchies of Codes （Ⅰ）

The weight hierarchy of a binary linear [n,κ] code C is the sequence (d1,d2,...,dκ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ).     

Finite Dimensional Lie Superalgebras W(m,n,t) and S(m,n,t) of Cartan Type

设Ｌ＝Ｗ或Ｓ，Ｆ是特征数大于２的域．本文证明了Ｆ上的有限维单李超代数Ｌ（ｍ，ｎ，ｔ）的自然滤过是不变的．进而得出了Ｌ（ｍ，ｎ，ｔ）与Ｌ（ｍ′，ｎ′，ｔ′）同构的充要条件是ｍ＝ｍ′，ｎ＝ｎ′和ｔｉ＝τ（ｔ′ｉ），ｉ＝１，２，…，ｍ，这里τ是｛１，２，…，ｍ｝的一个置换．     

A FrameWork for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to C 0 IP Methods

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of C ⁰ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings.