QSym over Sym has a stable basis

We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia-Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is n!.     

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

$D_4$型李代数包络代数旋模的范畴化

本文我们定义复数域$C$上一般线性李代数${\rm gl}_n$ BGG 范畴的若干子范畴及其上的投射函子,利用这些子范畴和投射函子范畴化了$D_4$型李代数包络代数旋模的$n$-次张量积.     

Memory estimation of inverse operators

We use methods of harmonic analysis and group representation theory to estimate the memory decay of the inverse operators in Banach spaces. The memory of the operators is defined using the notion of the Beurling spectrum. We obtain a general continuous non-commutative version of the celebrated Wiener's Tauberian Lemma with estimates of the “Fourier coefficients” of inverse operators. In particular, we generalize various estimates of the elements of the inverse matrices. The results are illustrated with a variety of examples including integral and integro-differential operators.     

Decomposition of local cohomology tables of modules with large E-depth

We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich.     

Frobenius扩张上的$n$-Gorenstein投射模和维数

In this paper, we study n-Gorenstein projective modules over Frobenius extensions and n-Gorenstein projective dimensions over separable Frobenius extensions. Let R ■ A be a Frobenius extension of rings and M any left A-module. It is proved that M is an n-Gorenstein projective left A-module if and only if A ■_RM and Hom_R(A, M) are n-Gorenstein projective left A-modules if and only if M is an n-Gorenstein projective left R-module. Furthermore, when R ■ A is a separable Frobenius extension, n-Gorenstein projective dimensions are considered.     

Gorenstein FP-内射复形

引给出了Gorenstein FP-内射复形的概念,进而研究了它的一些性质.     

On the existence of a single test module

We revisit two questions concerning the existence of a single test module by comparing them with similar questions (see Theorem 3.3). As a corollary, we identify domains over which strongly flat modules and torsion-free Whitehead modules coincide (see Corollary 3.6). We obtain several analogous results to the main theorem under stronger hypotheses (see section 4). In particular, we settle a long-standing question concerning a characterization of almost perfect domains (see Corollary 4.4). We also look into the case when the character module of K and the Matlis-dual of K are isomorphic (see Theorem 5.2).     

A semi-abelian extension of a theorem by Takeuchi

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.