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Abstract: We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two. [Copyright &y& Elsevier]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

Highlights • Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. • M. Herzog et al. in [J. Algebra, 2018] give two new criteria for solvability of finite groups. • They proved that if G is a group of order n and ψ (G) ≥ ψ (C n) / 6.68 , then G is solvable. • They conjectured that: Conjecture. If G is a group of order n and ψ (G) > 211 1617 ψ (C n) , then G is solvable. • As the main result of this paper we prove the validity of this conjecture. Abstract Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. In [M. Herzog, et al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: Conjecture. If G is a group of order n and ψ (G) > 211 ψ (C n) / 1617 , where C n is the cyclic group of order n, then G is solvable. In this paper we prove the validity of this conjecture. [ABSTRACT FROM AUTHOR]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]

This note is in the nature of an appendix to the recent paper by Paul Sobaje, [5] . In that paper, Section 2.2 the question of the uniqueness of G structures on (or liftings of) principal indecomposable G 1 -module in the case G = SL 3 is raised. We here answer this question positively as well as making some more general observations. [ABSTRACT FROM AUTHOR]

Let R be an integral domain. In this paper, we study the kernels of derivations of R [ 2 ] and closed polynomials in R [ n ] . Let D be an R -derivation of R [ 2 ] having divergence zero. In the first half of this paper, when R is a Q -domain, we give a necessary and sufficient condition for Ker D to be generated by one element over R . In the second half, we give some sufficient conditions for a non-constant polynomial to be closed in R [ n ] . [ABSTRACT FROM AUTHOR]

Let A → B be a homomorphism of commutative rings. The squaring operation is a functor Sq B / A from the derived category D ( B ) of complexes of B -modules into itself. This operation is needed for the definition of rigid complexes (in the sense of Van den Bergh), that in turn leads to a new approach to Grothendieck duality for rings, schemes and even DM stacks. In our paper with J.J. Zhang from 2008 we introduced the squaring operation, and explored some of its properties. Unfortunately some of the proofs in that paper had severe gaps in them. In the present paper we reproduce the construction of the squaring operation. This is done in a more general context than in the first paper: here we consider a homomorphism A → B of commutative DG rings . Our first main result is that the square Sq B / A ( M ) of a DG B-module M is independent of the resolutions used to present it . Our second main result is on the trace functoriality of the squaring operation . We give precise statements and complete correct proofs. In a subsequent paper we will reproduce the remaining parts of the 2008 paper that require fixing. This will allow us to proceed with the other papers, mentioned in the bibliography, on the rigid approach to Grothendieck duality. The proofs of the main results require a substantial amount of foundational work on commutative and noncommutative DG rings, including a study of semi-free DG rings , their lifting properties, and their homotopies. This part of the paper could be of independent interest. [ABSTRACT FROM AUTHOR]

Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to verify the inductive AM condition stated in [22] (Definition 7.2) for the simple alternating groups in characteristic 2. Such a condition, if checked for all simple groups in all characteristics would prove the Alperin–McKay conjecture (see [22]). We first check the Alperin–McKay conjecture for double cover of symmetric and alternating groups, and . The proof of this will extend known results about blocks of symmetric and alternating groups of a given weight. The first two parts are notations and basic results on blocks of symmetric, alternating groups and of their double covers that are recalled for convenience. In the third part we determine the number of height zero spin characters of a given block using certain height preserving bijections between blocks of a given weight. Then we determine the number of height zero spin characters in a given block of the normalizer of a defect group. We finish by showing that the inductive AM condition is true for the alternating groups in characteristic 2, the cases and being treated separately from the rest. I would like to thank Marc Cabanes and Britta Späth for taking the time to discuss all the mathematical issues I encountered while writing this paper. [Copyright &y& Elsevier]

Abstract: In an earlier paper (Turull, 2012 [12]) the author introduced the concept of endoisomorphisms and showed its natural connection with Clifford theory of finite groups. Associated with Clifford theory are module and character correspondences. In the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results. [Copyright &y& Elsevier]

Abstract: The study of the equivalence relations of particular multialgebras for which the factor multialgebras are universal algebras satisfying certain identities is a very important and intensively studied topic in multialgebra theory and not only. Our paper provides a general multialgebraic approach for the construction of all these relations and also some important and interesting properties concerning the construction of the corresponding factor universal algebra. One of the purposes of our approach is to improve some results concerning -relations of hyperrings and Krasner hyperrings and we will do this in the last sections of this paper. [Copyright &y& Elsevier]

Abstract: This is the second paper in a series devoted to studies of twisted tensor products of nonlocal vertex algebras. In this paper we introduce and study iterated twisted tensor products of nonlocal vertex algebras. Among the main results, we find conditions for constructing an iterated twisted tensor product of three factors, and prove that those conditions are enough for building an iterated twisted tensor product of any number of factors. And we also establish a universal property and give a characterization of an iterated twisted tensor product. Furthermore, we give an example of iterated twisted tensor product nonlocal vertex algebra. [Copyright &y& Elsevier]

Abstract: For a set of primes π, a group X is said to be π-decomposable if is the direct product of a π-subgroup and a -subgroup , where is the complementary of π in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: “Let π be a set of odd primes. If the finite group is a product of two π-decomposable subgroups and , then and this is a Hall π-subgroup of G.” We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. [Copyright &y& Elsevier]

Abstract: In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevichʼs topology, but irreducible with respect to Kambayashiʼs topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in Shafarevich (1981) [Sha81] which is different from the counter-example given by Homma in Kambayashi (1996) [Kam96]. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich. [Copyright &y& Elsevier]

Abstract: This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=G-Hilb) to the same level of difficulty as the toric case. [Copyright &y& Elsevier]

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G , whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper, [14] , S. Doty and J. M. Douglass consider the degree r component L r (E) of the free Lie algebra L (E) on an n -dimensional vector space E over a suitable field k , of characteristic not dividing r. The authors discuss, amongst other things, the question of when this space, regarded as a module for the general linear group of degree n (equivalently the Schur algebra S (n , r)) has the double centraliser property. Some general classes are described but the question in general is left open. Here we show that at most one block component of L r (E) can fail to have the double centraliser property and give some examples of the failure of the property for that block (and therefore for the Lie module). [ABSTRACT FROM AUTHOR]

Abstract Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin L -series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras. [ABSTRACT FROM AUTHOR]

Abstract Given a commutative ring R with unity, we prove in this paper that every coordinate of R [ 2 ] is stably tame. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we describe the K -module H H 1 (L K (Γ)) of outer derivations of the Leavitt path algebra L K (Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of H H 1 (L K (Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C ⁎ -algebra. [ABSTRACT FROM AUTHOR]

Abstract Olivieri, del Río and Simón defined strongly monomial groups and a significant result proved by them is the explicit description of the simple components of the rational group algebra Q G of a strongly monomial group G. In this paper, generalized strongly monomial groups are defined, which is a generalization of strongly monomial groups. Beside strongly monomial groups, an extensive list of important families of groups which are generalized strongly monomial is produced. A description of the simple components of Q G , when G is a generalized strongly monomial group, is also provided. Furthermore, the work by Jespers, Olteanu, del Río and Van Gelder on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has also been generalized. [ABSTRACT FROM AUTHOR]

Abstract Let l be a prime number. In the present paper, we discuss a pro-l version of the congruence subgroup problem for mapping class groups of genus one. Our main result is that the pro-2 version has an affirmative answer, but the pro- l version for l ≥ 11 has a negative answer. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies semigroup algebras A over a domain R for a certain family of semigroups which includes the symmetric groups, the full-transformation semigroups, and the rook semigroups. For each algebra A an A-module M on which A acts is introduced and an algebra B is defined as the commuting algebra of A acting on M. It is shown that A is also the commuting algebra of B acting on M. When A is a symmetric group algebra , B is the corresponding Schur algebra . The representation theory of the semigroup algebras A is well known. This paper analyzes the irreducible representations of the B algebras. In many cases, including the symmetric group, full-transformation semigroup, and rook semigroup cases, a complete set of inequivalent irreducible representations of the B algebra is obtained. In particular, this gives what appears to be a new description of a complete set of irreducible representations for the Schur algebra . [Copyright &y& Elsevier]

Abstract: It is well known that the Cartan matrix of a block of a finite group cannot be arranged as a direct sum of smaller matrices. In this paper we address the question if this remains true for equivalent matrices. The motivation for this question comes from the work of Külshammer and Wada (2002) , which contains certain bounds for the number of ordinary characters in terms of Cartan invariants. As an application we prove such a bound in the special case, where the determinant of the Cartan matrix coincides with the order of the defect group. In the second part of the paper we show that Brauer''s -conjecture holds for 2-blocks under some restrictions on the defect group. For example, the -conjecture holds for 2-blocks if the corresponding defect group is a central extension of a metacyclic group by a cyclic group. The same is true if the defect group contains a central cyclic subgroup of index 8. In particular the -conjecture holds for 2-blocks with defect at most 4. The paper is a part of the author''s PhD thesis. [Copyright &y& Elsevier]

Abstract: The goal of this paper is two-fold. First we provide the information needed to study Bol, or Bruck loops by applying group theoretic methods. This information is used in this paper as well as in Baumeister and Stein (2010) and in Stein (2009) . Moreover, we determine the groups associated to Bruck loops of 2-power exponent under the assumption that every non-abelian simple group S is either passive or isomorphic to , a 2-power. In a separate paper it is proven that indeed every non-abelian simple group S is either passive or isomorphic to , a 2-power (Stein, 2009) . The results obtained here are used in Baumeister and Stein (2010) , where we determine the structure of the groups associated to the finite Bruck loops. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give details of the proofs performed with GAP of the theorems of our paper [N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin, Pointed Hopf algebras over the sporadic simple groups, J. Algebra 325 (1) (2011) 305–320]. [ABSTRACT FROM AUTHOR]

Abstract: We show in this paper that given any reduced, cancellative, torsion-free, atomic monoid, it is possible to construct a possibly non-atomic domain with atomic factorization structure isomorphic to the given monoid. This is significant, since atomic monoids are known to have more freedom in the factorization properties they may possess than atomic domains. This construction is motivated by the paper written by Coykendall and Zafrullah (2004) , in which a non-atomic domain was constructed with factorization structure isomorphic to a singly-generated monoid. [ABSTRACT FROM AUTHOR]

Abstract Let D be a division ring with characteristic different from 2. In this paper we will describe an additive map f satisfying the identity f (x) y + x f (y) = l for every x , y ∈ D such that x y = a , where l , a ∈ D and a is nonzero. Additionally, for nonzero m , k ∈ D , we will describe the additive bijection f satisfying the identity f (x) f (y) = m for every x , y ∈ D such that x y = k ; this description is the solution to a problem posed by Chebotar, Ke, Lee, and Shiao in 2005. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find the number of classes of conjugate π-Hall subgroups in all finite almost simple groups. We also complete the classification of π-Hall subgroups in finite simple groups and correct some mistakes from our previous paper. [Copyright &y& Elsevier]

Abstract: This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we develop a theory of what we call (weak) quantum vertex -algebras with a field of characteristic zero and t a formal variable, and we give a conceptual construction of (weak) quantum vertex -algebras and their modules. As an application, we associate weak quantum vertex -algebras to quantum affine algebras, providing a solution to a problem posed by Frenkel and Jing. We also explicitly construct an example of quantum vertex -algebras from a certain quantum βγ-system. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γ-graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a -graded version of a Hartwig–Larsson–Silvestrov Theorem which leads us to a construction of a q-deformed Witt superalgebra. [ABSTRACT FROM AUTHOR]

Abstract: This paper offers an exhaustive study of certain 2-signalizers in known non-sporadic finite simple groups. The main result of this paper is relevant to the Generation-2 proof of the Classification of Finite Simple Groups. [Copyright &y& Elsevier]

Abstract: This paper is a continuation of the paper [G. Parmeggiani, Pushing up point stabilizers, I, J. Algebra 319 (9) (2008) 3854–3884]. Under the same hypotheses, we determine those amalgams which involve , , as factors. [Copyright &y& Elsevier]

Abstract: Suppose that Q is a connected quiver without oriented cycles and σ is an automorphism of Q. Let k be an algebraically closed field whose characteristic does not divide the order of the cyclic group . The aim of this paper is to investigate the relationship between indecomposable kQ-modules and indecomposable -modules. It has been shown by Hubery that any -module is an isomorphically invariant kQ-module, i.e., ii-module (in this paper, we call it -equivalent kQ-module), and conversely any -equivalent kQ-module induces a -module. In this paper, the authors prove that a -module is indecomposable if and only if it is an indecomposable -equivalent kQ-module. Namely, a method is given in order to induce all indecomposable -modules from all indecomposable -equivalent kQ-modules. The number of non-isomorphic indecomposable -modules induced from the same indecomposable -equivalent kQ-module is given. In particular, the authors give the relationship between indecomposable -modules and indecomposable kQ-modules in the cases of indecomposable simple, projective and injective modules. [Copyright &y& Elsevier]

Abstract: This paper is a sequel to the paper [E. Guardo, B. Harbourne, Resolutions of ideals of six fat points in , J. Algebra 318 (2) (2007) 619–640]. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at essentially distinct points of the projective plane. Each type gives rise to a surface X obtained by blowing up the points. We classify those types such that and is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well known in characteristic 0 [J.W. Bruce, C.T.C. Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19 (2) (1979) 245–256]. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme such that the points are essentially distinct and is nef, given only the configuration type of the points and the coefficients . [Copyright &y& Elsevier]

Abstract: Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander–Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results. [Copyright &y& Elsevier]

Abstract: Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincaré–Birkhoff–Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Möbius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one ordering restriction. [Copyright &y& Elsevier]

Abstract: In our previous paper [J.Y. Shi, Formula for the reflection length of elements in the group , J. Algebra 316 (1) (2007) 284–296], we get a closed formula for the reflection length functions on the reflection group with and . In the present paper, we introduce a partial ordering, called reflection ordering, on the elements of . By applying this formula, we give an explicit description for the reflection ordering on the groups and , . [Copyright &y& Elsevier]

Abstract: In this paper we study division algebras over the function fields of curves over . The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring . A previous paper showed such division algebras had index bounded by assuming the exponent was n and n was prime to p. In this paper we consider algebras of prime degree (and hence exponent) and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index q. [Copyright &y& Elsevier]

Abstract: In the first part of the paper we give a characterization for an associative algebra to be standardly stratified in the sense of Cline, Parshall and Scott, generalizing a theorem of V. Dlab. In the second part of the paper we construct characteristic tilting modules for standardly stratified algebras and use them to estimate the finitistic dimension of such algebras. These tilting modules give rise to the Ringel duality concept for stratified algebras. We also define and investigate a generalization of the notion of properly stratified algebras to the above setup. [Copyright &y& Elsevier]

Abstract: This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras. [Copyright &y& Elsevier]

Abstract: Rickard proved in his paper [J. Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002) 460–481] that if Λ is a finite-dimensional symmetric k-algebra and if there is a set of objects in satisfying some conditions, then there is a derived equivalence taking these objects to the simple modules of another algebra Γ. In this paper we generalize Rickard''s results to finite-dimensional selfinjective k-algebras by adding an extra condition. We use the techniques of Rickard''s paper in this paper. [Copyright &y& Elsevier]

Abstract: Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras , , and . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616]. In the present paper we refine and extend the results of [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616] to the case of a general root-reductive Lie algebra . We prove that the Cartan subalgebras of are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras , , and . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras , , , and with respect to the group of automorphisms of the natural representation which preserve the Lie algebra. [Copyright &y& Elsevier]

Abstract: Let A be a symmetric k-algebra over a perfect field k. Külshammer defined for any integer n a mapping on the degree 0 Hochschild cohomology and a mapping on the degree 0 Hochschild homology of A as adjoint mappings of the respective p-power mappings with respect to the symmetrising bilinear form. In an earlier paper it is shown that is invariant under derived equivalences. In the present paper we generalise the definition of to higher Hochschild homology and show the invariance of κ and its generalisation under derived equivalences. This provides fine invariants of derived categories. [Copyright &y& Elsevier]

Abstract: Let k be a field and let H be a rigid braided Hopf k-algebra. In this paper we continue the study of the theory of braided Hopf crossed products began in [J.A. Guccione, J.J. Guccione, Theory of braided Hopf crossed products, J. Algebra 261 (2003) 54–101]. First we show that to have an H-braided comodule algebra is the same that to have an -braided module algebra, where is a variant of , and then we study the maps and , that appear in the Morita context introduced in the above cited paper. [Copyright &y& Elsevier]

Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras and are not PI equivalent in characteristic . Furthermore we show that the following algebras are not PI equivalent in positive characteristic: and ; and when , , and ; and finally, and . Here E stands for the infinite-dimensional Grassmann algebra with 1, and is the subalgebra of of the block matrices with blocks and on the main diagonal with entries from , and off-diagonal entries from ; is the natural grading on E. [Copyright &y& Elsevier]

Abstract: This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal of R, there is a power Q of p, depending on , such that the Qth Frobenius power of the Frobenius closure of is equal to the Qth Frobenius power of . The paper addresses the question as to whether there exists a uniform which ‘works’ in this context for all parameter ideals of R simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform when R is Cohen–Macaulay. The purpose of this paper is to show that such a uniform exists when R is a generalized Cohen–Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne–Speiser–Lyubeznik Theorem employed by Katzman and Sharp in the Cohen–Macaulay case. [Copyright &y& Elsevier]