Abstract: Publication date: Available online 30 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Diego Lobos Maturana, Steen Ryom-HansenAbstractWe give a concrete construction of a graded cellular basis for the generalized blob algebra Bn introduced by Martin and Woodcock. The construction uses the isomorphism between KLR-algebras and cyclotomic Hecke algebras, proved by Brundan-Kleshchev and Rouquier. It gives rise to a family of Jucys-Murphy elements for Bn.

Abstract: Publication date: Available online 29 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): A.A. Ambily, Ravi A. RaoAbstractWe consider the Dickson–Siegel–Eichler–Roy's (DSER) subgroup of the orthogonal group OR(Q⊥H(R)m) of a quadratic space with a hyperbolic summand over a commutative ring in which 2 is invertible, rankQ=n≥1 and m≥1. We show that it is a normal subgroup of the orthogonal group OR(Q⊥H(R)m), for m≠2. In particular, when Q≅H(R)r for r≥1 and m≥2, the DSER elementary orthogonal group EOR(Q,H(R)m) coincides with the usual elementary orthogonal group EO2(r+m)(R) and it is a normal subgroup in OR(H(R)r+m). We also prove that the DSER elementary orthogonal group EOR(Q,H(P)) is a normal subgroup of OR(Q⊥H(P)), where Q is a quadratic space and H(P) is the hyperbolic space of the finitely generated projective module over a commutative ring R, with P a finitely generated projective module with rank(P)≥2 and rank(Q)≥1.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Fanggui Wang, Hwankoo Kim, Tao XiongAbstractWe study the finististic weak dimensions of certain pullbacks. As an application, it is proved that the finististic weak dimension of a pseudo-valuation domain is 0, 1, or 2.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Lixin MaoAbstractLet T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Let C1 and C2 be two classes of left A-modules, D1 and D2 be two classes of left B-modules, we prove that: (1) If ToriA(U,C1)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (PD1C1,AD2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. (2) If ExtBi(U,D2)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (AD1C1,ID2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. In additio...

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Pascal Koiran, Mateusz SkomraAbstractLet F(x,y)∈C[x,y] be a polynomial of degree d and let G(x,y)∈C[x,y] be a polynomial with t monomials. We want to estimate the maximal multiplicity of a solution of the system F(x,y)=G(x,y)=0. Our main result is that the multiplicity of any isolated solution (a,b)∈C2 with nonzero coordinates is no greater than 52d2t2. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of F and G, and we briefly review some connections between sparse polynomials and algebraic complexity theory.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Yuanlin Li, Qinghai ZhongAbstractA ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring Z(p)[Cn] is clean, where Z(p) is the localization of the integers at the prime p. In this paper, we consider a more general setting. Let K be an algebraic number field, OK be its ring of integers, and R be a localization of OK at some prime ideal. We investigate when R[G] is clean, where G is a finite abelian group, and obtain a complete characterization for such a group ring to be clean for the case when K=Q(ζn) is a cyclotomic field or K=Q(d) is a quadratic field.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Alana Cavalcante, Mauricio Corrêa, Simone MarchesiAbstractThis paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds X which is threedimensional and with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on X. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): M. Flores, D. GoundaroulisAbstractWe extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra of type B. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type B to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Stephen Lack, Giacomo TendasAbstractRegular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosický, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Working over a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosický.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): S. El Baghdadi, L. Izelgue, A. TamoussitAbstractWe investigate some ring theoretic properties of almost Krull domains. By using the language of star operations, we shed new light on the work of Pirtle on almost Krull domains. That allows us to give a new characterization of these domains. On the other hand, we show that Int(D) issued from an almost Krull domain is D–locally free and we give a complete characterization of when Int(D) is an almost Krull domain.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ilaria CastellanoAbstractFor a topological flow (V,ϕ) - i.e., V is a linearly compact vector space and ϕ a continuous endomorphism of V - we gain a deep understanding of the relationship between (V,ϕ) and the Bernoulli shift: a topological flow (V,ϕ) is essentially a product of one-dimensional left Bernoulli shifts as many as ent⁎(V,ϕ) counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of (V,ϕ). As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Luca Barbieri-Viale, Mike PrestAbstractFollowing Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced by the cartesian tensor structure on the base category via a cohomological Künneth formula.

Abstract: Publication date: Available online 11 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): D. CulverAbstractIn previous work, the author analyzed the co-operations algebra for the second truncated Brown-Peterson spectrum at the prime p=2. The purpose of this paper is to carry out the necessary modifications to odd primes.

Abstract: Publication date: Available online 4 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Onofrio M. Di Vincenzo, Vincenzo NardozzaAbstractLet F be any field, G a finite abelian group and let A, B be F-algebras graded by subgroups of G. If M is a G-graded free (A,B)-bimodule, we describe the G-graded polynomial identities of the triangular algebra of M and, in case the field F has characteristic zero, we provide the description of its G-graded cocharacters by means of the graded cocharacters of A and B.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ido EfratAbstractFor a prime number p and a free profinite group S on the basis X, let S(n,p), n=1,2,…, be the lower p-central filtration of S. For p>n, we give a combinatorial description of H2(S/S(n,p),Z/p) in terms of the Shuffle algebra on X.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Boštjan Gabrovšek, Eva HorvatAbstractWe present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Taku SuzukiAbstractA generalization of S. Mukai's conjecture says that if X is a Fano n-fold with Picard number ρX and pseudo-index iX, then ρX(iX−1)≤n, with equality if and only if X≅(PiX−1)ρX. In this paper, we prove that this conjecture holds if n=6 and either X admits an extremal contraction of fiber type or X admits no small extremal contractions.

Abstract: Publication date: Available online 24 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel Panario, Lucas Reis, Qiang WangAbstractLet Fq be the finite field with q elements, where q is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over Fq of high degree using rational transformations. In particular, given a divisor D>2 of q+1 and an irreducible polynomial f∈Fq[x] of degree n such that n is even or D≢2(mod4), we show how to obtain from f a sequence {fi}i≥0 of irreducible polynomials over Fq with deg(fi)=n⋅Di.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Lixin MaoAbstractLet T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. We prove that, if T is a right coherent ring, UB has finite flat dimension, UA has finite flat or injective dimension, then a left T-module (M1M2)φM is Gorenstein flat if and only if M1 is a Gorenstein flat left A-module, M2/im(φM) is a Gorenstein flat left B-module and the morphism φM:U⊗AM1→M2 is a monomorphism. This result extends an earlier result in this direction. In addition, we give an estimate of Gorenstein flat dimension of a left T-module.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Christoph Schweigert, Lukas WoikeAbstractBased on a detailed definition of extended homotopy quantum field theories we develop a field-theoretic orbifold construction for these theories when the target space is the classifying space of a finite group G, i.e. for G-equivariant topological field theories. More precisely, we use a recently developed bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. One main motivation is the 3-2-1-dimensional case where our orbifold construction allows us to describe the orbifoldization of equivariant modular categories by a geometric construction. As an important ingredient of this result, we prove that a 3-2-1-dimensional G-equivariant topological field theory yields a G-multimodular category by evaluation on the circle. The orbifold construction is a special case of a pushforward operation along an arbitrary morphism of finite groups and provides a valuable tool for the construction of extended homotopy quantum field theories.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Thomas Brüstle, Souheila Hassoun, Denis Langford, Sunny RoyAbstractExamples of exact categories in representation theory are given by the category of Δ-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset (Ex(A),⊂) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean.We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A,E). We investigate the Gabriel-Roiter measure for (A,E), and further study how these notions change when the exact structure varies.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Nik StoparAbstractWe define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we prove that every element in the socle of a unital semiprime ring is unit-regular.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Mallika Roy, Enric VenturaAbstractThe classical result by Dyer–Scott about fixed subgroups of finite order automorphisms of Fn being free factors of Fn is no longer true in Zm×Fn. Within this more general context, we prove a relaxed version in the spirit of Bestvina–Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of m,n. We also study periodic points of endomorphisms of Zm×Fn, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of Zm×Fn. On the way, we prove the analog of Day's Theorem for real elements in Zm×Fn, contributing a modest step into the project of doing so for any right angled Artin group (as McCool did with respect to Whitehead's Theorem in the free context).

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): María de la Paz Tirado HernándezAbstractLet k be a commutative ring of characteristic p>0. We prove that leaps of chain formed by modules of integrable derivations in the sense of Hasse-Schmidt of a k-algebra only occur at powers of p.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Neena Gupta, Sourav SenAbstractExtensive studies are being made on the family of Danielewski surfaces — they provide counter-examples to the Cancellation Problem. In [2], the authors investigated another family of non-cancellative surfaces which were named “double Danielewski surfaces”.In this note, we determine all the locally nilpotent derivations of a double Danielewski surface.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Amnon NeemanAbstractIn a 1973 article Lawvere defined (among many other things) metrics on categories—the article has been enormously influential over the years, spawning a huge literature. In recent work, which is surveyed in the current note, we pursue a largely-unexplored angle: we complete categories with respect to their Lawvere metrics.This turns out to be particularly interesting when the category is triangulated and the Lawvere metric is good; a metric is good if it is translation invariant and the balls of radius ε>0 shrink rapidly enough as ε decreases. The definitions are all made precise at the beginning of the note. And the main theorem is that a certain natural subcategory S(S), of the completion of S with respect to a good metric, is triangulated.There is also a theorem which, under restrictive conditions, gives a procedure for computing S(S). As examples we discuss the special cases (1) where S is the homotopy category of finite spectra, and (2) where S=Db(R–mod), the derived category of bounded complexes of finitely generated R–modules over a noetherian ring R.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Karl-Hermann Neeb, Malihe YousofzadehAbstractWe identify the universal central extension of g=A⊗k, where k is a finite dimensional perfect Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form κ which is invariant under all derivations and A is a unital supercommutative associative (super)algebra.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Louise SuttonAbstractWe continue the study of Specht modules labelled by hook bipartitions for the Iwahori–Hecke algebra of type B with e∈{3,4,…} via the cyclotomic Khovanov–Lauda–Rouquier algebra HnΛ. Over an arbitrary field, we explicitly determine the graded decomposition submatrices for HnΛ comprising rows corresponding to hook bipartitions.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Tobias RossmannAbstractDefine a module representation to be a linear parameterisation of a collection of module homomorphisms over a ring. Generalising work of Knuth, we define duality functors indexed by the elements of the symmetric group of degree three between categories of module representations. We show that these functors have tame effects on average sizes of kernels. This provides a general framework for and a generalisation of duality phenomena previously observed in work of O'Brien and Voll and in the predecessor of the present article. We discuss applications to class numbers and conjugacy class zeta functions of p-groups and unipotent group schemes, respectively.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Friedrich WehrungAbstractWe construct a completely normal bounded distributive lattice D in which for every pair (a,b) of elements, the set {x∈D a≤b∨x} has a countable coinitial subset, such that D does not carry any binary operation ∖ satisfying the identities x≤y∨(x∖y), (x∖y)∧(y∖x)=0, and x∖z≤(x∖y)∨(y∖z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex ℓ-subgroups of any (not necessarily Abelian) ℓ-group. It has ℵ2 elements. This solves negatively a few problems stated by Iberkleid, Martínez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal λ, the class of Stone duals of spectra of all Abelian ℓ-groups with order-unit is not closed under L∞λ-elementary equivalence.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Daniel DaigleAbstractWe investigate the structure of commutative integral domains B of characteristic zero by studying the kernels of locally nilpotent derivations D:B→B.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): V. Carmona Sánchez, C. Maestro Pérez, F. Sancho de Salas, J.F. Torres SanchoAbstractWe provide base change theorems, projection formulae and Verdier duality for both cohomology and homology in the context of finite topological spaces.

Abstract: Publication date: April 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 4Author(s): Mircea VoineaguWe prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and σ. We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.

Abstract: Publication date: Available online 22 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Michael PerlmanAbstractWe study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra DX of differential operators on the space of skew-symmetric matrices X=⋀2Cn. The simple composition factors of these modules are known by the work of Raicu-Weyman, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by computing local cohomology with support in the origin of the indecomposable summands referred to above.

Abstract: Publication date: Available online 22 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Gunnar Carlsson, Benjamin FilippenkoAbstractGiven finite metric spaces (X,dX) and (Y,dY), we investigate the persistent homology PH⁎(X×Y) of the Cartesian product X×Y equipped with the sum metric dX+dY. Interpreting persistent homology as a module over a polynomial ring, one might expect the usual Künneth short exact sequence to hold. We prove that it holds for PH0 and PH1, and we illustrate with the Hamming cube {0,1}k that it fails for PHn,n≥2. For n=2, the prediction for PH2(X×Y) from the expected Künneth short exact sequence has a natural surjection onto PH2(X×Y). We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes {0,1}k={0,1}k−1×{0,1}. For all n≥0, the interleaving distance between the prediction for PHn(X×Y) and the true persistent homology is bounded above by the minimum of the diameters of X and Y. As preliminary results of independent interest, we establish an algebraic Künneth formula for simplicial modules over the ring κ[R+] of polynomials with coefficients in a field κ and exponents in R+=[0,∞), as well as a Künneth formula for the persistent homology of R+-filtered simplicial sets – both of these Künneth formulas hold in all homological dimensions

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Guillermo AlesandroniAbstractLet S be a polynomial ring in n variables, over an arbitrary field. Let M be the family of all monomial ideals in S. Using combinatorial methods, we give an explicit characterization of all M∈M, such that pd(S/M)=n. In addition, we give the total, graded, and multigraded Betti numbers of S/M in homological degree n, for all M∈M. Finally, we show that for each M∈M, with pd(S/M)=n, the sum of the total Betti numbers of S/M is at least 2n.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Dimitra Kosta, Apostolos ThomaAbstractComputing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d∈N such that m(C)≤d for all monomial curves C in A4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in An, where n≥4.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Isaac BirdAbstractOver a Cohen-Macaulay local ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After presenting these categories, we compare them and consider which properties they inherit from the maximal Cohen-Macaulay modules. We then consider some further properties of these classes and how they interact with the entire module category.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): A.S. SivatskiAbstractLet k be a field, chark≠2, let φ be an anisotropic quadratic form over k, dimφ≥2, V the underlying linear space of φ. As usual, denote by D(φ) the set of nonzero values of φ. Given a positive integer m, we say that φ is m-essential if there exists a nonzero polynomial p∈k[x1,…,xm] such that p∈D(φk(x1,…,xm)), but p∉D(ψk(x1,…,xm))) for any anisotropic form ψ over k with dimψ

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Claus Michael Ringel, Pu ZhangAbstractLet k be a field and q a non-zero element of k. In Part I, we have exhibited a 6-dimensional k-algebra Λ=Λ(q) and we have shown that if q has infinite multiplicative order, then Λ has a 3-dimensional local module which is semi-Gorenstein-projective, but not torsionless, thus not Gorenstein-projective. This Part II is devoted to a detailed study of all the 3-dimensional local Λ-modules for this particular algebra Λ. If q has infinite multiplicative order, we will encounter a whole family of 3-dimensional local modules which are semi-Gorenstein-projective, but not torsionless.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Maarten SolleveldAbstractWe investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate.These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): A.-H. NokhodkarAbstractWe study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Rubén A. Hidalgo, Saúl QuispeAbstractLet Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥3 odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which Gn acts purely-non-free is σp(Gn)=n (this coincides with the strong symmetric genus of Gn when n is even). For each of the triangular conformal actions, every non-trivial subgroup of Gn has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of Gn, of its jacobian variety has only one component. We also study conformal/anticonformal actions of Gn, on closed Riemann surfaces, with the property that Gn admits anticonformal elements. It is known that Gn always acts on a genus one Riemann surface with such a property. We observe that the next genus σhyp(Gn)≥2 over which Gn acts in that way is n+1 for n≥2 even, and 2n−2 for n≥3 odd. We also provide examples of pseudo-real Riemann surfaces admitting Gn as the full group of conformal/anticonformal automorphisms.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Bekir Danış, Müfit SezerAbstractWe study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all cases an explicit generating set is known we compute the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also consider the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all situations we consider, it is possible to select a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Jaiung Jun, Kalina Mincheva, Louis RowenAbstractWe develop the basic theory of projective modules and splitting over semirings, within the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.

Abstract: Publication date: March 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 3Author(s): Jaya N.N. Iyer, Roy JoshuaAbstractIn this paper we show that the ℓn-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when ℓ is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves.

Abstract: Publication date: March 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 3Author(s): José Manuel Casas, Xabier García-MartínezAbstractIn this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Abstract: Publication date: March 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 3Author(s): Chahrazed Benouaret, Alain SalinierAbstractThe aim of the paper is to show the existence of some ingredients for an umbral calculus on some Ore extensions, in a manner analogous to Rota's classical umbral calculus which deals with a univariate polynomial ring on a field of characteristic zero. For that, we introduce the notion of a quasi-derivation in order to specify Ore extensions on which building up this umbral calculus is possible. This allows in particular to define an action of the Ore extension on tensor products of modules. We develop also a Pincherle calculus for operators and we define a coalgebra structure on the Ore extension.

Abstract: Publication date: Available online 17 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Heng XieAbstractIn this paper, we consider the Hermitian K-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian K-theory of P1 with involution given by [X:Y]↦[Y:X]. We also prove the C2-equivariant A1-invariance of Hermitian K-theory, which confirms the representability of Hermitian K-theory in the C2-equivariant motivic homotopy category of Heller, Krishna and Østvær [14].

Abstract: Publication date: Available online 11 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Kamal AziziherisAbstractLet cd(G) be the set of the degrees of all complex irreducible characters of a finite group G. For a finite nonabelian simple group S and a positive integer k, let Sk be the direct product of k copies of S. In [2], we conjectured that all finite groups G with cd(G)=cd(Sk) are quasi perfect groups (that is; G′=G″) and hence nonsolvable groups. Then we proved that this conjecture holds for some sporadic simple groups as well as for some simple groups of Lie type (see [1] and [2]). In this paper, we verify this conjecture for some alternating groups and for the simple groups Psp4(q)(q=2m≥2) and G22(q2)(q=32m+1≥27). Indeed, we show that if G is a finite group with cd(G)=cd(H), where H∈{A7k,S7k(k≥1),Psp4(q)k(q=2m≥2,k≥1),G22(q2)k(q=32m+1≥27,1≤k≤6560),A8k(1≤k≤5),S8,A9k,S9k,A10k,S10k(1≤k≤

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Eva BelmontAbstractLet Φ→Γ→Σ be a conormal extension of Hopf algebras over a commutative ring k, and let M be a Γ-comodule. The Cartan-Eilenberg spectral sequenceE2=ExtΦ(k,ExtΣ(k,M))⇒ExtΓ(k,M) is a standard tool for computing the Hopf algebra cohomology of Γ with coefficients in M in terms of the cohomology of Φ and Σ. We construct a generalization of the Cartan-Eilenberg spectral sequence converging to ExtΓ(k,M) that can be defined when Φ=Γ□Σk is compatibly an algebra and a Γ-comodule; this is related to a construction independently developed by Bruner and Rognes. We show that this spectral sequence is isomorphic, starting at the E1 page, to both the Adams spectral sequence in the stable category of Γ-comodules as studied by Margolis and Palmieri, and to a filtration spectral sequence on the cobar complex for Γ originally due to Adams. We obtain a description of the E2 term under an additional flatness assumption. We discuss applications to computing localizations of the Adams spectral sequence E2 page.

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Giovanni Caviglia, Javier J. GutiérrezAbstractGeneralizing a classical result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding categories of algebras.

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): V.H. Jorge Pérez, P.H. LimaAbstractIn this paper we study properties of the coefficient ideals of the powers of an arbitrary ideal in a quasi-unmixed local ring.

Abstract: Publication date: Available online 4 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Rebecca BlackAbstractFor a complex algebraic variety X, we show that triviality of the degree three unramified cohomology H0(X,H3) (occurring on the second page of the Bloch-Ogus spectral sequence [1]) follows from a condition on the integral Chow group CH2X and the integral cohomology group H3(X,Z). In the case that X is an appropriate approximation to the classifying stack BG of a finite p-group G, this result states that the group G has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order p3 for odd prime p have no degree three cohomological invariants.

Abstract: Publication date: Available online 30 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Cristian Camilo Cárdenas, Ivan StruchinerAbstractWe discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results

Abstract: Publication date: Available online 26 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Francesco Catino, Ilaria Colazzo, Paola StefanelliAbstractIn this work, we develop a novel construction technique for set-theoretical solutions of the Yang-Baxter equation. Our technique, named the matched product, is an innovative tool to construct new classes of involutive solutions as the matched product of two involutive solutions is still involutive, and vice versa. This method produces new examples of idempotent solutions as the matched product of other idempotent ones. We translate the construction in the context of semi-braces, which are algebraic structures tightly linked with solutions that generalize the braces introduced by Rump. In addition, we show that the solution associated to the matched product of two semi-braces is indeed the matched product of the solutions associated to those two semi-braces.

Abstract: Publication date: Available online 25 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Linquan Ma, Karl SchwedeAbstractWe prove that a local domain R, essentially of finite type over a field, is regular if and only if for every regular alteration π:X→SpecR, we have that Rπ⁎OX has finite (equivalently zero in characteristic zero) projective dimension.

Abstract: Publication date: Available online 25 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Tohru NakashimaAbstractWe consider several methods for constructing μ-stable reflexive sheaves on a smooth projective threefold. In particular, we prove the existence of μ-stable reflexive sheaves on certain fibered threefolds or blown-up threefolds.

Abstract: Publication date: Available online 25 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel Bulacu, Blas TorrecillasAbstractWe introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one.

Abstract: Publication date: Available online 24 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Stephen Lack, Ross StreetAbstractWe make several corrections and improvements to the published paper “Combinatorial categorical equivalences of Dold–Kan type”, mostly relating to the standing assumptions of the paper. In particular we have had to add one new assumption, but have been able to remove another.

Abstract: Publication date: Available online 23 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Fritz HörmannAbstractWe show that the theory of derivators (or, more generally, of fibered multiderivators) on all small categories is equivalent to this theory on partially ordered sets, in the following sense: Every fibered multiderivator defined on partially ordered sets has an enlargement to all small categories that is unique up to equivalence. Furthermore, extending a theorem of Cisinski, we show that every collection of model categories with Quillen adjunctions in several variables between them gives rise to a left and right fibered multiderivator on all small categories.

Abstract: Publication date: Available online 22 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Sunil Khanal, Rishi Raj Subedi, Gerard ThompsonAbstractWe obtain a matrix representation for each of the indecomposable 9-dimensional real Lie algebras that have a non-trivial Levi decomposition.

Abstract: Publication date: Available online 22 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Marialaura Noce, Gareth Tracey, Gunnar TraustasonAbstractWe give an example of a locally nilpotent group G containing a left 3-Engel element x where 〈x〉G is not nilpotent.

Abstract: Publication date: Available online 19 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel SmolkinAbstractWe exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to [9] and [16]. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras [1]. We also show that our subadditivity containment is sharper than ones shown previously in [32] and [11]. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas from [33] and [11] to show that the adjoint ideal JX(A,Z) reduces mod p to Takagi's adjoint test ideal, even when the ambient space is singular, provided that A is regular at the generic point of X. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.

Abstract: Publication date: Available online 19 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Rodrigo Gondim, Francesco Russo, Giovanni StaglianòAbstractWe present a general construction of hypersurfaces with vanishing hessian, starting from any irreducible non-degenerate variety whose dual variety is a hypersurface and based on the so called Dual Cayley Trick. The geometrical properties of these hypersurfaces are different from the series known until now. In particular, their dual varieties can have arbitrary codimension in the image of the associated polar map.

Abstract: Publication date: Available online 19 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ayten Koç, Murad ÖzaydınAbstractWe study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L-modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, FΓ-modules) via a new colimit model for M⊗FΓL.

Abstract: Publication date: Available online 18 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Edoardo Ballico, Claudio Fontanari, Changho KeemAbstractLet Hd,g,r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in Pr. We denote by Hd,g,rL the union of those components of Hd,g,r whose general element is linearly normal and we show that any non-empty Hd,g,rL (d≥g+r−3) is irreducible for an extensive range of triples (d,g,r) beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme Hd,g,rL of linearly normal curves for g+r−3≤d≤g+r, r≥3, and g≥2r+3 if d=g+r−3.

Abstract: Publication date: Available online 18 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Emil Sköldberg, Nghia T.H. TranAbstractLet a and b be two coprime positive integers and k an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras k[sa,sb] of embedding dimension two (thus also complete intersections) in terms of generators and relations. In addition, we compute the Hilbert series of these cohomology rings.

Abstract: Publication date: Available online 18 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Jorge MelloAbstractWe study families of varieties endowed with dynamical eigensystems of several maps, inducing canonical heights on the dominating variety as well as on the "good" fibers of the family. We show explicitely the dependence on the parameter for global and local canonical heights defined by Kawaguchi when the fibers change, extending previous works of Call and Silverman in dynamical systems formed by just one morphism.

Abstract: Publication date: Available online 18 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Christoforos NeofytidisAbstractWe study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.

Abstract: Publication date: Available online 18 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ching Hung Lam, Xingjun LinAbstractIn this paper, a holomorphic vertex operator algebra U of central charge 24 with the weight one Lie algebra A8,3A2,12 is proved to be unique. Moreover, a holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra F4,6A2,2 is obtained by applying a Z2-orbifold construction to U. The uniqueness of such a vertex operator algebra is also established. By a similar method, we also established the uniqueness of a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra E7,3A5,1. As a consequence, we verify that all 71 Lie algebras in Schellekens' list can be realized as the weight one Lie algebras of some holomorphic vertex operator algebras of central charge 24. In addition, we establish the uniqueness of three holomorphic vertex operator algebras of central charge 24 whose weight one Lie algebras have the type A8,3A2,12, F4,6A2,2, and E7,3A5,1.

Abstract: Publication date: Available online 12 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Luiz Gustavo Cordeiro, Viviane BeuterAbstractGiven an inverse semigroup S endowed with a partial action on a topological space X, we construct a groupoid of germs S⋉X in a manner similar to Exel's groupoid of germs, and similarly a partial action of S on an algebra A induces a crossed product A⋊S. We then prove, in the setting of partial actions, that if X is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs S⋉X is isomorphic to the crossed product AR(X)⋊S, where AR(X) is the Steinberg algebra of X. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form AR(X)⋊S are Steinberg algebras of appropriate groupoids of germs of the form S⋉X. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the second-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs.