Abstract: Publication date: Available online 17 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Heng Xie In 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 17 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Leonardo M. Cabrer, Hilary A. Priestley This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics—the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this ‘admissibility algebra’ for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to also provide a valuable new tool for studying the structure of Sugihara algebras more widely.

Abstract: Publication date: Available online 13 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): João Gouveia, Antonio Macchia, Rekha R. Thomas, Amy Wiebe The slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. The simplest slack ideals are toric and have connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non-incidences of the polytope. The slack ideal of a polytope is contained in this toric ideal if and only if the polytope is morally 2-level, a generalization of the 2-level property in polytopes. We show that polytopes that do not admit rational realizations cannot have toric slack ideals. A classical example of a projectively unique polytope with no rational realizations is due to Perles. We prove that the slack ideal of the Perles polytope is reducible, providing the first example of a slack ideal that is not prime.

Abstract: Publication date: Available online 12 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Erik Insko, Julianna Tymoczko, Alexander Woo Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here.

Abstract: Publication date: Available online 12 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Yucai Su, Chunguang Xia, Lamei Yuan We classify extensions between finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. We find that although certain finite irreducible conformal modules over B(p) are simply conformal modules over its Virasoro conformal subalgebra Vir, there exist more nontrivial extensions between these conformal B(p)-modules. For extensions between other conformal modules, the situation becomes rather different. As an application, we also solve the extension problem for a series of finite Lie conformal algebras b(n) for n≥1.

Abstract: Publication date: Available online 12 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Paul J. Truman Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive pth root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether's theorem, and Gómez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive pth root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K, we show that OL is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in Gómez Ayala's criterion for the normal case.

Abstract: Publication date: Available online 11 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Kamal Aziziheris Let 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): Cintya Wink de Oliveira Benedito, Carina Alves, Nelson Gomes Brasil Junior, Sueli Irene Rodrigues Costa In this paper we propose a framework to construct algebraic lattices in dimensions 4n via ideals from maximal orders of a quaternion algebra whose center is a totally real number field. For n=1,2,3,4 and 6 it was possible to construct rotated versions of the densest lattices in their dimensions, D4,E8,K12,Λ16 and Λ24. We also present a family of lattices in dimension 2r from A=(−1,−1)Q(ζ2r+ζ2r−1) and a characterization of a maximal quaternion order of A by using the Chebyshev polynomials.

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Eva Belmont Let 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. Gutiérrez Generalizing 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): Ngoc Phu Ha In a recent paper the authors Beliakova, Blanchet and Gainutdinov have shown that the modified trace on the category H-pmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra H. We generalize this fact to the context of G-graded categories and Hopf G-coalgebra studied by Turaev-Virelizier. We show that the symmetrised G-integral on a finite type pivotal Hopf G-coalgebra induces a modified trace in the associated G-graded category.

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): V.H. Jorge Pérez, P.H. Lima In 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 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Viktor Bekkert, Hernán Giraldo, José A. Vélez-Marulanda Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ,V). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ,V) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ,V) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k〚t〛/(t2).

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Yiqiang Li A canonical basis was constructed by Wang and the author in [6] inside Letzter's coideal subalgebra in quantum sl2. In this article, we provide an explicit description for the canonical bases and show that the bases coincide with the one defined algebraically by Bao-Wang in [1].

Abstract: Publication date: Available online 10 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Vyacheslav Futorny, Luis Enrique Ramirez, Jian Zhang We construct explicitly a large family of new simple modules for an arbitrary finite W-algebra of type A. A basis of these modules is given by the Gelfand-Tsetlin tableaux whose entries satisfy certain sets of relations. Characterization and an effective method of constructing such admissible relations are given. In particular we describe a family of simple infinite dimensional highest weight relation modules. We also prove a sufficient condition for the simplicity of tensor product of two highest weight relation modules and establish the simplicity of the tensor product any number of relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest weight modules.

Abstract: Publication date: Available online 4 September 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Rebecca Black For 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 29 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): V. Carmona Sánchez, C. Maestro Pérez, F. Sancho de Salas, J.F. Torres Sancho We provide base change theorems, projection formulae and Verdier duality for both cohomology and homology in the context of finite topological spaces.

Abstract: Publication date: Available online 28 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Friedrich Wehrung We 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: Available online 26 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Neena Gupta, Sourav Sen Extensive 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: Available online 20 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Mircea Voineagu We 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 19 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Karl-Hermann Neeb, Malihe Yousofzadeh We 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: Available online 14 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Mallika Roy, Enric Ventura The 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: Available online 13 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel Daigle We 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Louise Sutton We 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Nik Stopar We 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Christoph Schweigert, Lukas Woike Based 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Tobias Rossmann Define 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Lixin Mao Let 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): María de la Paz Tirado Hernández Let 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: Available online 12 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Thomas Brüstle, Souheila Hassoun, Denis Langford, Sunny Roy Examples 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: Available online 9 August 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Amnon Neeman In 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: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Luca Reggio We show that, if S is a finite semiring, then the free profinite S-semimodule on a Boolean Stone space X is isomorphic to the algebra of all S-valued measures on X, which are finitely additive maps from the Boolean algebra of clopens of X to S. These algebras naturally appear in the logic approach to formal languages as well as in idempotent analysis. Whenever S is a (pro)finite idempotent semiring, the S-valued measures are all given uniquely by continuous density functions. This generalises the classical representation of the Vietoris hyperspace of a Boolean Stone space in terms of continuous functions into the Sierpiński space.We adopt a categorical approach to profinite algebra which is based on profinite monads. The latter were first introduced by Adámek et al. as a special case of the notion of codensity monads.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Takato Uehara In this paper, we consider automorphism groups of rational surfaces which admit cuspidal anticanonical curves and have certain nontrivial automorphisms. By applying Coxeter theory, we show that the automorphism groups of the surfaces are isomorphic to the infinite cyclic group.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Dinh Thanh Trung We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated m-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud and Ulrich which states that these regularities are equal.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Shiquan Ruan, Haicheng Zhang Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Daniele Garzoni, Andrea Lucchini A subset S of a group G invariably generates G if, when each element of S is replaced by an arbitrary conjugate, the resulting set generates G. An invariable generating set X of G is called minimal if no proper subset of X invariably generates G. We will address several questions related to the behaviour of minimal invariable generating sets of a finite group.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Ph. Ellia A closed subscheme of codimension two T⊂P2 is a quasi complete intersection (q.c.i.) of type (a,b,c) if there exists a surjective morphism O(−a)⊕O(−b)⊕O(−c)→IT. We give bounds on deg(T) in function of a,b,c and r, the least degree of a syzygy between the three polynomials defining the q.c.i. (see Theorem 6). As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves (see Theorem 20) and some other related results.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Marcela Hanzer We complete our previous results on the generalized injectivity conjecture for classical p–adic groups. In our previous work, we proved that any irreducible generic subquotient of a standard representation parabolically induced from a maximal parabolic subgroup is, in fact, a subrepresentation. We now generalize this results to any standard representation, i.e. induced from a representation of any standard parabolic subgroup and, in this way, prove the generalized injectivity conjecture for classical p–adic groups completely.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Chih-Whi Chen, Ngau Lam We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a,b,c,d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O‾ over infinite rank Lie superalgebra of types a,b,c,d through the super duality is also a Koszul category.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Amit Shah We generalise some of the theory developed for abelian categories in papers of Auslander and Reiten to semi-abelian and quasi-abelian categories. In addition, we generalise some Auslander-Reiten theory results of S. Liu for Krull-Schmidt categories by removing the Hom-finite and indecomposability restrictions. Finally, we give equivalent characterisations of Auslander-Reiten sequences in a quasi-abelian, Krull-Schmidt category.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Yanjiong Yang, Xiaoguang Yan, Xiaosheng Zhu In this paper, weak tilting modules are introduced and investigated. We show that all tilting modules are weak tilting modules, but the converse is not true. We prove that a module is weak tilting if and only if its character module is cotilting. We also discuss the question when a weak tilting module is of finite type. As an application, we characterize when Gorenstein projective modules are Gorenstein flat.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Daniel J. Hernández, Pedro Teixeira, Emily E. Witt In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and F-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Ryo Kanda We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Manolis C. Tsakiris Let S=k[x1,…,xr] be a polynomial ring over an infinite field k, and I a homogeneous ideal of S generated in degree d. We prove the existence of an integer nI, such that for a set X of at least nI general points in Pr−1, the ideal I∏p∈XI(p) has a linear resolution, where I(p) is the vanishing ideal of the point p∈Pr−1. We also prove that nI≤r(reg(I)−d), which can be seen as a generalization of the well-known fact that Ims always has a linear resolution for s≥reg(I)−d where m=(x1,…,xr).

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): L.N. Bertoncello, D. Levcovitz Let R=K[X1,…,Xn] be a polynomial ring in n variables over a field K of characteristic zero and d a K-derivation of R. Consider the isotropy group of d: Aut(R)d:={ρ∈AutK(R) ρdρ−1=d}. In his doctoral thesis ([1]), Baltazar proved that if d is a simple Shamsuddin derivation of K[X1,X2], then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan ([13]) generalized this result to certain derivations of K[X1,X2] proving that, under certain hypothesis, a derivation of K[X1,X2] is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring R=K[X1,…,Xn] is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of K[X1,X2].

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Marcelo Flores We introduce an algebra of braids and ties (or bt-algebra) of type B. In analogy to the construction of the bt-algebra of type A, we define this bt–algebra of type B through a framization of the Hecke algebra of type B. We find a basis for it, a faithful tensorial representation, and we prove that it supports a Markov trace, from which we derive invariants of classical links in the solid torus.

Abstract: Publication date: January 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 1Author(s): Kenji Hashimoto, JongHae Keum, Kwangwoo Lee If an automorphism of a projective K3 surface with Picard number 2 is of infinite order, then the automorphism corresponds to a solution of Pell equation. In this paper, by solving this equation, we determine all Salem polynomials of symplectic and anti-symplectic automorphisms of projective K3 surfaces with Picard number 2.

Abstract: Publication date: Available online 30 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Enrico Carlini, Maria Virginia Catalisano, Elena Guardo, Adam Van Tuyl It remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function H of a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2 of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that H is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.

Abstract: Publication date: Available online 30 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Cristian Camilo Cárdenas, Ivan Struchiner We 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 7 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Wouter Castryck, Filip Cools, Jeroen Demeyer, Alexander Lemmens In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.

Abstract: Publication date: Available online 7 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Pierre-Alain Jacqmin As a first objective, we characterise those essentially algebraic categories which satisfy properties like being unital, strongly unital, n-permutable, subtractive or protomodular. For each such property, we obtain a Mal'tsev condition as an equivalent condition. Using the language of Janelidze matrix conditions, we treat many of these properties together.As a second objective, using these characterisations, we prove some embedding theorems for those properties in a regular context in the same style as we did in the companion paper [23]. Concrete examples of how to use these embedding theorems are given. Finally, to extend those embedding theorems to the exact context, we show that these properties are stable under the exact completion of a regular category.

Abstract: Publication date: Available online 26 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Francesco Catino, Ilaria Colazzo, Paola Stefanelli In 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): Alexander Baranov, Hasan M. Shlaka We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

Abstract: Publication date: Available online 25 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Linquan Ma, Karl Schwede We 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 Nakashima We 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 Torrecillas We 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 Street We 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 24 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Lars Winther Christensen, Oana Veliche, Jerzy Weyman While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen–Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.

Abstract: Publication date: Available online 23 July 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Fritz Hörmann We 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 Thompson We 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 Traustason We 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 Smolkin We 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 Staglianò We 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 Özaydın We 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 Keem Let 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. Tran Let 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 Mello We 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 Neofytidis We 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 Lin In 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 Beuter Given 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.

Abstract: Publication date: Available online 20 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Teresa Cortadellas Benítez, David A. Cox, Carlos D'Andrea We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano, and Idá, and we use this framework to relate the defining equations.

Abstract: Publication date: Available online 20 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Cătălin Ciupercă We study several aspects of the weak normalization in a graded extension of commutative rings. Applied to the Rees algebra of an ideal in a noetherian ring, the results obtained allow us to prove several properties of the weak subintegral closure of an ideal, a concept introduced by Vitulli and Leahy. The same methods are also used to eliminate one of the hypotheses in a theorem of Gaffney and Vitulli.

Abstract: Publication date: Available online 19 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Edoardo Lanari The goal of this paper is to address the problem of building a path object for the category of Grothendieck (weak) ∞-groupoids. This is the missing piece for a proof of Grothendieck's homotopy hypothesis. We show how to endow the putative underlying globular set with a system of composition, a system of identities and a system of inverses, together with an approximation of the interpretation of any map for a theory of ∞-categories. Finally, we introduce a coglobular ∞-groupoid representing modifications of ∞-groupoids, and prove some basic properties it satisfies, that will be exploited to interpret all 2-dimensional categorical operations on cells of the path object PX of a given ∞-groupoid X.

Abstract: Publication date: Available online 19 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): J.P.C. Greenlees For a compact Lie group G we show that the representing spectrum for Borel cohomology generates its category of modules if G is connected or if coefficients are of characteristic p and π0(G) is a p-group. For a closed subgroup H of G we consider the map C⁎(BG)⟶C⁎(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.

Abstract: Publication date: Available online 19 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): G. Bezhanishvili, P.J. Morandi, B. Olberding By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras. In our recent article, we have extended de Vries duality to completely regular spaces by generalizing de Vries algebras to de Vries extensions. To illustrate the utility of this point of view, we show how to use this new duality to obtain algebraic counterparts of normal and locally compact Hausdorff spaces in the form of de Vries extensions that are subject to additional axioms which encode the desired topological property. This, in particular, yields a different perspective on de Vries duality. As a further application, we show that a duality for locally compact Hausdorff spaces due to Dimov can be obtained from our approach.

Abstract: Publication date: Available online 18 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Maria Manuel Clementino, Dirk Hofmann, Willian Ribeiro Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of (T,V)-categories and show that, under suitable conditions, every injective (T,V)-category is exponentiable in (T,V)-Cat.

Abstract: Publication date: Available online 17 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Levon Haykazyan Simmons (2010) [10] introduced a pre-nucleus and its associated nucleus that measure the subfitness of a frame. Here we continue the study of this pre-nucleus. We answer the questions posed by Simmons.

Abstract: Publication date: Available online 13 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Yuta Kimura We show that for the path algebra A of an acyclic quiver, the singularity category of the derived category Db(modA) is triangle equivalent to the derived category of the functor category of mod_A, that is, Dsg(Db(modA))≃Db(mod(mod_A)). This extends a result in [14] for the path algebra A of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.

Abstract: Publication date: Available online 13 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Liena Colarte, Rosa M. Miró-Roig The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese 3-fold projections. More precisely, for any integer d≥4 and any d-th root e of 1 we denote by Xd the toric variety defined as the image of the morphism φTd:P3⟶Pμ(Td)−1 where Td are all monomials of degree d in k[x,y,z,t] invariant under the action of the diagonal matrix M(1,e,e2,e3). In this work, we describe a Z-basis of the lattice Lη associated to I(Xd) as well as a minimal binomial set of generators of the lattice ideal I(Xd)=I+(η).

Abstract: Publication date: Available online 13 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Colin Crowley, Noah Giansiracusa, Joshua Mundinger Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.

Abstract: Publication date: Available online 12 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Parangama Sarkar Let J⊂I be ideals in a formally equidimensional local ring with λ(I/J)

Abstract: Publication date: Available online 12 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Shiro Goto, Ryotaro Isobe, Shinya Kumashiro Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties.

Abstract: Publication date: Available online 12 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Reuben Green We apply the method of iterated inflations to show that the wreath product of a cellular algebra with a symmetric group is cellular, and obtain descriptions of the cell and simple modules together with a semisimplicity condition for such wreath products.

Abstract: Publication date: Available online 11 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Zeinab Akhlaghi, Silvio Dolfi, Emanuele Pacifici, Lucia Sanus Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. The degree graph Δ(G) of G is defined as the simple undirected graph whose vertex set V(G) consists of the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this note, we provide an upper bound on the size of V(G) in terms of the clique number ω(G) (i.e., the maximum size of a subset of V(G) inducing a complete subgraph) of Δ(G). Namely, we show that V(G) ≤max{2ω(G)+1,3ω(G)−4}. Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3], [4], [9], and answers a question posed by the first author and H.P. Tong-Viet in [4].

Abstract: Publication date: Available online 11 June 2019Source: Journal of Pure and Applied AlgebraAuthor(s): J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, M.P. López López In this paper we develop a descent theory for morphisms α between a monoid B and a unital magma A in a monoidal category with equalizers and coequalizers. We introduce the category of strong descent data for α and we prove that under faithfully flat conditions this category is equivalent to the one of right B-modules. As an application we prove that the category of strong Hopf modules, introduced by us for Hopf quasigroups and weak Hopf quasigroups, is equivalent to a suitable category of strong descent data.

Abstract: Publication date: Available online 17 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Leo Herr Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category A-mod of modules over a fixed ring A in a topos E. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site A-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by Illusie.

Abstract: Publication date: Available online 14 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Herivelto Borges, Mariana Coutinho Let G be the projective plane curve defined over Fq given byaXnYn−XnZn−YnZn+bZ2n=0, where ab∉{0,1}, and for each s∈{2,…,n−1}, let DsP1,P2 be the base-point-free linear series cut out on G by the linear system of all curves of degree s passing through the singular points P1=(1:0:0) and P2=(0:1:0) of G. The present work determines an upper bound for the number Nq(G) of Fq-rational points on the nonsingular model of G in cases where DsP1,P2 is Fq-Frobenius classical. As a consequence, when Fq is a prime field, the bound obtained for Nq(G) improves in several cases the known bounds for the number nP of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point P distinct from its vertices.

Abstract: Publication date: Available online 13 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Seyedeh Narges Hosseini, Behrouz Edalatzadeh, Ali Reza Salemkar Let (g,n) be a pair of Leibniz algebras, where n is an ideal of g. In this article, we develop the concepts of the second relative homology and the relative stem cover of the pair (g,n). By constructing a version of the Hopf's formula for HL2(g,n), we prove the existence of relative stem covers for (g,n). We also give some inequalities and a certain upper bound for the dimension of HL2(g,n). In addition, we classify all pairs of finite dimensional nilpotent Leibniz algebras that have one or two steps distance to this upper bound.

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Benjamin Drabkin, Lorenzo Guerrieri Let I be an ideal whose symbolic Rees algebra is Noetherian. For m≥1, the m-th symbolic defect, sdefect(I,m), of I is defined to be the minimal number of generators of the module I(m)Im. We prove that sdefect(I,m) is eventually quasi-polynomial as a function in m. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian.

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daeyeol Jeon, Chang Heon Kim, Andreas Schweizer We determine which of the modular curves XΔ(N), that is, curves lying between X0(N) and X1(N), are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all XΔ(N) that have infinitely many quadratic points over Q.

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Muriel Livernet, Sarah Whitehouse, Stephanie Ziegenhagen A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Anuj Jakhar, Sudesh K. Khanduja, Neeraj Sangwan Let v be a Krull valuation of a field K with valuation ring Rv and K1,K2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of Rv in the composite field K1K2 is a free Rv-module. For a given pair of prolongations v1,v2 of v to K1,K2 respectively, it is shown that there exists a unique prolongation w of v to K1K2 which extends both v1,v2. Moreover with Si as the integral closure of Rv in Ki, if the ring S1S2 is integrally closed and the residue field of v is perfect, then f(w/v)=f(v1/v)f(v2/v), where f(v′/v) stands for the degree of the residue field of a prolongation v′ of v over the residue field of v. As an application, it is deduced that if K1,K2 are algebraic number fields which are linearly disjoint over K=K1∩K2, then the number of prime ideals of the ring AK1K2 of algebraic integers of K1K2 lying over a given prime ideal ℘ of AK equals the product of the numbers of prime ideals of

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Cristian D. González-Avilés Let f:S′→S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r≥1, let and be, respectively, the restriction and corestriction maps in étale cohomology induced by f. For certain pairs (f,G), we construct maps αr:KerCoresG(r)→CokerResG(r) and βr:CokerResG(r)→KerCoresG(r) such that αr∘βr=βr∘αr=n. In the simplest nontrivial case, namely when f is a quadratic Galois covering, we identify the kernel and cokernel of βr with the kernel and cokernel of another map CokerCoresG(r−1)→KerResG(r+1). We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme S to that of a quadratic Galois cover S′ of S.

Abstract: Publication date: Available online 8 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Paul Barajas, Daniel Duarte We give an explicit presentation of the module of differentials of order n of a finitely generated algebra via a higher-order Jacobian matrix. We use the presentation to study some aspects of this module in the case of hypersurfaces. More precisely, we prove higher-order versions of known results relating freness and torsion-freness of the module of differentials with the regularity and normality of the hypersurface. We also study its projective dimension.

Abstract: Publication date: Available online 7 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Luis Narváez Macarro Let k be a commutative ring and A a commutative k-algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators DA/k is an isomorphism. This result generalizes the characteristic 0 case in which the ring DA/k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k-derivations of A provided that A is smooth over k.

Abstract: Publication date: Available online 7 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Sonia L'Innocente, Françoise Point Let B be a commutative Bézout domain and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class Mod-B of all (right) B-modules. We analyse the definable sets in terms, on the one hand, of the definable sets in the classes Mod-BM, where BM ranges over the localizations of B at M, M∈MSpec(B), and on the other hand, of the constructible subsets of MSpec(B). This allows us to derive decidability results for the class Mod-B, in particular when B is the ring Z˜ of algebraic integers or one of the rings Z˜∩R,Z˜∩Qp.

Abstract: Publication date: Available online 7 May 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Jacques Boulanger, Jean-Luc Chabert To study the question of whether every two-dimensional Prüfer domain possesses the stacked bases property, we consider the particular case of the Prüfer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and in this case we reduce the study of the stacked bases property to questions concerning 2×2-matrices.