Journal of Pure and Applied Algebra

Published by Elsevier
Print ISSN: 0022-4049
Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The authors explore the possibility of using different condition coherence, which has its origin in topology and logic. In particular, they concentrate on posets with principal ideas that are algebraic lattices and with coherent topologies. These form a Cartesian closed category which has fixed points for domain equations. It is shown that a universal domain exists. A categorical treatment of the construction of this domain is provided, and its relationship to other applications discussed.< >
We establish model category structures on algebras and modules over operads and non-Σ operads in monoidal model categories. The results have applications in algebraic topology, stable homotopy theory, and homological algebra.
We classify rank two Fano bundles over the Grassmannian of lines $\G(1,4)$. In particular we show that the only non-split rank two Fano bundle over $\G(1,4)$ is, up to a twist, the universal quotient bundle $\cQ$. This completes the classification of rank two Fano bundles over Grassmannians of lines.
A Gorenstein sequence H is a sequence of nonnegative integers H=(1,h1,…,hj=1) symmetric about j/2 that occurs as the Hilbert function in degrees less or equal j of a standard graded Artinian Gorenstein algebra A=R/I, where R is a polynomial ring in r variables and I is a graded ideal. The scheme PGor(H) parametrizes all such Gorenstein algebra quotients of R having Hilbert function H and it is known to be smooth when the embedding dimension satisfies h1⩽3. The authors give a structure theorem for such Gorenstein algebras of Hilbert function H=(1,4,7,…) when R=K[w,x,y,z] and I2≅〈wx,wy,wz〉 (Theorems 3.7 and 3.9). They also show that any Gorenstein sequence H=(1,4,a,…),a⩽7 satisfies the condition ΔH⩽j/2 is an O-sequence (Theorems 4.2 and 4.4). Using these results, they show that if H=(1,4,7,h,b,…,1) is a Gorenstein sequence satisfying 3h-b-17⩾0, then the Zariski closure of the subscheme C(H)⊂PGor(H) parametrizing Artinian Gorenstein quotients A=R/I with I2≅〈wx,wy,wz〉 is a generically smooth component of PGor(H) (Theorem 4.6).They show that if in addition 8⩽h⩽10, then such PGor(H) have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain PGor(H) having several components in embedding dimension four or more (Pacific J. Math. 187(1) (1999) 1–11).The proofs use properties of minimal resolutions, the smoothness of PGor(H′) for embedding dimension three (J.O. Kleppe, J. Algebra 200 (1998) 606–628), and the Gotzmann Hilbert scheme theorems (Math. Z. 158(1) (1978) 61–70).
According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.
Sufficient conditions for an ideal $\mathcal I$ in $R\Mod$ to be covering are proved. This allows to obtain an alternative proof of the existence of phantom covers of modules. Our approach is inspired by an extension of the standard deconstructibility techniques used in Approximation Theory.
Up until now, it was recognized that a large number of 2-torsion points was a technical barrier to improve the bounds for the symmetric tensor rank of multiplication in every extension of any finite field. In this paper, we show that there are two exceptional cases, namely the extensions of $\mathbb{F}_2$ and $\mathbb{F}_3$. In particular, using the definition field descent on the field with 2 or 3 elements of a Garcia-Stichtenoth tower of algebraic function fields which is asymptotically optimal in the sense of Drinfel'd-Vladut and has maximal Hasse-Witt invariant, we obtain a significant improvement of the uniform bounds for the symmetric tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$.
Let R be a one-dimensional noetherian domain containing the field Q of rational numbers. Let A be an A2-fibration over R. Then there exists HϵA such that A is an A1-fibration over R[H]. As a consequence, if is free then A = R[2].
We use the Z-algebra theory developed by J. Lepowsky and R. Wilson to study the structure of level three standard modules for the affine Lie algebra A2(2) in the principal picture. We get their linear bases in terms of Z-operators. As a consequence we give Z-algebra proofs of the conjectures of Capparelli (1993) on two combinatorial identities.
We define the tensor product of filtered $A_\infty$-algebras. establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical $A_\infty$-algebras our definition recovers the one given by Markl and Shnider. We also give a criterion that implies that a given $A_\infty$-algebra is quasi-isomorphic to the tensor product of two subalgebras. This will be used in a sequel to prove a K\"unneth Theorem for the Fukaya algebra of a product of Lagrangian submanifolds.
Recently, the first Abel map for a stable curve of genus g≥2 has been constructed. Fix an integer d≥1 and let C be a stable curve of compact type of genus g≥2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map.
We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular we show that a semi-abelian category is peri-abelian if and only if for each normal subobject $K\leq X$, the Higgins commutator of $K$ with itself coincides with the normalisation of the Smith commutator of the denormalisation of $K$ with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).
In this paper, we study precompact abelian groups G that contain no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0. We characterize groups with this property in the following classes of groups: (a) bounded precompact abelian groups; (b) minimal abelian groups; (c) totally minimal abelian groups; (d) \omega-bounded abelian groups. We also provide examples of minimal abelian groups with this property, and show that there exists a minimal pseudocompact abelian group with the same property; furthermore, under Martin's Axiom, the group may be chosen to be countably compact minimal abelian.
Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.
Let $A$ be a unital associative algebra over a field $k$, $E$ a vector space and $\pi : E \to A$ a surjective linear map with $V = {\rm Ker} (\pi)$. All algebra structures on $E$ such that $\pi : E \to A$ becomes an algebra map are described and classified by an explicitly constructed global cohomological type object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such algebra is isomorphic to a Hochschild product $A \star V$, an algebra introduced as a generalization of a classical construction. We prove that ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is the coproduct of all non-abelian cohomologies ${\mathbb H}^{2} \, \, (A, \, (V, \cdot))$. The key object ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ responsible for the classification of all co-flag algebras is computed. All Hochschild products $A \star k$ are also classified and the automorphism groups ${\rm Aut}_{\rm Alg} (A \star k)$ are fully determined as subgroups of a semidirect product $A^* \, \ltimes \bigl(k^* \times {\rm Aut}_{\rm Alg} (A) \bigl)$ of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra $P$, all Poisson algebras having a Poisson algebra surjection on $P$ with a $1$-dimensional kernel are described and classified.
For any finite abelian group G, we study the moduli space of abelian $G$-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M_{1,n}, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric.
We study the geometry of abelian subgroups relative to combings (normal forms) of automatic groups and groups which act properly and cocompactly on spaces of nonpositive curvature. We prove that in general combings which arise in the latter context are incompatible (in a sense which we make precise) with regular combings, the fundamental objects of automatic group theory.
We prove that for each ℓ-group G, the topological space Spec(G) satisfies a condition Idω. Generalising a previous construction of Delzell and Madden we show that for each nondenumer-able cardinal there is a completely normal spectral space that is not homeomorphic to Spec(G) for any ℓ-group G. We show also that a stronger form of property Idω, called Id, suffices to ensure that a completely normal spectral space is homeomorphic to Spec(G) for some ℓ-group G.
We show that every Abelian group G with r0(G)=|G|=|G|ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet–Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet–Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm–Kaplansky invariant fp,k of G satisfies (fp,k)ω=fp,k provided that fp,k is infinite and fp,k>fp,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.
For a fixed proper subgroup R (of type t) of the group of rational numbers, a torsion-free group A is called an R-group if it satisfies Bext1(A, R) = 0, where Bext stands for the set of balanced extensions. Those R-groups whose nonzero elements are of types ≤t are investigated. In the constructible universe L, these R-groups (up to cardinality ℵω) turn out to coincide with those A for which the group is a Butler group; here R0 denotes the largest subgroup of R of idempotent type . This claim is false in models of set theory in which Shelah's Proper Forcing Axiom holds.
In this paper we show that a direct summand of a simply presented mixed abelain group is an almost affable group. As a consequence, the classification theorem due to the author is extended to the largest possible class.
In the context of semi-abelian categories, we develop some new xaspects of the categorical theory of central extensions by Janelidze and Kelly. If is a semi-abelian category and is any admissible subcategory we give several characterizations of trivial and central extensions. The notion of central extension becomes intrinsic when is the subcategory of the abelian objects in . We apply these results to the category of internal groupoids in a semi-abelian category. As a very special case, we get the known description of central extensions for crossed modules.
We study the locus of abelian Galois covers of $\mathbb{P}^{1}$ in $A_{g}$ and the problem of occurrence of Shimura (special) subvarieties generated by these covers in the Torelli locus $T_{g}$ inside $A_{g}$. We first investigate the existence of Shimura subvarieties in the mentioned locus by some computational methods based on Moonen-Oort works and then exclude many cases using both characteristic $p$ methods and monodromy computations.
We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.
A ring R is said to be a unique addition ring (UA-ring) if any semigroup isomorphism R∗ = (R, ∗) ∼- (S, ∗) = S∗ of multiplicative semigroups with another ring S is always a ring isomorphism. See [5, 7–9] for earlier work on UA-rings. Depending on the context, we may or may not regard 0 as an element of R∗. An abelian group G is called a UA-group if its endomorphism ring E(G) is a UA-ring. Given an abelian group G, denote by E∗(G) the semigroup of all endomorphisms of G and let RG be the collection of all rings R such that R∗ ∼- E∗(G). The group G is said to be an if for every ring (E∗(G), ⊕), where ⊕ is an addition on the semigroup E∗(G), there is an abelian group H such that (E∗(G), ⊕) is (isomorphic to) the endomorphism ring of H. Equivalently, G is an if for every ring R in RG there is an abelian group H such that R is (isomorphic to) the endomorphism ring of H.Section 1 is a study of separable torsion-free abelian UA-groups. In Section 2 we develop necessary and sufficient conditions for a torsion-free separable group to be an . All groups are abelian.
The class of finitely presented algebras over a field K with a set of generators a_1,...,a_n and defined by homogeneous relations of the form a_1a_2...a_n = a_{sigma(1)}a_{sigma(2)}...a_{sigma(n)}, where sigma runs through an abelian subgroup H of Sym_{n}, the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid S_n(H), with the "same" presentation as the algebra, is cancellative in terms of the stabilizer of 1 and the stabilizer of n in H. This work is a continuation of earlier work of Cedo, Jespers and Okninski. Comment: 15 pages
In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $\pi_A$ satisfying $\pi_A^2=-p$. Comment: 12 pages, revised version
We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97–113, A characterization of the circle group and the p-adic integers via sequential limit laws, preprint), and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint).
We obtain necessary and sufficient conditions for abelian varieties to acquire semistable reduction over fields of low degree. Our criteria are expressed in terms of torsion points of small order defined over unramified extensions.
We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K for an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253–281] and Pendergrass in [J.W. Pendergrass, The 2-part of the Schur group, J. Algebra 41 (1976) 422–438].
In this paper we discuss multiplicative relations between eigenvalues of Frobenius endomorphism of abelian varieties of small dimension over finite fields.
We define admissible abelian categories and compute the K-theory of such categories, with the aim to study and compute the K-groups of noncommutative rings and other noncommutative situations. One of the main results of this dissertation is the localization theorem.
We prove a Dold–Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold–Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.
Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications. Comment: 14 pages
The quasi-Baer-splitting property is extended from self-small torsion free groups to arbitrary self-small abelian groups. The self-small group A has this property iff it is almost-faithful as an E-module. This fact is reflected in the structure of A/t(A) as a module over the Walk-endomorphism ring of A. A self-small group A is almost E-flat and has the quasi-Baer-splitting property iff the class of almost A-adstatic modules is closed with respect to submodules and iff A is “almost-projective” with respect to the class of almost A-static groups.
We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Quillen homology, which, in turn, can be easily computed from the homotopy groups of the associated simplicial Dieudonné module.
We show that if A1, A2,..., An, B1, B2,...,Bt are uniform objects of an abelian category C, then A1 ⊕ A2 ⊕⋯⊕ An and B1 ⊕ B2 ⊕⋯⊕ Bt are in the same monogeny class if and only if n = t and there is a permutation σ of 1, 2,...,n} such that Ai and Bσ(i) are in the same monogeny class for every i = 1, 2,..., n. This is proved by showing that strong components of bipartite digraphs with enough edges intersect the two independent sets of vertices of a bipartition of the digraph in sets of the same cardinality.
We prove that direct and inverse limits of sequences of reflexive Abelian groups that are metrizable or kω-spaces, but not necessarily locally compact, are reflexive and dual of each other provided some extra conditions are satisfied by the sequences.
Some categories of groups (typically involving groups, possibly infinite, with abelian Frobenius kernels) are shown to be equivalent to categories whose objects are modules over groups and rings. The rings in question are associated with algebraic number fields, and the category equivalences allow some applications of number theory to group theory. In particular the number of isomorphism classes of metabelian Frobenius groups with kernel of order n and complement of order m is calculated in terms of the prime factorization of n and the structure of the Euler group .
Let G be a connected reductive group defined over an algebraically closed field k of characteristic p>0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the “order formula” of Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (ρ,V) with the property that the exponential of dρ(X) lies in ρ(G) for each p-nilpotent . This property permits a simplification of the description given by Suslin et al. of the (even) cohomology ring for the Frobenius kernels . The previous authors already observed that the natural representation of a classical group has the above property (with no restriction on p). Our methods apply to any Chevalley group and hence give the result also for quasisimple groups with “exceptional type” root systems. The methods give explicit sufficient conditions on p; for an adjoint semisimple G with Coxeter number h, the condition p>2h−2 is always good enough.
An abelian group G is almost A-solvable if the natural map θG:Hom(A,G)⊗E(A)A→G is a quasi-isomorphism. Two strongly indecomposable torsion-free abelian groups A and B of finite rank are quasi-isomorphic if and only if the classes of almost A-solvable and almost B-solvable groups coincide. Homological properties of almost A-solvable groups are described, and several examples are given. In particular, there exists a torsion-free almost A-solvable group which is not quasi-isomorphic to an A-solvable group.
Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,\sigma)]t^n$ with coefficients in the Grothendieck ring. The main result of this paper asserts that the motivic zeta-function for an algebraic curve with a finite abelian group action is rational. This is a partial generalization of Weil's First Conjecture.
We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.
In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.
We prove that any abelian variety with CM by OL of characteristic p is defined over a finite field, where OL is the ring of integers of the CM field L. This generalizes a theorem of Grothendieck on isogeny classes of CM abelian varieties. We also provide a direct proof of the Grothendieck theorem, which does not require several ingredients based on Weil's foundation as the original proof does. A description of the isomorphism classes is given. We analyze the reduction map modulo p for the abelian varieties concerned and solve the lifting and algebraization problem.
We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology. (c) 2007 Elsevier B.V. All rights reserved.
We give results on when homomorphisms between abelian varieties are or are not defined over fields obtained from division points on the varieties. For example, if A and B are abelian varieties defined over a field F, of dimensions d and e, respectively, and L is the intersection of the fields F(AN, BN) for all integers N prime to the characteristic of F and greater than 2, then every element of Hom(A, B) is defined over L, is unramified at the discrete places of good reduction for A × B, and [L : F] divides H(d,e), where H(d,e) is a number given by an explicit formula and is less than 4(9d)2d(9e)2e.
Let φ be a Drinfeld module defined over a finite extension K of the rational function field , we show that the submodule φ(Kab)tors of all torsion points in the maximal abelian extension Kab is infinite if and only if φ is of complex multiplication type over K.
We study the moduli space Sg(a) of principally polarized supersingular abelian varieties of dimension g with a-number a. We determine the dimension of each irreducible component of Sg(a) and the number of irreducible components.
For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closuregK(H) of H in K as the subgroup consisting of χ∈K such that χ(an)⟶0 in T=R/Z for every sequence {an} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the Gδ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.
Top-cited authors
Ross Street
  • Macquarie University
John Power
  • Macquarie University
D. F. Anderson
  • University of Tennessee
Henrik Holm
  • University of Copenhagen
W. G. Dwyer
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