Journal of Pure and Applied Algebra最新文献

We examine the power series ring $R\left[\right[X\left]\right]$ over a valuation ring R of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R\left[\right[X\left]\right]$, i.e. an $R\left[\right[X\left]\right]$-module C that is flat over R and has flat dimension at least 2 over $R\left[\right[X\left]\right]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R\left[\right[X\left]\right]$. We also use this theory to give a new proof that $R\left[\right[X\left]\right]$ is not a coherent ring, a fact which is essential in our construction of the module C.

New presentations of Specht modules of symmetric groups over fields of characteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanley and Wachs. These involve generators that are column tabloids and relations that are Garnir relations with maximal number of exchanges between consecutive columns or symmetrization of Garnir relations with minimal number of exchanges between consecutive columns. In this paper, we examine Garnir relations and their symmetrization with any number of exchanges. In both cases, we provide sufficient arithmetic conditions so that the corresponding quotient is a Specht module. In particular, in the first case this yields new presentations of Specht modules if the parts of the conjugate partition that correspond to maximal number of exchanges greater than 1 are distinct. These results generalize the presentations mentioned above and offer an answer to a question of Friedmann, Hanlon and Wachs. Our approach is via representations of the general linear group.

We show that if a countably generated Lie algebra H does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras A and B (satisfying some mild conditions), then H embeds into a quotient of $A⁎B$ that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.

This article studies the reduction maps in the higher K-theory of the ring of integers in a number field arising from the canonical reduction maps at nonzero prime ideals. It proves an explicit density estimate for the subset of primes where the images of a fixed collection of elements vanish. Our result applies to a collection of elements possibly having different degrees and suggests that the linearly independent elements of global K-theory exhibit mutually independent reduction patterns. We also relate the reduction map in K-theory to the reduction map in stable cohomology of general linear groups. This connection allows us to examine the pullback of Quillen's e-classes in the cohomology of the stable general linear group over a finite field. During the proof of the main result, we construct the smallest Galois extension which trivializes a Galois cohomology class of degree one, and show that the linear independence of classes results in disjointness of corresponding field extensions.

In this paper, we give a characterization of locally nilpotent derivations on $A^2$-fibrations having kernels isomorphic to $A^1$-fibrations over Noetherian normal domains containing $Q$.

Recently, Wang and the second author constructed a bar involution and canonical basis for a quasi-permutation module of the Hecke algebra associated to a type B Weyl group W, where the basis is parameterized by left cosets of a quasi-parabolic reflection subgroup in W. In this paper we provide an alternative approach to these constructions, and then generalize these constructions to Coxeter groups which contain a product of type B Weyl groups as a parabolic subgroup.

We prove that for every prime p algebraically clean graphs of groups are virtually residually p-finite and cohomologically p-complete. We also prove that they are cohomologically good. We apply this to certain 2-dimensional Artin groups.

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the relationships as well as the discriminative power of such amplitude distances arising in topological data analysis scenarios.

The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak–Delgado lattice of a central product contains the product of the Chermak–Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak–Delgado lattice relative to their heights in the Chermak–Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak–Delgado lattices.

We establish an embedding from the Hecke algebra associated with the edge contraction of a Coxeter system along an edge to the Hecke algebra associated with the original Coxeter system.