Augmentation Ideals Research Articles

The Modular Isomorphism Problem and Abelian Direct Factors

Let p be a prime and let G be a finite p-group. We show that the isomorphism type of the maximal abelian direct factor of G, as well as the isomorphism type of the group algebra over Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p$$\\end{document} of the non-abelian remaining direct factor, if existing, are determined by FpG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_p G$$\\end{document}, generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra 636, 533-559 (2023)) over the prime field. To do this, we address the problem of finding characteristic subgroups of G such that their relative augmentation ideals depend only on the k-algebra structure of kG, where k is any field of characteristic p, and relate it to the modular isomorphism problem, extending and reproving some known results.

Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case, for example the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory, i.e., we study the simultaneous behavior for all compact Lie groups at once.

Deformations of wreath products

Connectivity is a homotopy invariant property of a separable C ∗ -algebra A, which has three important consequences: absence of nontrivial projections, quasidiagonality and realization of the Kasparov group K K ( A , B ) as homotopy classes of asymptotic morphisms from A to B ⊗ K if A is nuclear. Here we give a new characterization of connectivity for separable exact C*-algebras and use this characterization to show that the class of discrete countable amenable groups whose augmentation ideals are connective is closed under generalized wreath products. In a related circle of ideas, we give a result on quasidiagonality of reduced crossed-product C*-algebras associated to noncommutative Bernoulli actions.

On p–almost direct products and residual properties of pure braid groups of nonorientable surfaces

We prove that the n th pure braid group of a nonorientable surface (closed or with boundary, but different from RP2) is residually 2-finite. Consequently, this group is residually nilpotent. The key ingredient in the closed case is the notion of p-almost direct product, which is a generalization of the notion of almost direct product. We prove therefore also some results on lower central series and augmentation ideals of p-almost direct products.

VECTOR INVARIANT IDEALS OF ABELIAN GROUP ALGEBRAS UNDER THE ACTION OF THE SYMPLECTIC GROUPS

Let F be a finite field and let Sp 2ν(F) be the symplectic group over F. If Sp 2ν(F) acts on the F-vector space F2ν, then it can induce an action on the vector space F2ν ⊕ F2ν, defined by (x, y)A = (xA, yA), ∀ x, y ∈ F2ν, A ∈ Sp 2ν(F). If K is a field with char K ≠ char F, then Sp 2ν(F) also acts on the group algebra K[F2ν ⊕ F2ν]. In this paper, we determine the structures of Sp 2ν(F)-stable ideals of the group algebra K[F2ν ⊕ F2ν] by augmentation ideals, and describe the relations between the invariant ideals of K[F2ν] and the vector invariant ideals of K[F2ν ⊕ F2ν].

On the structure of the augmentation quotient group for some nonabelian 2-groups

Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups Qn(G) = Δn(G)/Δn+1(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.

Invariant ideals of abelian group algebras under the torus action of a field, II

Let V = V 1 ⊕ V 2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F • by defining ( v 1 ⊕ v 2 ) g = v 1 g − 1 ⊕ v 2 g . If K is a field of characteristic different from that of F, then G acts on the group algebra K [ V ] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we show that, for almost all fields F, the G-stable ideals are uniquely writable as finite irredundant intersections of augmentation ideals of subspaces W 1 ⊕ W 2 , with W 1 ⊆ V 1 and W 2 ⊆ V 2 . As a consequence, the set of all G-stable ideals is Noetherian.

Simplicial cohomology of augmentation ideals in l1(G)

AbstractLet G be a discrete group.We give a decomposition theorem for the Hochschild cohomology of l1(G) with coefficients in certain G-modules. Using this we show that if G is commutative-transitive, the canonical inclusion of bounded cohomology of G into simplicial cohomology of l1(G) is an isomorphism.

Primeness, semiprimeness and localisation in Iwasawa algebras

Necessary and sufficient conditions are given for the completed group algebras of a compactpp-adic analytic group with coefficient ring thepp-adic integers or the field ofppelements to be prime, semiprime and a domain. Necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coateset al.

The Stable Class of the Augmentation Ideal

The Stable Class of the Augmentation Ideal

Presenting Powers of Augmentation Ideals of Elementary p-Groups

Presenting Powers of Augmentation Ideals of Elementary p-Groups

Presenting Powers Of Augmentation Ideals and Pfister Forms Dedicated to Professor Daniel Quillen on his Sixtieth Birthday

Presenting Powers Of Augmentation Ideals and Pfister Forms Dedicated to Professor Daniel Quillen on his Sixtieth Birthday

Subgroups Determined by Certain Products of Augmentation Ideals

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals γ3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)In(H) (resp., In(H)I(G)I(H)) for n > 1 equals Dn+2(H), the (n + 2)th dimension subgroup of H.

Subgroups Determined by Certain Products of Augmentation Ideals

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals γ3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)I n (H) (resp., I n (H)I(G)I(H)) for n > 1 equals Dn+2(H), the (n + 2)th dimension subgroup of H.

Residual properties of unit loops and augmentation ideals of alternative loop rings

Residual properties of unit loops and augmentation ideals of alternative loop rings

Augmentation ideals of equivariant cohomology rings

The purpose of this note is to establish a number of useful results about the augmentation ideal J for the coefficient ring F ∗ G of a Noetherian complex stable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions.

Injective modules, localization and completion in group algebras

Let k be an uncountable field of characteristic zero and G a polycyclic-by-finite group. If P is a prime ideal of kG and L = L(P) the largest Ore set of elements of kG which are regular mod P, it is shown that the injective hull of any simple kG L -module is artinian. The technique used is to construct normal elements in completions of kG at the augmentation ideals of certain abelian normal subgroups. We also show that if N is a normal subgroup of G such that N is torsion free nilpotent of finite rank and G N is polycyclic-by-finite, then the completion ▪ is Noetherian.

On the isomorphism problem for group rings and completed augmentation ideals

On the isomorphism problem for group rings and completed augmentation ideals

Group rings whose augmentation ideals are residually Lie solvable

Group rings whose augmentation ideals are residually Lie solvable

On the structure of augmentation ideals in group rings

On the structure of augmentation ideals in group rings