The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

The weak conjecture on spherical classes

Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, , to and is the inclusion of the Dickson algebra, , into . This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in }, (2) or a conjecture on ${\mathcal{A}}$ -decomposability of the Dickson algebra in $\Gamma_k^{\wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2). Received November 27, 1996; in final form March 6, 1998     

Arnon completion of the Dyer-Lashof algebra

In his PhD thesis, Arnon [1] builds a completion of the Dickson algebras which contains a “free root” algebraD _fin on the top Dickson classes. Hu’ng [5] has shown that this algebra is in fact isomorphic to a similar completion (A _μ)* of the dual of the Steenrod algebraA*. Arnon also completed the Steenrod algebraA with respect to its halving homomorphism to obtainA _μ. Here we study an analogous completion of the Dyer-Lashof algebraR to obtainR _μ with canonical subcoalgebrasR _μ[n]. Unlike the Steenrod algebra, we may further completeR _μ with respect to length to obtain . It turns out, somewhat surprisingly, that the dual ( ) contains (A _μ)* as a dense subalgebra. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Division of the Dickson algebra by the Steinberg unstable module

We compute the division of the Dickson algebra by the Steinberg unstable module in the category of unstable modules over the mod-2 Steenrod algebra.     

On the action of the Iwahori–Hecke algebra on modular invariants

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of the Takayasu modules.     

The relative trace ideal and the depth of modular rings of invariants

We prove that for a modular representation, the depth of the ring of invariants is the sum of the dimension of the fixed point space of the p-Sylow subgroup and the grade of the relative trace ideal. We also determine which of the Dickson invariants lie in the radical of the relative trace ideal and we describe how to use the Dickson invariants to compute the grade of the relative trace ideal.     

On the ring of invariants ofF_{2^n }^*

In [1] the first and last authors studied a decomposition ofH ^*(R P ^∞×…×R P ^∞;F ₂) into modules over the Steenrod algebra obtained from an action of the cyclic group . Here a minimal set of generators for the ring of invariants is characterized and counted by analyzing the associated ring of Laurent polynomials. A structure theorem for the ring of invariant Laurent polynomials is given and a ‘destabilisation cancels localisation’ theorem is obtained. The authors gratefully acknowledge the support of NSERC. 1980 Mathematics Subject classification, 13F20, 55. Keywords: Invariant theory, Steenrod algebra.     

論Понтрягин示性類(Ⅳ)

<正> 前言 在前一文中,我們曾應用Steenrod冪以證明一個可微分閉流形上法示性類的拓撲不變性.本文的目的則在提供一個不同的方法,不假助於Steenrod冪以證明同一事實.同樣的方法,亦可用於Stiefel-Whitney示性類,而由此獲得這些類的拓撲不變性的一個不同證明,而在證明中避免用到Steenrod     

A counterexample for lightning flash modules over E(e1, e2)

We give a counterexample to Theorem 5 in Section 18.2 of Margolis’ book, “Spectra and the Steenrod Algebra” and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.     

Actions of finite groups on Amitsur rings

Action of finite groups on Amitsur rings is considered. We prove that for an Amitsur ring without |G|-torsion, the ring of invariants is approximated by finitely many Amitsur rings. It is also shown that the class of Amitsur rings is invariant under Montgomery equivalence.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 514–522, September-October, 1995.Supported by the Russian Foundation for Fundamental Research (RFFR), grant No. 93-01-16171.     

Conditional identity calculus and conditioned rational equivalence

We construct a conditional identity calculus (similar to the Birkhoff identity calculus), which complies with the concept of truth for a conditional identity on a universal algebra. The relationship is studied between the isomorphism of embedding categories of conditional varieties and the conditioned rational equivalence of these varieties. As applications, we describe invariants for the relations ‘is conditional rational equivalent’ and ‘is similar’ on finite universal algebras. Supported by RFFR grant No. 93-01-01520. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 432–459, July–August, 1998.     

Automorphism bases for degrees of unsolvability

A set of degrees of unsolvability is said togenerate the degrees if every degree is in the closure of the set under the lattice operations on degrees. (Of course the inf operation is only partially defined.) It is shown that every set of degrees which is comeager or of measure one generates the degrees and that the set of minimal degrees generates the degrees. (Thus any automorphism of degrees is determined by its action on, for example, the minimal degrees). Also the degrees below0′ which have the same jump as any given degree below0′ and those which cup any given nonzero degree ≤0′ to0′ are shown to generate the degrees below0′. The second author's research was performed while he was a Dickson instructor at the University of Chicago. The authors' research was supported by Nationals Science Foundation grants MCS 77-03452 and MCS 76-07033.     

Products of regular cardinals and cardinal invariants of products of Boolean algebras

We answer some questions of Monk, and give some information on others concerning cardinal invariants of Boolean algebras under ultraproducts and products. The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research, and Alice Leonhardt for the beautful typing. Publication no. 345. §1–4 of this paper are essentially the letters which the author sent in December 1987 to Monk solving problems from his notes on cardinal invariants of B.A.; §8 and §9 were written for §4; the other sections, §§5, 6 and 7, were completed in March 1988. Concerning §§5–9, for further results see Abstracts of AMS and subsequent papers. §10 was written during the Arcata meeting, summer 1985, and §11 in January 1986, after questions of Todorcevic.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)     

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.     

A semantics for the first quartet by T. S. Eliot

We construct a semantics for tense logic based on the following central concepts: (a) state of affairs, (b) is simultaneous or earlier than (not later than), (c) is accessible, and (d) is realized, in which a bit of T. Eliot’s lyrics is interpreted. Supported by the Science Foundation of Serbia (grant No. 04M02A) through the Belgrade Mathematical Institute. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 383–408, July–August, 1999.     

A representation of the dual of the Steenrod algebra

In this paper we show how to embed \(A_*\), the dual of the mod 2 Steenrod algebra, into a certain inverse limit of algebras of invariants of the general linear group. The prime 2 is fixed throughout the paper.     

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.     

On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.