<i>A</i>-generators for certain polynomial algebras

Peterson, Franklin P.

doi:10.1017/s0305004100067803

Cited by 48 publications

(60 citation statements)

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“…The hit problem was first studied by Peterson [22,23], Wood [38], Singer [28], and Priddy [24], who showed its relation to several classical problems respectively in cobordism theory, modular representation theory, Adams spectral sequence for the stable homotopy of spheres, and stable homotopy type of classifying spaces of finite groups. The vector space QP k was explicitly calculated by Peterson [22] for k = 1, 2, by Kameko [14] for k = 3.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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On the Peterson hit problem

Sum¹

2015

Advances in Mathematics

487

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Abstract. We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k := F 2 [x 1 , x 2 , . . . , x k ] as a module over the mod-2 Steenrod algebra, A. In this paper, we study a minimal set of generators for A-module P k in some so-called generic degrees and apply these results to explicitly determine the hit problem for k = 4. Dedicated to Prof. N. H. V. Hưng on the occasion of his sixtieth birthday Introduction and statement of resultsLet V k be an elementary abelian 2-group of rank k. Denote by BV k the classifying space of V k . It may be thought of as the product of k copies of the real projective space RP ∞ . ThenHere the cohomology is taken with coefficients in the prime field F 2 of two elements.Being the cohomology of a space, P k is a module over the mod-2 Steenrod algebra A. The action of A on P k is explicitly given by the formulaand subject to the Cartan formulafor f, g ∈ P k (see Steenrod and Epstein [30]). A polynomial f in P k is called hit if it can be written as a finite sum f = i>0 Sq i (f i ) for some polynomials f i . That means f belongs to A + P k , where A + denotes the augmentation ideal in A. We are interested in the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k as a module over the Steenrod algebra. In other words, we want to find a basis of the F 2 -vector spaceThe hit problem was first studied by Peterson [22,23]

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We are interested in the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra P k as a module over the Steenrod algebra. In other words, we want to find a basis of the F 2 -vector spaceThe hit problem was first studied by Peterson [22,23] …”

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confidence: 99%

On the Peterson hit problem

Sum¹

2015

Advances in Mathematics

487

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“…Some of the motivation for studying these problems is mentioned in [9]. It stems from the Peterson conjecture proved in [4] and various other sources [10,11]. The following result is useful for determining Agenerators for P( ).…”

Section: Introductionmentioning

confidence: 99%

Some Relations between Admissible Monomials for the Polynomial Algebra

Mothebe

Uys

2015

International Journal of Mathematics and Mathematical Sciences

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Let P( ) = F 2 [ 1 , . . . , ] be the polynomial algebra in variables , of degree one, over the field F 2 of two elements. The mod-2 Steenrod algebra A acts on P( ) according to well known rules. A major problem in algebraic topology is of determining A + P( ), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q( ) = P( )/A + P( ). Q( ) has been explicitly calculated for = 1, 2, 3, 4 but problems remain for ≥ 5.Both P( ) = ⨁ ≥0 P ( ) and Q( ) are graded, where P ( ) denotes the set of homogeneous polynomials of degree . In this paper,−1 ∈ P ( −1) is an admissible monomial (i.e., meets a criterion to be in a certain basis for Q( −1)), then, for any pair of integers ( , ), 1 ≤ ≤ , and ≥ 0, the monomial ℎ ( ) = +1 ⋅ ⋅ ⋅ −1 ∈ P +(2 −1) ( ) is admissible. As an application we consider a few cases when = 5.

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“…For p D 2 the hit problem for symmetric algebras has been studied by Janfada and Wood [9; 10] (see below); and quotients had earlier been studied by Peterson [18]. The infinite symmetric algebra over a prime p can be regarded as the cohomology H .BUI F p / (or H .BOI F 2 / if p D 2).…”

Section: Introduction and Theoremmentioning

confidence: 99%

“…Most work on the hit problem has been done on the cohomology of products of projective spaces (the canonical polynomial algebras over A on one-dimensional generators), where the problem is quite difficult. Initial results for such products were obtained by Peterson [17], and he made a conjecture [18] for p D 2 about exactly which degrees contain A-generators (albeit not their rank), which was proven by Wood [22]. A good reference list of the literature on attempts to solve the problem for projective spaces is given by Nam [13].…”

Section: Introductionmentioning

confidence: 99%

A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

Pengelley

Williams

2011

Algebr. Geom. Topol.

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A new action of the Kudo-Araki-May algebra on the dual of the symmetric algebras, with applications to the hit problem DAVID PENGELLEY FRANK WILLIAMSThe hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebras over the field F p , for p an arbitrary prime, and treat the equivalent problem of determining the set of A -primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo-ArakiMay algebra K, and consider the K-module structure of the primitives, which form a sub K-algebra of the dual of the infinite symmetric algebra. Our examples show that the K-action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.

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A-generators for certain polynomial algebras

Abstract: R. M. W. Wood has proved a conjecture we made about dimensions of Agenerators of H*(RP CO x ... xRP°°). In this note we give some of the consequences of this conjecture.

Cited by 48 publications

References 2 publications

On the Peterson hit problem

On the Peterson hit problem

Some Relations between Admissible Monomials for the Polynomial Algebra

A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

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