On the homology of elementary Abelian groups as modules over the Steenrod algebra

Ault, Shaun V.; Singer, William M.

doi:10.1016/j.jpaa.2011.04.004

Cited by 24 publications

(57 citation statements)

References 6 publications

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“…, x n ] = H * B(C n 2 ) + . See [4] for a recent paper on the problem, and [3] for work on the problem using the results we prove here. One approach to it is to consider the analogous problem over subalgebras A(n).…”

Section: A Final Examplementioning

confidence: 99%

Idempotents, localizations and Picard groups of 𝒜(1)-modules

Bruner¹

2014

Contemporary Mathematics

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We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the subalgebra A(1) = Sq 1 , Sq 2 of the mod 2 Steenrod algebra. Since their augmentation ideals are Q 1 -local, we do this by studying the Q i -local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category of boundedbelow A(1)-modules. We show that the Picard groups of the whole category are detected in the local Picard groups, and show that every bounded-below A(1)-module is uniquely expressible as an extension of a Q 0 -local module by a Q 1 -local module, up to stable equivalence. Contentsc 0000 (copyright holder)We next show that the localizations L i F 2 and their suspensions and loops account for the whole Picard group of the Q i -local subcategories (with no local finiteness hypotheses needed). We show that if Pic (i) denotes the Picard group of the category of bounded-below Q i -local modules, thenwith the Z/(4) due to the four-fold periodicity of the loops of P 0 (Theorems 6.8 and 6.9 and Propositions 8.1 and 8.2). Next we show that the global Picard group

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Section: A Final Examplementioning

confidence: 99%

Idempotents, localizations and Picard groups of 𝒜(1)-modules

Bruner¹

2014

Contemporary Mathematics

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“…Remark 5. The author learned about this relationship between images and kernels from William Singer, and significant progress was made along these lines by Singer and the author [3,2]. Singer's approach to the Hit Problem, by comparing images and kernels of Steenrod squares, to our knowledge, has not been considered elsewhere.…”

Section: Proof the Inclusionmentioning

confidence: 99%

“…It is easy to see that U P (0) = 0 [3], but the structure of U P (1) is more delicate. First, we observe that if the cohomological degree of x ∈ M is less than 4, then x may be in the kernel of Sq 3 with no chance of being in the image of Sq 2 .…”

Section: Proof the Inclusionmentioning

confidence: 99%

“…Each of these has a bigrading inherited from Γ. The space of partially A +annihilateds, ∆(k), is in fact a free subalgebra of Γ (see [3]), and I(k) is a two-sided ideal of ∆(k), making U (k) an algebra (see [2], though the fact that I(k) is a twosided ideal was known to Singer before the author included it in his publication). In our study of the A + -annihilated problem, we recognized that finding elements of kernels (i.e., ∆(k)) may be a more difficult task than finding elements of images (i.e., I(k)), and so we consider the following two problems: I.…”

Section: The Dual Problemmentioning

confidence: 99%

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Bott periodicity in the Hit Problem

Ault

2014

Math. Proc. Camb. Phil. Soc.

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Abstract. In this short note, we use Robert Bruner's A(1)-resolution of P = F 2 [t] to shed light on the Hit Problem. In particular, the reduced syzygies Pn of P occur as direct summands of P ⊗n , where P is the augmentation ideal of the map P → F 2 . The complement of Pn in P ⊗n is free, and the modules Pn exhibit a type of "Bott Periodicity" of period 4: P n+4 = Σ 8 Pn. These facts taken together allow one to analyze the module of indecomposables in P ⊗n , that is, to say something about the "A(1)-hit Problem." Our study is essentially in two parts: First, we expound on the approach to the Hit Problem begun by William Singer, in which we compare images of Steenrod Squares to certain kernels of Squares. Using this approach, the author discovered a nontrivial element in bidegree (5, 9) that is neither A(1)-hit nor in kerSq 1 + kerSq 3 . Such an element is extremely rare, but Bruner's result shows clearly why these elements exist and detects them in full generality. Second, we describe the graded F 2 -space of A(1)-hit elements of P ⊗n by determining its Hilbert series.

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“…The problem is to describe the subspace of all P -anihilated or βP -annihilated elements. Following ideas of Ault and Singer [2,3], we study subspaces of partially annihilated elements (∆ M (k) and ∆ β M (k), see below). It appears that there is a structure (homotopy system, see Definitions 2.1 and 2.2) which relates these spaces with the spaces of so called spike images (I M (k) and I β M (k), see below).…”

Section: Introductionmentioning

confidence: 99%

Relations between kernels and images of reduced powers for some right Ap-modules

Popelensky

2018

Publ Inst Math (Belgr)

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We investigate the right action of the mod p Steenrod algebra Ap on the homology H * (L ∧s , Zp) where L = BZp is the lens space. Following ideas of Ault and Singer we investigate the relation between intersection of kernels of the reduced powers P p i and Bockstein element β and the intersection of images of P p i+1 −1 and of β. Namely one can check that k i=0 im P p i+1 −1 ⊂ k i=0 ker P p i and k i=0 im P p i+1 −1 ∩ im β ⊂ k i=0 ker P p i ∩ ker β. We generalize Ault's homotopy systems to p > 2 and examine when the reverse inclusions are true.

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On the homology of elementary Abelian groups as modules over the Steenrod algebra

Cited by 24 publications

References 6 publications

Idempotents, localizations and Picard groups of 𝒜(1)-modules

Idempotents, localizations and Picard groups of 𝒜(1)-modules

Bott periodicity in the Hit Problem

Relations between kernels and images of reduced powers for some right Ap-modules

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