The Categories of Unstable Modules and Unstable Algebras Over the Steenrod Algebra Modulo Nilpotent Objects

Henn, Hans-Werner; Lannes, Jean; Schwartz, Lionel

doi:10.2307/2375065

Cited by 93 publications

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“…The role of the Singer functors in calculating these derived functors on general modules over the Steenrod algebra has been clarified (at odd primes) by the author in [17], establishing the relationship between the approach of Lannes and Zarati [11] and that of Singer. For these applications, it is important to understand the behaviour of the functor Hom ᐁ .R s M; /, for M 2 Obᐁ. Restricting to the full subcategory of ᐁ with objects of the form H V , this is equivalent to understanding R s M in the category ᐁ=ᏺil of unstable modules localized away from the nilpotent unstable modules, using the work of Henn, Lannes and Schwartz [8].…”

Section: Autv S / Smentioning

confidence: 99%

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On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s

Powell

2012

Algebr. Geom. Topol.

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On unstable modules over the Dickson algebras, the Singer functors R s and the functors Fix s GEOFFREY M L POWELLThe category D s ᐁ of unstable modules over the Steenrod algebra equipped with a compatible module structure over the Dickson algebra D s is studied at the prime 2, with applications to the Singer functor R s , considered as a functor from unstable modules ᐁ to D s ᐁ. An explicit copresentation of R s M is given using Lannes' Tfunctor when M is a reduced unstable module; applying Lannes' functor Fix s , this is used to show that R s gives a fully-faithful embedding of ᐁ in D s ᐁ. In addition, the right adjoint Z s to R s is introduced and is related to the indecomposables functor and the functor Fix s .

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Section: Autv S / Smentioning

confidence: 99%

“…An element of K ᐁ is nilclosed if and only if the underlying unstable module is nilclosed; in this case, the unstable K -module structure is the restriction of the induced unstable r l.K/-module structure (r l.K/ has a canonical unstable algebra structure [8]). …”

Section: Nillocalization Of the Category Of Modules Over A Noetherianmentioning

confidence: 99%

On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s

Powell

2012

Algebr. Geom. Topol.

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“…By filtering U (M ) one then verifies that if M is nilclosed, so is U (M ). Thus to identify U (r(F )) with r(I • F ), it suffices to check that l(U (r(F ))) = I • F , where l : U − → F is left adjoint to r. The functor l is exact, preserves tensor products, and can be regarded as localization away from nilpotent modules [HLS,K6]. Thus it carries S * (r(F ))/(Sq |x| x − x 2 ) to the functor that sends V to…”

Section: Proof Of Propositionmentioning

confidence: 99%

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Kuhn

2001

Cohomological Methods in Homotopy Theory

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Abstract. Let T (j) be the dual of the j th stable summand of Ω 2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) → T (2j) → . . . is an infinite wedge of stable summands of K(V, 1)'s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one gets K(Z/2, 1) = RP ∞ as one of the summands.I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T (n, j) for n, j ≥ 0 (with T (1, j) = T (j)), dual to summands of Ω n+1 S N , conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T (n, j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T (n, j) → T (n, 2j) → . . . agrees with the cohomology of the wedge of stable summands of K(V, n)'s corresponding to the wedge occurring in the n = 1 case above.One can also map the T (n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T (∞, 2 k ), for k ≥ 0, map to each other, giving rise to a fitration of HZ/2 which is equivalent to the mod 2 symmetric powers of spheres filtration.Our homotopical constructions use Hopf invariant methods and loopspace technology. These are quite general and should be of independent interest.To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how χ(Sq i ) acts on Dyer-Lashof operations. This should be of use in other settings.In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.

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“…A strong form of the conjecture is true when p is odd. The situation is much more complex when p = 2, but is reduced to a question about 2-central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2-groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less.The the odd prime theorem can be deduced using the approach to U , the category of unstable modules over the Steenrod algebra, initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2].…”

mentioning

confidence: 99%

“…The the odd prime theorem can be deduced using the approach to U , the category of unstable modules over the Steenrod algebra, initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2].…”

mentioning

confidence: 99%

The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

Kuhn

2008

Math. Proc. Camb. Phil. Soc.

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Abstract. We study H * (P ), the mod p cohomology of a finite p-group P , viewed as an Fp[Out(P )]-module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e S ∈ Fp[Out(P )] is a primitive idempotent associated to an irreducible Fp[Out(P )]-module S, then the Krull dimension of e S H * (P ) equals the rank of P . The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Fp[Out(P )]-module occurs as a composition factor in H * (P ) with similar frequency.In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p = 2, but is reduced to a question about 2-central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2-groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less.The the odd prime theorem can be deduced using the approach to U , the category of unstable modules over the Steenrod algebra, initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2]. Also featured are unstable algebras of cohomology primitives associated to central group extensions.

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The Categories of Unstable Modules and Unstable Algebras Over the Steenrod Algebra Modulo Nilpotent Objects

Cited by 93 publications

References 18 publications

On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s

On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

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The Categories of Unstable Modules and Unstable Algebras Over the Steenrod Algebra Modulo Nilpotent Objects

Cited by 93 publications

References 18 publications

On unstable modules over the Dickson algebras, the Singer functors Rsand the functors Fixs

On unstable modules over the Dickson algebras, the Singer functors Rsand the functors Fixs

New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

Contact Info

Product

Resources

About

On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s

On unstable modules over the Dickson algebras, the Singer functors R_sand the functors Fix_s