An Integral Version of the Brown-Gitler Spectrum

Shimamoto, Don

doi:10.2307/1999138

Cited by 5 publications

(4 citation statements)

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“…Brown-Gitler spectra [BJG73] have many applications in classical algebraic topology, including Mahowald's analysis of the bo-resolution [Mah81, Shi84], Cohen's solution of the Immersion Conjecture [Coh85], and more [Mah77, HK99, Goe99]. The classical lambda algebra was essential for constructing and analyzing Brown-Gitler spectra; see [BJG73,Shi84] as above, as well as [GJM86]. In [CQ21], the last two authors introduced a motivic analog of the bo-resolution, the kq-resolution, and analyzed it over algebraically closed fields of characteristic zero.…”

Section: 21mentioning

confidence: 99%

The motivic lambda algebra and motivic Hopf invariant one problem

William¹,

Culver²,

Quigley³

2021

Preprint

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We introduce the F -motivic lambda algebra for any field F of characteristic not equal to 2. This is an explicit differential graded algebra whose homology is the E 2 -page of the F -motivic Adams spectral sequence. Using the R-motivic lambda algebra, we compute the cohomology of the R-motivic Steenrod algebra through filtration 3. This is the algebraically universal case, yielding information about the cohomology of the F -motivic Steenrod algebra for any base field F .We then study the 1-line of the F -motivic Adams spectral sequence in detail. In particular, we produce differentials d 2 (h a+1 ) = (h 0 + ρh 1 )h 2 a valid over any base field F , as well as the following computations in the F -motivic Adams spectral sequence for particular base fields F . For F of the form R, Fq with q an odd prime-power, Qp with p any prime, or Q, we determine the 1-line of the E 3 -page of the F -motivic Adams spectral sequence, as well as all higher differentials in stems s ≤ 7; for F = R, we determine all permanent cycles on the 1-line; and for F = Fq or F = Qp with q, p ≡ 1 (mod 4), we determine all differentials out of the 1-line.These computations are stable motivic analogues of the classic Hopf invariant one problem. We also consider the unstable motivic analogue, showing that it reduces to known results with a finite number of exceptions. As an application, we classify which unstable motivic spheres may be represented by smooth schemes admitting a unital product. * , * is a fundamental computational tool in motivic stable homotopy theory. Here, A F is the F -motivic Steenrod algebra, which acts on M F , the mod 2 motivic cohomology of Spec(F ). This spectral sequence converges to π F * , * , the homotopy groups of the (2, η)-completed F -motivic sphere. This spectral sequence has been used frequently over the past decade. Dugger, Isaksen, Wang, and Xu [DI10, Isa19, IWX20a] have made deep computations for F = C, leading to the current best results on classical stable stems [IWX20b]. Belmont, Dugger, Guillou, and Isaksen [DI16a, DI17, GI20, BI20b, BGI21] made extensive computations for F = R, with applications to C 2 -equivariant stable stems. Wilson and Østvaer [Wil16,WØ17] analyzed the spectral sequence for F = F q to study motivic stable stems over finite fields in weight zero.

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Section: 21mentioning

confidence: 99%

The motivic lambda algebra and motivic Hopf invariant one problem

William¹,

Culver²,

Quigley³

2021

Preprint

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“…With topological applications in mind and to fix conventions, recall the families B(k), B 0 (k) indexed by N, where B(k) is the kth Brown-Gitler spectrum [BG73,GLM93] and B 0 (k) is used to denote the kth integral Brown-Gitler spectrum (cf. [Shi84], where a different indexing is used); the notation B 0 follows [Pea14]. The indexing below follows that of [GLM93], in particular there are homotopy equivalences B(2k) ≃ B(2k + 1) and B 0 (2k) ≃ B 0 (2k + 1), so consideration is limited to the even-indexed spectra.…”

Section: Dual Brown-gitler Modulesmentioning

confidence: 99%

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

Powell

2015

Topology and its Applications

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A structure theorem for bounded-below modules over the subalgebra A (1) of the mod 2 Steenrod algebra generated by Sq 1 , Sq 2 is proved; this is applied to prove a classification theorem for a family of indecomposable A (1)-modules. The action of the A (1)-Picard group on this family is described, as is the behaviour of duality.The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A (1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown-Gitler spectrum).2000 Mathematics Subject Classification. 19L41; 55S10. Key words and phrases. Steenrod algebra, indecomposable module, truncated projective space, Brown-Gitler spectra.The author is very grateful to an anonymous referee for remarks which have lead to an improvement in the presentation of the paper.

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“…To calculate Dieudonné ring and Hopf ring for a ring spectrum E , we use the composite We also need integral versions of Brown-Gitler spectra for our later calculations. The n th integral Brown-Gitler spectrum, which was originally denoted B 1 .n/ and indexed by n 2 1 2 N in Shimamoto [23], and Goerss, Jones and Mahowald [10], will be denoted B 0 .4n/. For all n 2 N and 1 Ä i Ä 3, set B 0 .4n/ D B 0 .4n C i / and then index B 0 .n/ by n 2 N .…”

Section: Brown-gitler Spectra and Dieudonné Ringsmentioning

confidence: 99%