Brave new Hopf algebroids and extensions of $MU$-algebras

Baker, Andrew; Jeanneret, Alain

doi:10.4310/hha.2002.v4.n1.a9

Cited by 18 publications

(20 citation statements)

References 13 publications

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“…See Section 6. Lastly, we can also show the collapse at the E 4 -term of the homological homotopy fixed point spectral sequence for R = THH (BP ), where BP is the p-local Brown-Peterson S -algebra [BJ02], without making the (presently uncertain) assumption that BP can be realized as a commutative S -algebra. See Theorem 6.4(b).…”

Section: Introductionmentioning

confidence: 92%

Differentials in the homological homotopy fixed point spectral sequence

Bruner

Rognes

2005

Algebr. Geom. Topol.

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We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S -algebra R. There is a homological homotopy fixed point spectral sequence with E 2 s,t = H −s gp (T; H t (R; F p )), converging conditionally to the continuous homology H c s+t (R hT ; F p ) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations β ǫ Q i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2r -term of the spectral sequence there are 2r other classes in the E 2r -term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E ∞ -term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH (B) of many S -algebras, including B = M U , BP , ku, ko and tmf . Similar results apply for all finite subgroups C ⊂ T, and for the Tate-and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K -theory of commutative S -algebras.

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Section: Introductionmentioning

confidence: 92%

Differentials in the homological homotopy fixed point spectral sequence

Bruner

Rognes

2005

Algebr. Geom. Topol.

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“…We expect that this will turn out to be true and even that the tower is one of R-algebras. This should involve techniques similar to those of [12,6]. It is also worth noting that our proofs make no distinction between the cases where I * R * is infinitely or finitely generated.…”

Section: Discussionmentioning

confidence: 99%

“…This work grew out of a preprint [4] and the work of [6]; it is also related to ongoing collaboration with Alain Jeanneret on Bockstein operations in cohomology theories defined on R-modules [7].…”

Section: Introductionmentioning

confidence: 99%

On the Adams spectral sequence forR–modules

Baker

Lazarev

2001

Algebr. Geom. Topol.

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We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S -algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E 2 -term involves the cohomology of certain 'brave new Hopf algebroids' E R * E . In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum.We show that the Adams Spectral Sequence for S R based on a commutative localized regular quotient R ring spectrum E = R/I[X −1 ] converges to the homotopy of the E -nilpotent completionWe also show that when the generating regular sequence of I * is finite,the Bousfield localization of S R with respect to E -theory. The spectral sequence here collapses at its E 2 -term but it does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve the construction of an I -adic towerwhose homotopy limit is L R E S R . We describe some examples for the motivating case R = M U .

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“….]. Using for instance Angeltveit's result [1, theorem 4.2], one can prove that the BP n are A ∞ spectra and from [4] it is known that this S-algebra structure can be improved to an MU -algebra structure. On the other hand, Strickland showed in [14] that BP n with n 2 is not a homotopy commutative MU -ring spectrum for p = 2.…”

Section: Connective Lubin-tate Spectramentioning

confidence: 99%

Uniqueness of $E_\infty$ structures for connective covers

Baker

Richter

2007

Proc. Amer. Math. Soc.

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Abstract. We refine our earlier work on the existence and uniqueness of E ∞ structures on K-theoretic spectra to show that the connective versions of real and complex K-theory as well as the connective Adams summand at each prime p have unique structures as commutative S-algebras. For the pcompletion p we show that the McClure-Staffeldt model for p is equivalent as an E ∞ ring spectrum to the connective cover of the periodic Adams summand L p . We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum E n and BP n .

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Brave new Hopf algebroids and extensions of $MU$-algebras

Cited by 18 publications

References 13 publications

Differentials in the homological homotopy fixed point spectral sequence

Differentials in the homological homotopy fixed point spectral sequence

On the Adams spectral sequence forR–modules

Uniqueness of $E_\infty$ structures for connective covers

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