The homology of $\mathrm{tmf}$

Mathew, Akhil

doi:10.4310/hha.2016.v18.n2.a1

Cited by 40 publications

(58 citation statements)

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“…(3) p = 2. Here we argue similarly as in (2), using an eight cell complex DA(1) and the equivalence TMF (2) ⊗ DA(1) ≃ TMF 1 (3) (2) due to Hopkins-Mahowald (see [Mat16b,§ 4]).…”

Section: The Galois Condition) It Follows That We Have a G-galois Exmentioning

confidence: 73%

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

et al. 2020

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Let A → B be a G-Galois extension of rings, or more generally of E∞-ring spectra in the sense of Rognes. A basic question in algebraic K-theory asks how close the map K(A) → K(B) hG is to being an equivalence, i.e., how close algebraic K-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization.We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on K 0 (−) ⊗ Q. As applications, we prove various descent results in the periodically localized K-theory, TC , THH , etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes. ContentsDUSTIN CLAUSEN, AKHIL MATHEW, NIKO NAUMANN, AND JUSTIN NOEL Appendix A.Étale descent for spectral algebraic spaces 35 Appendix B. Descent for higher real K-theories by Lennart Meier, Justin Noel, and Niko Naumann 38 References 43 1. Introduction 1.1. Motivation. Let X be a noetherian scheme. A subtle and important invariant of X is given by the algebraic K-theory groups {K n (X)} n≥0 . As X varies, the groups {K n (X)} behave something like a cohomology theory in X. For example, they form a contravariant functor in X and there is an analog of the classical Mayer-Vietoris sequence thanks to the localization properties of algebraic K-theory [TT90, Thm. 10.3].A highbrow formulation of this property is that the groups {K n (X)} arise as the homotopy groups of a spectrum K(X), and that the contravariant functorforms a sheaf of connective spectra on the Zariski site of X. 1 As is well-known, however, the Zariski topology of X is too coarse to have a strong analogy with algebraic topology: a more appropriate topology is given by theétale topology. One might hope that K is a sheaf (i.e., behaves 'like a cohomology theory') for theétale topology; if so, one could then hope for a local-to-global spectral sequence (an analog of the Atiyah-Hirzebruch spectral sequence for topological K-theory) beginning withétale cohomology and ultimately converging to algebraic K-theory. Indeed, the convergence properties of such a spectral sequence are the subject of the Quillen-Lichtenbaum conjecture [Lic73, Qui75], which is a consequence of the Rost-Voevodsky Norm Residue theorem (see [Kol15] for a recent survey).The problem is that K-theory is not a sheaf for theétale topology. If E → F is a G-Galois extension of fields, one has a G-action on K(F ) and a canonical mapbut this need not be an equivalence 2 , contradictingétale descent. In fact, since algebraic K-theory satisfies Nisnevich descent (cf.[Nis89] and [TT90, Thm. 10.8]), the failure of descent along Galois extensions of commutative rings is the only obstruction to satisfyingétale descent [Lur11, Cor. 4.24].In the foundationa...

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“…(3) p = 2. Here we argue similarly as in (2), using an eight cell complex DA(1) and the equivalence TMF (2) ⊗ DA(1) ≃ TMF 1 (3) (2) due to Hopkins-Mahowald (see [Mat16b,§ 4]).…”

Section: The Galois Condition) It Follows That We Have a G-galois Exmentioning

confidence: 73%

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

et al. 2020

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“…For p = 2, the homotopy groups of T (2) coincide with those of BP up to dimension 12. There is a map from the twelve dimensional even complex DA(1) to BP which induces an isomorphism in π 0 (see for instance [Mat16]). It is unique in mod 2 homology and lifts to a map DA(1) −→ T (2) by what was just said.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Mat16]) looks encouraging when looking for new splittings of products of known spectra with M String into complex orientable spectra at the prime 2.…”

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confidence: 99%

Towards a splitting of the $K(2)$-local string bordism spectrum

Laures

Schuster

2018

Proc. Amer. Math. Soc.

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“…Example 5.6. The map KO → KU is a Z/2-Galois extension, as shown in [47, Chapter 5] using the following result, a proof of which appears in [41].…”

Section: Galois Extensionsmentioning

confidence: 98%

“…In fact, the idea of this paper arose, in part, from the analysis of the homology of connective tmf def = τ ≥0 Tmf by the first author in [Mat16b]. There, working over Z (2) rather than Z, it was shown that there is a 2-local eight cell complex DA(1) such that the homotopy group sheaf π 0 of O top ⊗ DA(1) ∈ QCoh(M ell ) is given by the pushforward of the structure sheaf via an eight-fold cover…”

Section: Applications To Topological Modular Formsmentioning

confidence: 99%

Affineness and chromatic homotopy theory

Mathew

Meier

2015

Journal of Topology

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Abstract. Given an algebraic stack X, one may compare the derived category of quasi-coherent sheaves on X with the category of dg-modules over the dg-ring of functions on X. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and vanishing results for Tate spectra.

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The homology of $\mathrm{tmf}$

Cited by 40 publications

References 13 publications

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

Towards a splitting of the $K(2)$-local string bordism spectrum

Affineness and chromatic homotopy theory

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