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2. Computations in algebraic K-theory

    1. The motivic filtration on the algebraic K-theory of the sphere spectrum

          Due to work of Rognes, Blumberg–Mandell, and others, we understand the algebraic K-theory of the sphere spectrum fairly well. Recent work of Hahn–Raksit–Wilson [arXiv:2206.11208] gives a motivic filtration on TC of \mathbb{E}_{\infty}-rings which is closely related to the motivic filtration on algebraic K-theory in motivic homotopy theory.

      Add remark

      Problem 2.1.

      [Jeremy Hahn] Can one understand the motivic filtration (or motivic spectral sequence) on the \mathrm{TC} of the sphere spectrum? What about for the motivic filtration on related invariants such as \mathrm{THH}, \mathrm{TP}?

        • The K-theory of Johnson–Wilson theory

              The algebraic K-theory of Johnson–Wilson theory E(n) is closely related to the topological cyclic homology of the connective cover of the two periodic Johnson–Wilson theory E'(n).

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          Problem 2.2.

          [Ishan Levy] Compute something about \mathrm{TC} (and related invariants such as \mathrm{TP}, \mathrm{THH}) of \tau_{\geq0}E'(n), where E'(n) is the 2-periodic Johnson–Wilson theory.

            • Detecting Greek letter families in algebraic K-theory


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              Problem 2.3.

              [Maxwell Johnson] Prove Conjecture 1.1 of [arXiv:1810.10088]. This is a version of redshift for the Greek letter family in the homotopy groups of spheres: it asks that the K-theory of an \mathbb{E}_{\infty}-ring detecting the \alpha^{(n)}-family detects the \alpha^{(n+1)}-family.

                • Atiyah–Segal completion thoerem for iterated K-theory


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                  Problem 2.4.

                  [Mark Behrens] Can one prove an Atiyah–Segal completion theorem for the algebraic K-theory of topological K-theory?

                    • Motivic filtered Poitou–Tate duality


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                      Problem 2.5.

                      [Jeremy Hahn] Is there a duality at the level of motivic filtered spectra \mathrm{fil}_{\text{mot}}\mathrm{TC}(\mathcal{O}_K) that recovers Poitou–Tate duality at the level of associated graded?

                        • Devissage for equivariant/hermitian K-theory

                              We understand devissage for the K-theory of stable categories fairly well now due to work of Quillen, Barwick, and Burklund–Levy (see [arXiv:2112.14723]). See section 2.2 of [arXiv:2009.07225] for what is known for hermitian K-theory.

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                          Problem 2.6.

                          [Ishan Levy] What is the optimal devissage result for equivariant/hermitian K-theory?

                            • Motivic filtration on \mathrm{TR}

                                  [arXiv:2206.11208] gives a motivic filtration on \mathrm{THH}, \mathrm{TC}, \mathrm{TC^{-}}, \mathrm{TP} of ring spectra generalizing previously defined motivic filtrations such as in [arXiv:1802.03261]. In remark 1.13 of loc. cit. it is mentioned that their techniques can be made to additionally define a motivic filtration on \mathrm{TR} of discrete rings.

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                              Problem 2.7.

                              [Dylan Wilson] Can one define a well behaved motivic filtration on the \mathrm{TR} of ring spectra? Can one compute it in good cases?

                                • Calculation of \mathrm{TP}(\mathbb{Z})

                                      Bokstedt and Madsen computed \mathrm{TC}(\mathbb{Z}), without calculating \mathrm{TP}(\mathbb{Z}). However, Hesselholt and Madsen calculated (in [MR1317575]) (L_{K(1)} \mathbb{S})^{tS^1}. Their results suggest the following:

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                                  Conjecture 2.8.

                                  [Sanath Devalapurkar] Fix a prime p. Let j = \tau_{\geq 0} L_{K(1)} \mathbb{S}. Then, \mathrm{TP}(\mathbb{Z}_p) \simeq j^{tS^1}, for the trivial S^1-action on j.
                                    1. Remark. [Sanath Devalapurkar] This would follow if, for instance, \mathrm{TR}(\mathbb{Z}_p) \simeq j.

                                        • L-theory of integers

                                              The symmetric and normal L-theory of the integers was computed in [arXiv:2004.06889], and the Tate cohomology of Q_8 was computed by Atiyah in [MR0148722]. The normal L-theory of \mathbb{Z} looks very similar to the Tate cohomology of Q_8; similarly, the symmetric L-theory of \mathbb{Z} looks very similar to the Tate cohomology of \mathrm{Pin}(2).

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                                          Conjecture 2.9.

                                          [Sanath Devalapurkar] Equip \mathrm{Pin}(2) (resp. Q_8) with the canonical \mathbb{Z}/2-action (resp. the action given by conjugation by i). Then, the E_\infty-map L^s(\mathbb{Z})_{(2)} \to L^n(\mathbb{Z}) can be identified by taking strict \mathbb{Z}/2-fixed points of the map \mathbb{Z}_{(2)}^{t\mathrm{Pin}(2)} \to \mathbb{Z}^{tQ_8}. (This map is induced by the inclusion Q_8\subseteq \mathrm{Pin}(2).)

                                              Cite this as: AimPL: Equivariant techniques in stable homotopy theory, available at http://aimpl.org/equivstable.