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5. Partially commutative ring spectra

    1. Recollement for \mathbb{E}_V-algebras

          The category of G-spectra can be reconstructed from Borel data glued along geometric fixed point data (see [arXiv:1710.06416]).

      Add remark

      Problem 5.1.

      [Ishan Levy] Given a G-representation V, can one similarly reconstruct the category of \mathbb{E}_V-algebras from Borel data and geometric fixed point data?
        1. Remark. [Ishan Levy] It seems like the correct statement should involve module structures over factorization homology, such as those used in [arXiv:2103.04457]. See Remark 4.3.1 of that paper for a comment about this.

            • The structure on \mathrm{BP}_{\mathbb{R}}\langle n \rangle

                  We now know that \mathrm{BP}{\langle n \rangle} can be made an \mathbb{E}_3-ring [arXiv:2012.00864].

              Add remark

              Problem 5.2.

              [Mike Hill] What multiplicative structure can be put on \mathrm{BP}_{\mathbb{R}}\langle n \rangle? Is it \mathbb{E}_{2\sigma+1}?
                1. Remark. [Sanath Devalapurkar] Generalizing this, does \mathrm{BP}_\mathbb{R}\langle n\rangle admit the structure of an \mathbb{E}_\rho \rtimes \mathrm{U}(1)_\mathbb{R}-algebra? (This is the \mathbb{Z}/2-equivariant analogue of a framed \mathbb{E}_2-structure.)

                    • Improved quotients of even rings

                          [arXiv:2203.14787] shows that quotients of powers of elements in rings are more structured than previously thought: if f \in \pi_{2*}R and \mathrm{cof} (f) is a left unital ring, then \mathrm{cof} (f^n) is an \mathbb{E}_{n-1}-algebra.

                      On the other hand, we know that when R is even, then one can do a bit better for n=1: \mathrm{cof} (f) can be made \mathbb{E}_1 for example (see [arXiv:1809.04723]).

                      Add remark

                      Problem 5.3.

                      [Ishan Levy] Given an even \mathbb{E}_\infty-ring R and f \in \pi_*R, is \mathrm{cof} (f^n) an \mathbb{E}_{n+1}-algebra?

                        • Quotients of \mathbb{E}_V-algebras


                          Add remark

                          Problem 5.4.

                          [Tomer Schlank] Given an \mathbb{E}_V-algebra, and an element f in its RO(G)-graded homotopy, are the quotients of powers (or norms) of f highly structured?

                            • Multiplicative structure on equivariant quotients of E_m

                                  We know that Morava K-theory, which is a quotient of E_m, is an \mathbb{E}_1-ring but not an \mathbb{E}_2-ring.

                              Add remark

                              Problem 5.5.

                              [Jeremy Hahn and Dylan Wilson] What multiplicative structures can be supported on C_{p^n}-equivariant quotients of E_m, such as those studied in [arXiv:2204.04366]?

                                • Primitive pth root of unity in \pi_0

                                      \mathrm{KU}[\zeta_p] does *not* admit an \mathbb{E}_3-algebra structure, an \mathbb{E}_2-Thom spectrum structure, or an \mathbb{E}_2-\mathbb{S}[\![q-1]\!]-algebra structure (Here, \mathbb{S}[\![q-1]\!] is the completion of \mathbb{S}[q^{\pm 1}] = \mathbb{S}[\mathbb{Z}] at the augmentation ideal induced by the map \mathbb{Z} \to \ast, and we ask that the map \mathbb{S}[\![q-1]\!] \to \mathrm{KU}[\zeta_p] sends q\mapsto \zeta_p). We also know that \mathrm{KU}[\zeta_{p^\infty}] doesn’t admit the structure of an \mathbb{E}_2-algebra.

                                  Add remark

                                  Problem 5.6.

                                  [Sanath Devalapurkar] Does \mathrm{KU}[\zeta_p] admit an \mathbb{E}_2-algebra structure?

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