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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. [org.aimpl.user:ishanl@mit.edu] 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 $p$th 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.