Algebraic chromatic homotopy theory for BP∗BPBP_*BP-comodules

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.Comment: All comments welcom

From the Editors - Diverse People, Diverse Approaches

We are proud to announce Praxis’ second volume as a peer-reviewed journal. Our call for articles addressing diversity in the writing center fielded a record number of submissions. We thank all of the authors who submitted careful, insightful, creative and challenging work. We also want to thank our external review board and our editorial team as well as the administrative staff at the Undergraduate Writing Center. Andrea Saathoff, who led Praxis into peerreview status last year and continues to work behind the scenes, deserves a special “Thank you.” This journal, like the writing-center scholarship and pedagogy it supports, exists because of the committed, collaborative work of a broad community of writers and educators. To our authors, reviewers, editors, readers and supporters—Thank you.University Writing Cente

Local duality for structured ring spectra

We use the abstract framework constructed in our earlier paper to study local duality for Noetherian $\mathbb{E}_{\infty}$-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.Comment: Revised version, to appear in Journal of Pure and Applied Algebr

The algebraic chromatic splitting conjecture for Noetherian ring spectra

We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum $R$, and prove it whenever $\pi_*R$ is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups.Comment: Final version to appear in Mathematische Zeitschrif