Mini-Workshop: Chromatic Phenomena and Duality in Homotopy Theory and Representation Theory

Abstract

This mini-workshop focused on chromatic phenomena and duality as unifying themes in algebra, geometry, and topology. The overarching goal was to establish a fruitful exchange of ideas between experts from various areas, fostering the study of the local and global structure of the fundamental categories appearing in algebraic geometry, homotopy theory, and representation theory. The workshop started with introductory talks to bring researches from different backgrounds to the same page, and later highlighted recent progress in these areas with an emphasis on the interdisciplinary nature of the results and structures found. Moreover, new directions were explored in focused group work throughout the week, as well as in an evening discussion identifying promising long-term goals in the subject. Topics included support theories and their applications to the classification of localizing ideals in triangulated categories, equivariant and homotopical enhancements of important structural results, descent and Galois theory, numerous notions of duality, Picard and Brauer groups, as well as computational techniques. Mathematics Subject Classification (2010): 13D, 14C, 16D, 18E, 20C, 55M, 55N, 55P, 55U. Introduction by the Organisers A spur of flourishing interactions between algebra, geometry, and homotopy theory was initiated by the seminal work of Devinatz, Hopkins, and Smith classifying the thick subcategories of the homotopy category of finite spectra. It was an unprecedented structural result that in particular organized and vastly generalized previous computational advancements in chromatic homotopy theory, and inspired analogous classifications in other settings, which aid the understanding of 508 Oberwolfach Report 9/2018 localizations. Neeman after Hopkins, for example, related the thick and localizing subcategories of the derived category of a ring to certain subsets of the prime spectrum of the ring. These classical results and their various analogues have been streamlined into tensor-triangulated geometry as developed by Balmer in the last decade. Balmer’s formalization of the theory has been crucial in the expansion of chromatic techniques to new areas, most notably modular representation theory. The aim of this mini-workshop was to gather researchers from homotopy theory, algebraic and triangulated geometry, and modular representation theory, in order to showcase recent advancements, as for example the development of stratifications of triangulated categories, new classification results in modular representation theory and stable equivariant homotopy theory, as well as advances in our understanding of various duality phenomena. A second goal was to facilitate new collaborations among the researches; discussions about the interactions among these fields also led to the coining of a new term to encompass them, prismatic algebra. The main activities during the first two days of the workshop were the introductory lectures by Dell’Ambrogio (on tensor-triangulated geometry), Pevtsova (on modular representation theory), Schlank (on chromatic homotopy theory), and Neeman (on Grothendieck duality’s latest conceptualization). The aim was to bring the researchers from various backgrounds to a common ground, so that deeper exchanges can happen as the week progresses. At the same time, four work groups were formed to explore open questions in depth and detail, and as the week progressed, more and more time was spent on group work. At the end of most days, we convened for a report on progress by each group. In addition to all this, we had five talks showcasing the latest developments in the area, and on Thursday evening we had an informal discussion about the past and future of this interactive field, sharing open problems and questions. Introductory talks. Dell’Ambrogio’s lectures introduced the machinery of tensor-triangulated (tt-)geometry, which starts with the Balmer spectrum, a space associated to a tt-category and is built from thick ideals. The geometry of this space is related to localizations of the category in question, as is made precise by a theory of supports. Pevtsova’s and Schlank’s lectures were, in a sense, case studies of the general theory from Dell’Ambrogio’s talk, although historically they came first and were the motivation behind the development of the general theory. Pevtsova talked about the stable module category of a finite group scheme in positive characteristic, where the thick subcategories are related to the prime spectrum of the cohomology ring of the group scheme in question. Schlank discussed chromatic homotopy theory, whereby the thick subcategories in stable homotopy theory are related to points on the moduli stack of formal groups. Neeman’s talk addressed the question of Grothendieck duality; while parts of the foundational theory are classically formal, other parts have been accepted as messy. Neeman presented the classical formal sides of the theory, and supplemented this with his recent results streamlining the sticky points. Mini-Workshop: Chromatic Phenomena and Duality 509 Research talks. Even though the case of the stable homotopy category was the original example of a determination of a Balmer spectrum, through the thick subcategory theorem of Devinatz, Hopkins, and Smith, classifying the thick tensor ideals of the equivariant stable homotopy category of a finite group is very recent work. Noel spoke about the latest advances in this problem, building on previous work by Balmer, Sanders, and Strickland. The talks of Greenlees and Castellana both discussed recent progress in duality in homotopy-theoretic settings. Greenlees explored various duality patterns appearing in the local cohomology spectral sequences for topological modular forms and A(2), an important subalgebra of the dual Steenrod algebra. Castellana presented a stratification result for homotopical (p-local compact) groups, generalizing a theorem of Benson, Iyengar, and Krause for finite groups. In a nutshell, the analogous structural results hold, but the proofs require additional tools from homotopy theory. Grodal gave a cohomological classification of endotrivial modules for arbitrary finite groups, obtained by homotopical methods. This amounts to computing the Picard group of the stable module category, and the results tied together extensive previous work of Alperin, Carlson, Dade, Thevenaz, and others. Balmer explored the notion of residue fields in tt-geometry, using for a compactly generated tensor triangulated category the embedding (modulo phantom maps) into a Grothendieck category via the restricted Yoneda functor. Group work. Inspired by questions from Neeman’s talk, Krause, Neeman, and Pevtsova formed a group to explore the question of strong generation in ttcategories using chromatic methods and the theory of supports. Finding an appropriate framework for Grothendieck’s local duality in the affine case for Gorenstein rings was another topic, because one wants to formulate an analogue for finite dimensional algebras via the action of Hochschild cohomology. Castellana, Greenlees, and Grodal did some computations of the singularity and cosingularity categories of the ring of cochains on a classifying space of a group, to get a feeling for some unexplored structural properties. This was mostly inspired by a recent preprint by Greenlees and Stevenson. Balmer, Dell’Ambrogio, Ricka, Sanders, and Stojanoska discussed questions related to Gorenstein and Anderson duality. One was to formalize Gorenstein duality in an abstract tensor-triangulated setting, in a way that lends itself to studying descent for dualizing modules. Another was to use techniques from relative homological algebra to establish vast generalizations of Anderson duality. Barthel, Beaudry, Heard, Noel, and Schlank introduced Brauer spectra into modular representation theory and worked on computations of Brauer groups of stable module categories using descent-theoretic methods among other techniques. Finally, in the informal evening discussion on Thursday, the victories as well as challenges of this interactive field were discussed, including several open problems of more long-term and open-ended nature than in the group work part of the workshop. 510 Oberwolfach Report 9/2018 Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1641185, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Paul Balmer in the “Simons Visiting Professors” program at the MFO. Mini-Workshop: Chromatic Phenomena and Duality 511 Mini-Workshop: Chromatic Phenomena and Duality in Homotopy Theory and Representation Theory