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Mini-workshop: Chromatic phenomena and duality in homotopy theory and representation theory. Abstracts from the mini-workshop held March 4–10, 2018. (English) Zbl 1409.00065
Summary: 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.
##### MSC:
 00B05 Collections of abstracts of lectures 00B25 Proceedings of conferences of miscellaneous specific interest 55-06 Proceedings, conferences, collections, etc. pertaining to algebraic topology 18-06 Proceedings, conferences, collections, etc. pertaining to category theory 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18Exx Categorical algebra 55Pxx Homotopy theory 14F42 Motivic cohomology; motivic homotopy theory 16D90 Module categories in associative algebras 20C20 Modular representations and characters
##### Software:
CHomP; GCLC; KILLING; LiE; Mod-p group cohomology; QuillenSuslin; REPSN; Z
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##### References:
 [1] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine angew. Math. 588, (2005) 149–168. [2] P. Balmer, Tensor triangular geometry, Proceedings ICM, Hyderabad (2010), Vol. II, 85– 112. [3] P. Balmer, B. Sanders, The spectrum of the equivariant stable homotopy category of a finite group, Invent. Math. 208 (2017), no. 1, 283–326. [4] T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, N. Stapleton, The Balmer spectrum of the equivariant homotopy category of a finite abelian group, preprint 2017, arXiv:1709.04828 516Oberwolfach Report 9/2018 [5] D. J. Benson, J. F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80. [6] D. Benson, S. Iyengar, H. Krause, Local cohomology and support for triangulated categories, Ann. Sci. ´Ec. Norm. Sup´er. (4) 41 (2008), no. 4, 573–619. [7] D. Benson, S. Iyengar, H. Krause, J. Pevtsova, Stratification for module categories of finite group schemes, J. Amer. Math. Soc. 31 (2018), no. 1, 265–302. [8] A. B. Buan, H. Krause, Ø. Solberg, Support varieties: an ideal approach, Homology, Homotopy Appl. 9 (2007), no. 1, 45–74. [9] M. J. Hopkins, J. H. Smith, Nilpotence and stable homotopy theory II, Ann. of Math. (2) 148(1998) no. 1, 1–49. [10] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997). [11] A. Neeman, The chromatic tower for D(R), With an appendix by Marcel Bkstedt. Topology 31(1992), no. 3, 519–532. [12] R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105, no. 1 (1997). Tensor triangular geometry for finite group schemes Julia Pevtsova The purpose of this extended abstract is to give a snapshot of what was covered in a two lecture survey given at the beginning of the mini-workshop on “Chromatic phenomena and duality in homotopy theory and representation theory”. My task was to focus on the (modular) representation part of the story. I narrowly interpreted the “chromatic phenomena in representation theory” as the description of the structure of thick and localizing subcategories in the stable module - and related - categories of a finite group scheme. Using the language introduced by Paul Balmer, one can reformulate this as a question about the spectra of the corresponding categories, or more generally, about their properties with respect to tensor triangular geometry. We start with some terminology. A finite group scheme G defined over a field k is a representable functor: G : comm k-algebras → groups such that the representing algebra k[G] is finite dimensional as a vector space over k. For what follows, we assume that k has positive characteristic p. Dualizing the coordinate algebra, we get the group algebra kG which is a finite dimensional cocommutative Hopf algebra. This correspondence gives an equivalence of categories  finite group∼finite dimensional coschemescommutative Hopf algebras Examples of these structures include finite groups, restricted Lie algebras and Frobenius kernels of algebraic groups. Representations of a finite group scheme G are equivalent to representations of its group algebra kG. Since the latter is Frobenius, one can construct the stable module categories Stmod G and stmod G Mini-Workshop: Chromatic Phenomena and Duality517 of all and finite dimensional representations of G which are tensor triangulated categories. Hence, one can study tensor triangular geometry, or tt-geometry, in this context. Following the fundamental principles of tt-geometry, we seek to construct a support map supp : Stmod G → X for a topological space X which captures the basic structure of our category. This is where cohomology enters into the picture as we set X = Proj H∗(G, k). By a theorem of Friedlander and Suslin, the (graded commutative) cohomology ring H∗(G, k) is a finitely generated k-algebra; hence, X is a projective variety of finite type. It is known, for example by the work of I. Dell’Ambrogio, that for the support map to classify the tensor ideals in stmod G, which is the category of compact objects in Stmod G, it suffices to show that it satisfies the following list of properties. (1) supp k = X, supp(0) = ∅; (2) “2 out of 3”. If M1→ M2→ M3→ is a triangle in Stmod G, then supp M2⊂ supp M1∪ supp M3; (3) ⊕. supp M ⊕ N = supp M ∪ supp N for any M, N ∈ Stmod G; (4) Shift. supp M = supp Ω−1M for M ∈ Stmod G, and Ω−1the Heller shift; (5) Realization. For any closed subset Y ⊂ X, ∃ M ∈ stmod G such that supp M = Y ; (6) Detection. supp M = ∅ ⇔ M ∼= 0 in StMod G; (7) Tensor product property. supp M ⊗ N = supp M ∩ supp N for any M, N ∈ Stmod G. In Balmer’s terminology, this would say that this support theory is universal for stmod G. To achieve the explicit construction of the universal support theory for Stmod G one constructs not one, but two support theories. The first one is the theory of πsupports of Friedlander and Pevtsova which relies on the notion of a π-point. This construction is inspired by Carlson’s rank variety for elementary abelian p-groups. The other approach, which takes its roots in the classical cohomological support variety, is the Benson-Iyengar-Krause theory of supports via local cohomology functors Γp: Stmod G → Stmod G for p ∈ Proj H∗(G, k). In a joint work with Benson, Iyengar and Krause we show that these two theories coincide for finite group schemes, thereby producing a universal support theory in that context. As an application, we classify localizing (and colocalizing) tensor ideals in Stmod G in the usual way: namely, we prove that there is one-to-one correspondence  subcategories of StMod G∼Projsubsets ofH∗(G, k) 518Oberwolfach Report 9/2018 given by support. For this result we need to develop one other new technique, that of a “reduction to a closed point”, relating the functors Γpand Γmfor a point p in X = Proj H∗(G, k) with a residue field K = k(p) and a closed point m in XK lying over p. This relationship, coming from commutative algebra, led to another application which was the last topic of my two lectures: namely, Gorenstein duality for stmod G. Categorical introduction to chromatic homotopy theory Tomer Schlank I gave two introductory talks on chromatic homotopy. The talks were given using the Balmer spectrum of the symmetric monoidal ∞-category of spectra as a starting point. We use this starting point to discuss types of finite complexes, and the notion of K(n)-local spectra. I then defined Morava E-theory as the Galois closure (in sense of Rognes) of the K(n)-local sphere. This way it is possible to present the main ingredients of chromatic homotopy theory from a purely categorical point of view (rather than using the Landweber exact functor theorem and the theory of formal groups). These ingredients include: (1) The En-local category and En-localisation. (2) The Morava stabiliser group and its action on Morava E(n)-theory. (3) The chromatic fracture square. (4) The chromatic convergence theorem. (5) Telescopic localisation. (6) The telescope conjecture. (7) vn-self maps and the K(n)-local sphere. (8) Ambidexterity of the K(n)-local category. Finally, using ambidexterity (more specifically the dualisability of the K(n)-localisation of the suspension spectra of classifying spaces of finite groups) I connected the theory back to the notion of a formal group law. Grothendieck duality made simple—a brief survey Amnon Neeman There are two classical paths to the foundations of Grothendieck duality: one due to Grothendieck and Hartshorne [3] and (much later) Conrad [1], and a second due to Deligne [2] and Verdier [8] and (much later) Lipman [5]. The consensus has been that both are unsatisfactory. Until the recent past no one knew a clean way to set up the theory. This changed dramatically about three years ago. However: even though the main articles have already appeared in print, only a few experts have been aware of the developments—the papers presenting the results have all focused not so much on the simple proofs of the old theorems, but rather on the technical advances made possible by the new insights. In the talk I took the opposite tack: the Mini-Workshop: Chromatic Phenomena and Duality519 theorems presented were relatively small technical improvements on what may be found in Hartshorne [3], the emphasis was on the clean, modern approach to the proofs. What is perhaps more remarkable is that, with one exception, the modern avenue to the foundations of Grothendieck duality was paved and ready for use already in the mid-1990s. The key new ingredient, which removed the last remaining obstacles, may well appear to be a small, minor step—especially when presented as part of the whole picture. In fact: in a talk I gave at Macquarie University in September 2017, presenting the results to a seminar of category theorists, Steve Lack reacted by asking why it took us so long to see our way through. In the Oberwolfach talk I tried to explain this. The new insight might seem small in hindsight, but required quite a leap of imagination. It hinged on studying a certain map using the chromatic tools that formed the core subject of the workshop, and applying these tools to a morphism which—on the face of it—seems totally worthless. To make this more concrete: the morphism—which proved to be key to the recent progress—has been around for 50 years now, and was dismissed as useless by some of the most eminent mathematicians of the era. The reader is referred to [7] for a more extensive survey, and to [4, 6] for the published accounts [written for the experts] of recent progress. References [13] Brian Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. [14] Pierre Deligne, Cohomology ‘a support propre en construction du foncteur f!, Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer–Verlag, 1966, pp. 404–421. [15] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966. [16] Srikanth B. Iyengar, Joseph Lipman, and Amnon Neeman, Relation between two twisted inverse image pseudofunctors in duality theory, Compos. Math. 151 (2015), no. 4, 735–764. [17] Joseph Lipman, Notes on derived functors and Grothendieck duality, Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, 2009, pp. 1–259. [18] Amnon Neeman, Traces and residues, Indiana Univ. Math. J. 64 (2015), no. 1, 217–229. [19] , Grothendieck duality made simple, (in progress). [20] Jean-Louis Verdier, Base change for twisted inverse images of coherent sheaves, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 393–408. 520Oberwolfach Report 9/2018 My current favourite duality pictures (the local cohomology theorems for tmf and H∗,∗(A(2))) J.P.C. Greenlees (joint work with R.R.Bruner, J.Rognes) The aim of the talk was to consider duality properties related to tmf at the prime 2 and especially to make explicit and pictorial the duality this implies for coefficient rings. The local cohomology theorem for tmf∗ Gorenstein duality for tmf gives a local cohomology spectral sequence for tmf∗. This takes the form HJ∗(tmf∗) ⇒ Σ−22π∗(Ztmf). Here Ztmfdenotes the Anderson dual of tmf and J = (β1, β2) is an ideal of tmf∗ with radical the ideal tmf>0of positive degree elements, where β1(of degree 8) is essentially the Bott element and β2(of degree 192) is a periodicity element. A picture was displayed showing explicitly what this means for the coefficient ring. The local cohomology theorem for H∗,∗(A(2)). The coefficient ring tmf∗can be calculated by an Adams spectral sequence, and the bigraded Ext group at the E2-term (i.e., the cohomology H∗,∗(A(2)) of the algebra A(2) = hSq1, Sq2, Sq4i) enjoys a precisely analogous duality. The consequence of this duality is a Local Cohomology Spectral Sequence of bigraded algebras HJ h∗(H∗,∗(A(2)) ⇒ Σ−(23,0)H∗,∗(A(2))∨, where Jh = (h0, g, w1, w2) is a bihomogeneous ideal whose radical is the augmentation ideal of H∗,∗(A(2)), and where (23, 0) refers to the Adams grading. A picture was displayed showing the striking duality this implies for the bigraded algebra H∗,∗(A(2)). References [21] R.R.Bruner, J.P.C.Greenlees and J.Rognes “The Local Cohomology Theorems for tmf and H∗,∗(A(2)) In preparation, 31pp [22] R.R.Bruner and J.Rognes “The coefficient ring tmf∗.” In preparation [23] J.P.C.Greenlees “Homotopy invariant commutative algebra over fields” CRM IRTATCA lectures (2015), (to appear) 61pp, arXiv:1601.024737 [24] J.P.C.Greenlees and V.Stojanoska “Anderson and Gorenstein duality” Preprint (2017), 21pp, arXiv: 1705.02664 Mini-Workshop: Chromatic Phenomena and Duality521 Stratification and duality for homotopical groups Nat‘alia Castellana (joint work with Tobias Barthel, Drew Heard, Gabriel Valenzuela) Let G be a finite group or a connected Lie group, and k a field of characteristic p. Benson–Iyengar–Krause (see [4]) developed the notion of stratification of a triangulated category by a Notherian commutative ring, using support theoretic techniques, which captures both the classification of thick and localizing subcategories. For G finite, this machinery was used to prove stratification results for StMod(kG), K(InjkG) and D(C∗(BG, k)). For G a connected compact Lie group, Benson and Greenlees [3] proved that D(C∗(BG, k)) is stratified by the canonical action of H∗(BG, k). In this talk, new examples of stratification results will be presented coming from homotopical generalizations of classifying spaces of compact Lie groups at a prime p called p-local compact groups, introduced by Broto-Levi-Oliver [6]. They generalize previous results showing the statement only depends on the p-local information (in a group theoretic sense) of G. For a commutative ring spectrum R we write ModRfor the category of Rmodules. For M, N ∈ ModRwe write M ⊗RN for the monoidal product of R, and HomR(M, N ) for the spectrum of R-module morphisms between M and N . Given a space X we write X+for the suspension spectrum Σ∞+X and C∗(X, R) for F (X+, R), the spectrum of R-valued cochains on X+. If R is a commutative ring spectrum, then so is C∗(X, R). Often R = Hk will be the Eilenberg–MacLane spectrum of a discrete commutative ring k; we simply write C∗(X, k). A subcategory T ⊆ C of a stable ∞-category is called thick if it is closed under finite colimits, retracts, and desuspensions, and T is called localizing (respectively colocalizing) if it is closed under all filtered colimits (respectively all filtered limits). Definition 1. A commutative ring spectrum R is called Noetherian if π∗R is Noetherian. A morphism f : R → S of commutative ring spectra induces a triple of adjoints (Ind, Res, Coind) between ModRand ModS, where Ind : ModR→ ModSis induction − ⊗RS, Res: ModS→ ModRis restriction along f , and Coind : ModR→ ModSis coinduction, given by HomR(S, −). We denote by res the induced morphismbetweenthehomogeneousprimeidealspectra Spech(π∗(S)) → Spech(π∗(R)). A functor F : C → D is said to be conservative if it reflects equivalences. Definition 2. A morphism of Noetherian commutative ring spectra f : R → S is said to satisfy Quillen lifting if for any two modules M, N ∈ ModRsuch that there is p ∈ res suppS(Ind M ) ∩ res cosuppS(Coind N ), there exists a homogeneous prime ideal q ∈ res−1(p) with q ∈ suppS(Ind M ) ∩ cosuppS(Coind N ). The motivating example of a morphism of ring spectra satisfying Quillen lifting is the following one. 522Oberwolfach Report 9/2018 Example 3. Let G be a compact Lie group, k a field of characteristic p, and let E(G) be a set of representatives of conjugacy classes of elementary abelian p-subgroups of G. It is a consequence of the strong form of Quillen stratification for group cohomology that the following morphism satisfies Quillen lifting, Y C∗(BG, k) →C∗(BE, k). E∈E(G) We isolate sufficient conditions for descent of stratification and costratification along a morphism f : R → S in the next theorem. Theorem 4. Suppose that f : R → S is a morphism of Noetherian ring spectra satisfying Quillen lifting and such that induction and coinduction along f are conservative. If ModSis canonically stratified, then so is ModR. If f additionally admits an R-module retract, then canonical costratification descends along f as well. In [6], Broto, Levi, and Oliver introduced the powerful concept of p-local compact groups as a common generalization of the notions of p-compact group [7] as well as fusion systems F on a finite group [5]. A p-local compact group G = (S, F ) consists of a saturated fusion system on a discrete p-toral group S. This definition provides a combinatorial model of the p-local structure of a compact Lie group (S, F ). In order to recover the p-completion of the classifying space, extra structure is needed. But, the latter is uniquely determined (see [8]) which makes it possible to construct a (p-completed) classifying space BG associated to G, thus making saturated fusion systems amenable to homotopical techniques. Broto, Levi and Oliver provide examples given by compact Lie groups with no restriction on the group of components as well as p-completions of finite loop spaces. Checking that the conditions of Theorem 4 are satisfied for the morphism φG: C∗(BG, Fp) → C∗(BS, Fp) crucially relies on the construction of a transfer morphism to prove that Ind and Coind are conservative functors. Theorem 5. Any p-local compact group G = (S, F ) admits a stable transfer C∗(BS, Fp) → C∗(BG, Fp) of C∗(BG, Fp)-modules. One consequence of this theorem is that the cohomology ring H∗(BG, Fp) is Noetherian for any p-local compact group, extending the classical result for finite groups and compact Lie groups, and for p-compact groups (see [7]). We then use Rector’s general formalism [9] to generalize the F -isomorphism to p-local compact groups. This allows us to deduce a strong form of Quillen stratification from the F -isomorphism theorem, following Quillen’s original argument. Theorem 6. For any p-local compact group G, there is an F -isomorphism H∗(BG, Fp) → limH∗(BE, F ←−p), Fe Mini-Workshop: Chromatic Phenomena and Duality523 where Feis the full subcategory of F on the elementary abelian subgroups of S. Moreover, the variety of G admits a strong form of Quillen stratification: VG∼=aVE,G+, E∈E(G) where E(G) denotes a set of representatives of F -isomorphism classes of elementary abelian subgroups of S. Our main result is then a combination of the previous three theorems. Theorem 7. If G is a p-local compact group, then ModC∗(BG,Fp)is canonically stratified and costratified. In particular, there are bijections  of ModC∗(BG,Fp)↔SpecSubsets ofh(H∗(BG, Fp))↔∼Colocalizing subcat.of ModC∗(BG,Fp) as well as  of ModcompactC∗(BG,F)↔Specialization closed subsets ofSpech(H∗(BG, Fp)). p Finally, Benson and Greenlees [2] show that C∗(BG, Fp) is an absolute Gorenstein ring spectrum for any finite group G. Using methods from [1], we extend this result to p-compact groups. As an immediate consequence this implies the existence of a local cohomology spectral sequence for p-compact groups. Theorem 8. Let G be a p-compact group of dimension w, then G is absolute Gorenstein, i.e., for each p ∈ Spech(H∗(BG, Fp)) of dimension d, the local cohomology at p is given by Hp∗C∗(BG, Fp) ∼= Ip[w + d], where Ipdenotes the injective hull Ipof (H∗(BG, Fp))/p. References [25] T. Barthel, D. Heard, G. Valenzuela,Local duality for structured ring spectra, Journal of Pure and Applied Algebra 222 2018, 433–463. [26] D. Benson, J. Greenlees, Localization and duality in topology and modular representation theory, J. Pure Appl. Algebra 212 (2008), 1716–1743. [27] D. Benson, J. Greenlees, Stratifying the derived category of cochains on BG for G a compact Lie group, J. Pure Appl. Algebra 218 (2014), 642–650. [28] D. Benson, S. Iyengar, H. Krause, Stratifying triangulated categories, J. Topol. 4 (2011), 641–666. [29] C. Broto, R. Levi, B. Oliver,The homotopy theory of fusion systems, J. Amer. Math. Soc. 16(2003), 779–856. [30] C. Broto, R. Levi, B. Oliver, Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups, Geom. Topol. 11 (2007), 315–427. [31] W.G. Dwyer, C. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Annals of Math. Second Series 139 (1994), 395–442. [32] R. Levi, A. Libman, Existence and uniqueness of classifying spaces for fusion systems over discrete p-toral groups, L. London. Math. Soc. 91 (2015), 47–70 [33] D.L. Rector, Noetherian cohomology rings and finite loop spaces with torsion, J. Pure Appl. Algebra 32 (1984), 191–217 524Oberwolfach Report 9/2018 The classification of thick tensor ideals in genuine A-spectra Justin Noel (joint work with Tobias Barthel, Markus Hausmann, Niko Naumann, Thomas Nikolaus, Nat Stapleton) In this joint project [BHNNNS], we classify the tt-ideals in genuine A-spectra [LMS86] when A is a finite abelian group. The original result of this type was the case when A is the trivial group, which is the celebrated thick subcategory theorem of Hopkins and Smith [HS98] (see also [Hop87, Rav92]). The primary prerequisites for the new classification result are: (1) The general classification of tt-ideals in a rigid tensor triangulated category C, in terms of the Thomason subsets of the Balmer spectrum Spc(C) of prime tensor ideals [Bal05, Bal10b]. (2) The identification of the underlying set of the Balmer spectrum Spc(SHG) for genuine G-spectra from [BS17]. To complete the classification of the tt-ideals for (compact) genuine G-spectra, it suffices to identify the standard basic opens of Spc(SHG). Namely, the basic opens can be enumerated by the compact G-spectra. For such a G-spectrum X, the corresponding open is identified by determining, for each prime p and each conjugacy class H ⊆ G of subgroup, the smallest n ≥ 0, such that K(n)∗(ΦHX) 6= 0 (if no such finite n exists, we will let this value be ∞). This last number, which we will denote tp(ΦHX), is called the type of ΦHX. Balmer and Sanders show that one can always reduce to the case when G is a p-group. Moreover, when G is an abelian p-group, which we will henceforth assume, they show that it suffices to just determine how the type of X can vary from the type of ΦGX. So we first establish an inequality which bounds how the type can vary. To show the inequality is sharp we need to find suitable examples. In more detail, we show that tp(ΦGX) + dim(H1(BG; Fp)) ≥ tp(X). This ends up being a consequence of a generalization of Kuhn’s blue shift theorem for Tate cohomology [Kuh04]. This generalization identifies the acyclics for the geometric fixed points of Borel equivariant Lubin-Tate theories. This refines the results of [MNN15], which showed that if the group was not an abelian p-group generated by n or fewer elements, then the geometric fixed points of a height n Borel equivariant Lubin-Tate theory E would be contractible. This new generalization builds on the moduli-theoretic description of E0(BG) from [HKR00]. To show that this inequality is sharp and complete the classification, we need to find an X for which the inequality is an equality. We observe that such complexes have already been constructed in the work of [Aro98, ADL16, AL17]. References [ADL16]Gregory Arone, William G. Dwyer, and Kathryn Lesh. Bredon homology of partition complexes. Doc. Math., 21:1227–1268, 2016. Mini-Workshop: Chromatic Phenomena and Duality525 [AL17]Gregory Arone and Kathryn Lesh. Fixed points of coisotropic subgroups of Γkon decomposition spaces. 2017. Available at http://front.math.ucdavis.edu/1701.06070. [Aro98]Gregory Arone. Iterates of the suspension map and Mitchell’s finite spectra with Ak-free cohomology. Math. Res. Lett., 5(4):485–496, 1998. [Bal05]Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005. [Bal10b]Paul Balmer. Tensor triangular geometry. In International Congress of Mathemati- cians, Hyderabad (2010), Vol. II, pages 85–112. Hindustan Book Agency, 2010. [BHNNNS] Tobias Barthel, Markus Hausmann, Niko Naumann, Thomas Nikolaus, Justin Noel, and Nathaniel Stapleton. The Balmer spectrum of the equivariant homotopy category of a finite abelian group. Preprint arXiv:1709.04828v2, 2017. [BS17]Paul Balmer and Beren Sanders. The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math., 208(1):283–326, 2017. [Hop87]Michael J. Hopkins. Global methods in homotopy theory. In Homotopy Theory-Proc. Durham Symposium 1985. Cambridge University Pres, 1987. [HKR00]Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc., 13(3):553–594, 2000. [HS98]Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998. [Kuh04]Nicholas J. Kuhn. Tate cohomology and periodic localization of polynomial functors. Invent. Math., 157(2):345–370, 2004. [LMS86]L. Gaunce Lewis, J. Peter May, and Mark Steinberger. Equivariant Stable Homotopy Theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, 1986. [MNN15]Akhil Mathew, Niko Naumann, and Justin Noel. Derived induction and restriction theory. 2015. Available at http://front.math.ucdavis.edu/1507.06867. [May96]J. Peter May. Equivariant homotopy and cohomology theory, volume 91 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. [Rav92]Douglas C. Ravenel. Nilpotence and periodicity in stable homotopy theory, volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith. Endotrivial modules via homotopy theory Jesper Grodal For G a finite group and k a field of characteristic p, an endotrivial module is a kG-module M such that M ⊗ M∗∼= k ⊕ (proj) as kG-modules. Isomorphism classes of indecomposable endotrivial modules form a group Tk(G), and identify with the Picard group of the stable module category Pic(stmod(kG)). They occur in many parts of representation theory as “almost 1-dimensional modules”. I started my talk by briefly explaining the classification of these modules for S a finite p-group, due to seminal work of Dade [Dad78a, Dad78b], Alperin [Alp01] and Carlson–Thevenaz [CT04, CT05]. I then went on to explain how to calculate this group for G an arbitrary finite group, based on my preprint [Gro16]. The image of the restriction map Tk(G) → Tk(S), for S a p-Sylow subgroup of G, is known, at least as an abstract group, by an elaboration of the above-mentioned calculation of Tk(S). So the main question in describing Tk(G) for an arbitrary 526Oberwolfach Report 9/2018 finite group G lies in understanding the kernel Tk(G, S) = ker(Tk(G) → Tk(S)). This subgroup consists of finitely generated kG–modules M , whose restriction to S have the form M |S∼= k ⊕ (free), i.e., “Sylow-trivial” modules. The following theorem describes this group: Theorem 1. [Gro16, Thm. A] Fix a finite group G and k a field of characteristic p dividing the order of G, and let Op∗(G) denote the orbit category on non-trivial p–subgroups. The group Tk(G, S) is described via the following isomorphism of abelian groups Φ : Tk(G, S)−∼→ H1(Op∗(G); k×). The inverse map, which to a 1–cocycle constructs an endotrivial module, is very explicit, in terms of the so-called twisted Steinberg complex, and can also be viewed as a “derived induction” map. I then went on to describe a number of explicit results about Tk(G, S) that can be obtained with Theorem 1 as starting point, also taken from [Gro16]; they transform the calculation Tk(G, S) to standard calculations in local group theory: I presented a positive solution to the so-called Carlson–Thevenaz conjecture, providing an explicit algorithm for computing Tk(G, S) purely in terms of normalizers of p–subgroups and their intersections, that can easily be put on a computer. I also deduced other consequences such as that if the p–subgroup complex Sp(G) is simply connected, then Tk(G, S) equals the one-dimensional characters of G, providing vanishing results for many classes of groups. The Carlson–Thevenaz conjecture comes out as a special case of more general “centralizer” and “normalizer” decompositions for Tk(G, S), that express Tk(G, S) in terms of p–local group theory packaged in different ways. E.g., the “centralizer decomposition” breaks Tk(G, S) up in two parts, one only depending on the p–fusion in G, and one depending on the 1–dimensional characters on the centralizers of elementary abelian p–subgroups of rank one and two. Finally, I described a number of explicit computations, both showing how existing results in the literature can be easily recovered by these methods, and computing Tk(G, S) for a range of new groups, e.g, as a test case, the Monster sporadic simple group for all primes p. References [Alp01]J. L. Alperin. A construction of endo-permutation modules. J. Group Theory, 4(1):3–10, 2001. [CT04]J. F. Carlson and J. Th´evenaz. The classification of endo-trivial modules. Invent. Math., 158(2):389–411, 2004. [CT05]J. F. Carlson and J. Th´evenaz. The classification of torsion endo-trivial modules. Ann. of Math. (2), 162(2):823–883, 2005. [Dad78a] E. C. Dade. Endo-permutation modules over p-groups. I. Ann. of Math. (2), 107(3):459–494, 1978. [Dad78b] E. C. Dade. Endo-permutation modules over p-groups. II. Ann. of Math. (2), 108(2):317–346, 1978. [Gro16] J. Grodal. Endotrivial modules for finite groups via homotopy theory. arXiv:1608.00499. Mini-Workshop: Chromatic Phenomena and Duality527 Homological residue fields Paul Balmer In Prismatic Algebra, one encounters a broad variety of tensor-triangulated categories T . The chromatic analysis of their compact-rigid objects Tcamounts to the determination of their triangular spectrum Spc(Tc), whose points are triangular primes P ⊂ Tc. We presented another approach to primes, by means of maximal Serre ⊗-ideal subcategories B of the module category mod − Tc(the Freyd envelope of Tc). We proved with Krause and Stevenson [2] that every triangular prime P is the preimage under Yoneda h : Tc→ mod − Tc of one of those new homological primes B. It is an open question whether this B is unique. Remarkably, one can prove that B is unique (for a given P ) in the standard examples from stable homoopy theory (including the equivariant versions), algebraic geometry (without noetherianity assumption), modular representation theory (including finite group schemes), etc. However, the proof is specific to each example and we do not know an abstract proof. The associated homological spectrum Spch(T ) consisting of all homological primes (all maximal Serre ⊗-ideals) of mod − Tc, can be used to define supports for big objects, unconditionally (i.e. without supposing noetherianity of the triangular spectrum Spc(Tc)) as was done in the joint work with Favi [1]. To do this, one considers the big category M od − Tcof modules over Tc, of which the above mod − Tcis the finitely presented part. There is a restricted Yoneda functor h : T → M od − Tc which is not faithful or full anymore (it kills phantom maps) but which remains a ⊗-functor. For every (big) object X ∈ T , one can then define its support Supph(X) ⊆ Spch(T ) as the set of homological primes B such that X does not vanish in the Gabriel quotient of M od − Tcby hBi. This ‘big support’ lacks the fundamental properties needed of a good theory of support, notably the general tensor formula, but it is a good theory for ring objects (even for weak ring objects: objects equipped with a possibly non-associative, non-commutative multiplication admitting a one-sided unit). It is work-in-progress to develop the properties of this big support and its relevance for the telescope property, i.e. for understanding the smashing subcategories of T . References [34] Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3), 102(6):1161–1185, 2011. [35] Paul Balmer, Henning Krause, and Greg Stevenson. Tensor-triangular fields: Rumina- tions. preprint, arXiv:1707.02167, 2017. Reporter: Drew K. Heard
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