A(2) Modules and their Cohomology

Transcription

1 A(2) Modules and their Cohomology Robert Bruner Department of Mathematics Wayne State University and University of Oslo Topology Symposium University of Bergen June 1, 2017 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 1 / 52

2 Outline 1 Introduction 2 tmf and theories lying under it 3 E(0) 4 A(1) and E(1) 5 E(2) 6 The stable module category 7 Localization and periodicity 8 Calculating cohomology 9 Summary Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 2 / 52

3 Introduction Joint work with John Rognes on THH(tmf ) and its circle action. Goal: Study the v 3 -periodic homotopy in S K(tmf ) THH(tmf ) ts1 p = 2 and H = HF 2 in this talk. Tool: Adams spectral sequence Ext A (H (tmf X ), F 2 ) = tmf (X ) for relevant compexes X. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 4 / 52

4 tmf and theories lying under it tmf and A(2) Since H tmf = A /A(2) = A A(2) F 2, the Adams spectral sequence takes the simpler form Ext A (H (tmf X ), F 2 ) = tmf (X ) Ext A(2) (H (X ), F 2 ) = tmf (X ) Sub-Hopf algebras of the Steenrod algebra like A(2) are classified by their profile functions: let (r 1, r 2,..., r k ) denote the sub-hopf algebra of A dual to F 2 [ξ 1,..., ξ k ]/(ξ 2r 1 1,..., ξ 2r k k ). A(2) = (3, 2, 1) is generated by Sq 1, Sq 2, Sq 4. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 6 / 52

5 tmf and theories lying under it Sub-Hopf algebras of A(2) (3, 2, 1) (2, 2, 1) (2, 1, 1) (1, 2, 1) A(2) (2, 2, 1) (2, 1, 1) (1, 2, 1) tmf tmf Cν (2, 1, 1) (1, 2, 1) (2, 1) (1, 1, 1) A(1) E(2) ko BP 2 = tmf 1 (3) (1, 1) E(1) ku (1) E(0) HZ Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 7 / 52

6 E(0) E(0) E(0) = E[Q 0 ] has finite representation type. There are only two indecomposable E(0)-modules: the simple module F 2, and the free module E(0). All other E(0)-modules are direct sums of these. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 9 / 52

7 A(1) and E(1) E(1) E(1) = E[Q 0, Q 1 ] and A(1) = Sq 1, Sq 2 have tame representation type: their indecomposable modules of finite-type fall into a small number of families of simply parameterized modules. Over E(1), the indecomposable modules of finite type are E(1) and the lightning flashes, parameterized by substrings of the bi-infinite string Q 1 Q0 1 Q 1Q0 1 For example, H RP 4 corresponds to the string Q 1 0 Q 1Q 1 0 : x 2 x 4 x x 3 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 11 / 52

8 A(1) and E(1) A(1) Over A(1) the classification has been carried out for ungraded modules over GF (4) by William Crawley-Boevey. To describe the modules topologists care about: Discard those which cannot be graded appropriately. Do the Galois descent from GF (4) to GF (2). Crawley-Boevey s classification uses admissable words in the alphabet where c i = c 1 i and b 1 = Sq 1, a1 = Sq 2, a2 = Sq 2 Sq 1 Sq 2, and a0 = Sq 2 Sq 2 = Sq 1 Sq 2 Sq 1. {a 2, a 1, a 0, a 1, a 2 } {b 1, b 1 }, Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 12 / 52

9 E(2) E(2) and larger algebras Unfortunately, E(2) has wild representation type: classifying even the finite modules over it would require solving the (unsolvable) word problem. This then also applies to all the algebras containing E(2). However, we are not interested in all modules, but only in certain special ones. For these, some progress may be possible. For example, see Benson, Representations of Elementary Abelian p-groups and Vector Bundles. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 14 / 52

10 The stable module category A finite dimensional cocommutative Hopf algebra B is a Frobenius algebra, so that projective free injective. We then define the stable module category St(B-Mod): objects are B-modules, and morphisms are stable equivalence classes of homomorphisms [M, N] = Hom B (M, N)/ where f g iff f g factors through a free module. Then Every B-module can be written (non-uniquely) as F M, where F is free and M is reduced (has no free sub-modules or quotient modules ). M is stably isomorphic to N iff M is isomorphic to N. For s > 0, Ext s B (M, N) depends only on M and N St(B-Mod) is tensor triangulated with a triangle ΩZ X Y Z for each exact sequence 0 X Y Z 0 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 16 / 52

11 Localization and periodicity Margolis homology In the Steenrod algebra we have elements P s t = (ξ 2s t ) for s < t which satisfy (P s t ) 2 = 0. We may then define the Margolis homology H(, P s t ) = ker(ps t ) im(p s t ) For a sub-hopf algebra B of the Steenrod algebra we have Theorem (Adams-Margolis) In B-Mod, f : M N is a stable isomorphism H(f, P s t ) is an isomorphism for each P s t B. The P s t s in B are totally ordered by degree. Let I B be the (ordered) set of P s t B. Definition For any subset J I B, we say that M B Mod is J-local if H(M, P s t ) = 0 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 18 / 52

12 Localization and periodicity Theorem (Margolis) If I B = J 0 J1, with J 0 an initial seqment and J 1 a final segment of I B, then there exists a unique triangle in which L J1 (M) L Ji (M) is J i -local, i = 0, 1 ɛ M ι L J0 (M) ɛ is an isomorphism in P s t homology for P s t J 1 ι is an isomorphism in P s t homology for P s t J 0 The functors L Ji are called Margolis localizations, though L J0 is really a colocalization. By composition, we can define L J (M) for any subinterval J of I B. For E(1) or A(1) modules, this reduces to a single exact triangle L 1 (M) ɛ M ι L 0 (M) in which L i (M) is Q i -local. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 19 / 52

13 Localization and periodicity v 0 periodicity Let B be any sub-hopf algebra of A containing Q 0. Theorem If M is a Q 0 -local B-module, then ΩM ΣM. Proof. Tensor M with the short exact sequence 0 F 2 E(0) ΣF 2 0 where E(0) has its unique B-module structure. Since M has only Q 0 homology and E(0) has no Q 0 homology, M E(0) is free. Remark If M is such a module, it follows that Ext B (M, F 2 ) = F 2 [h 0 ] Ext 0 B (M, F 2). Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 20 / 52

14 Localization and periodicity v 1 periodicity Theorem If M is a Q 1 -local E(1)-module then ΩM Σ 3 M. Proof. The same argument works, using 0 F 2 E[Q 1 ] Σ 3 F 2 0. Remark If M is Q 1 -local and reduced then Ext E(1) (M, F 2 ) = F 2 [v 1 ] Ext 0. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 21 / 52

15 Localization and periodicity v 4 1 periodicity Theorem If M is a Q 1 -local A(1)-module then Ω 4 M Σ 12 M. Proof. Use the exact sequence 0 F 2 A(1) /A(0) Σ 2 A(1) Σ 4 A(1) Σ 7 A(1) /A(0) Σ 12 F 2 0 as above, noting that both A(1) and A(1) /A(0) are Q 1 -acyclic A(1)-modules, so that they become free after tensoring them with a Q 1 -local module. Remark This arises from the first four stages of the Postnikov tower of ko. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 22 / 52

16 Localization and periodicity v 4 1 periodicity Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 23 / 52

17 Localization and periodicity ko RP 4 = Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 24 / 52

18 Localization and periodicity For general M, the sequence tensored with M and spliced gives a 4-periodic complex from which we get a spectral sequence converging to Ext A(1) (M, F 2 ). At E 1, the terms s 1, 2 (mod 4) are suspensions of M concentrated in homological degree s, while the terms s 0, 3 (mod 4) are Ext s 1 A(0) (M, F 2), in homological degrees s. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 25 / 52

19 Localization and periodicity v 2 periodicity Theorem If M is a Q 2 local E(2)-module then ΩM Σ 7 M. Proof. This is a general fact. If A = B E[Q] and M is an A-module which is free over B, then tensoring M with shows that ΩM Σ Q M. 0 F 2 E[Q] Σ Q F 2 0 Remark In this context, it is easy to construct a Q-localization M L Q M, a Q-colocalization C Q M, and a Tate module T Q M which sits in a triangle C Q T Q M L Q M. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 26 / 52

20 Localization and periodicity A(2) In A(2) the homologies which determine stable isomorphism are [q 0, q 1, q 2, q 3 ] = [Q 0 < Q 1 < P 1 2 < Q 2 ]. The Margolis localizations and colocalizations they determine are M M[0 3] M[0 2] M[0 1] M[0] M[1 3] M[1 2] M[1] M[2 3] M[2] M[3] Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 27 / 52

21 Localization and periodicity v 8 2 periodicity Theorem If M is a Q 2 -local A(2)-module then Ω 8 M Σ 56 M. Proof. There is a complex 0 F 2 P 0 P 1 P 2 P 3 in which each of the P i is Q 2 -acyclic. P 4 P 5 P 6 P 7 Σ 56 F 2 0 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 28 / 52

22 Localization and periodicity Let us write A(2) as A for brevity. P 0 = A/(Sq 1, Sq 2 ) so that Ext A (P 0, F 2 ) = Ext A(1) (F 2, F 2 ) ko P 1 = Σ 4 A/(Sq 1, Sq 2 Sq 3 ) giving P 2 = Σ 8 A/(Sq 1 ) giving Ext A (P 1, F 2 ) = Ext A(1) (Σ 4 F 2, F 2 ) Σ 4 ksp Ext A (P 1, F 2 ) = Ext A(0) (Σ 8 F 2, F 2 ) Σ 8 HZ P 3 = Σ 15 A Σ 18 A/(Sq 1, 0), (Sq 3, 0), (Sq 4, Sq 1 ), (Sq 4 Sq 2, Sq 3 ) so that 0 Σ 18 A /E[Q 0, Q 1 ] P 3 Σ 15 A /E[Q 0, Q 1 ] 0 Thus, Ext A (P 3, F 2 ) can be calculated as an extension of two copies of the Adams spectral sequence converging to ku. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 29 / 52

23 Localization and periodicity P 4 = (Σ 22 A Σ 24 A) / (Sq 1, 0), (Q 1, Sq 1 ), (0, Q 1 ), giving the first Adams cover of ku. P 5 = Σ 26 A/(Sq 1 ) Σ 24 A/(Sq 2 ), giving an HZ and the first Postnikov cover of ko. P 6 = Σ 33 A/(Sq 1, Q 1 ) Σ 36 A/(Sq (0,2) ), giving a ku and the bottom edge of a wedge: Ext of the second summand is F 2 [ 4 g], polynomial on a class in (t s, s) = (5, 1) whose fourth power is g. P 7 = (Σ 39 A Σ 39 A) / (Sq 1, Sq 1 ), (Q 1, 0), (Sq (0,2), 0), (0, Sq 2 ), 0 Σ 39 A /A(1) P 7 Σ 39 A /E[Sq (0,1), Sq (0,2) ] 0 Thus, Ext A (P 7, F 2 ) can be calculated as an extension of the Adams spectral sequence converging to ko and the wedge, which has Ext = F 2 [v 1, 4 g]. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 30 / 52

24 Localization and periodicity Advertisement - FPMods A sage package for doing calculations with finitely presented A-modules. Written by Mike Catanzaro as his master s thesis. Allowed to become moribund, as new versions of sage broke some things. Recently resuscitated by Sverre Lunøe-Nielsen. Here is the sage code that verifies my claim about P 7 : sage: A2 = SteenrodAlgebra(p=2,profile=(3,2,1)) sage: load( fpmods.py ) sage: P7 = FP_Module([0,0], [[Sq(1),Sq(1)],[Sq(0,1),0],[Sq(0,2),0],[0,Sq(2)]] algebra=a2) sage: ko = FP_Module([0],[[Sq(1)],[Sq(2)]],algebra=A2) sage: p = FP_Hom(P7,ko,[[0],[1]]) sage: K,j = p.kernel() sage: K.degs Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 31 / 52

25 Calculating cohomology Calculating Ext A(2) Figure: P 0 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 33 / 52

26 Calculating cohomology Calculating Ext A(2) Figure: P 0, P 1 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 34 / 52

27 Calculating cohomology Calculating Ext A(2) d Figure: P 1 0 P1 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 35 / 52

28 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 0 P1 ) Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 36 / 52

29 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 0 P1 ), P 2 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 37 / 52

30 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 0 P1 ) d1 P 2 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 38 / 52

31 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 d 0 1 P1 P2 ) Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 39 / 52

32 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 d 0 1 P1 P2 ), P 3 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 40 / 52

33 Calculating cohomology Calculating Ext A(2) d Figure: H(P 1 d 0 1 P1 P2 ) d2 P 3 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 41 / 52

34 Calculating cohomology Calculating Ext A(2) d Figure: E 3 (P 1 d 0 1 d P1 2 P2 P3 ) Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 42 / 52

35 Calculating cohomology Calculating Ext A(2) d Figure: E 3 (P 1 d 0 1 d P1 2 P2 P3 ), P 4 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 43 / 52

36 Calculating cohomology Ext A(2) Figure 1: Ext A(2), 0 n 52, 0 s 30 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 44 / 52

37 Calculating cohomology Similar calculations There is a v 4 2 sequence for H (C(ν)) giving a simple calculation of Ext A(2) (H Cν, F 2 ). v2 8 sequences with small, well understood terms, exist for the cofibers of 2 and η, as well as other complexes useful in computing THH (tmf ). Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 45 / 52

38 Summary Ext A(2) again Theorem (Iwai and Shimada) The cohomology of A(2) is Ext A(2) (F 2, F 2 ) = F 2 [h 0, h 1, h 2, c 0, d 0, e 0, g, α, β, γ, δ, w 1, w 2 ]/I. The ideal I has 56 generators: h 0 h 1, h 1 h 2, h0 2h 2 h1 3, h 0h2 2, h c 0 γ h 1 δ, βγ g 2, d0 2 gw 1, γδ h 1 c 0 w 2, γ 2 h1 2w 2 gβ 2, α 4 h0 4w 2 w 1 g 2 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 47 / 52

39 Summary Free over F 2 [w 1, w 2 ]; here w 1 and w 2 restrict to v 4 1 and v 8 2, resp. A sum of cyclic R = F 2 [g, w 1, w 2 ]-modules isomorphic to R, R/(g) and R/(g 2 ). Four infinite families, h i 0 αj, i 0, 0 j 3. Thirty-two other summands. E 3, E 4 and E 5 = E are then modules over R 1 = F 2 [g, w 1, w2 2] and R 2 = F 2 [g, w 1, w2 4 ] resp. Mostly cyclic. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 48 / 52

40 Summary Summary of periodicities v 8 2, g tmf v 4 2, 4 g tmf Cν v 2 v 4 1, v 1 ko (2, 1, 1) (1, 2, 1) BP 2 P 1 2 ku Q 2 h 0 HZ Q 1 Q 0 Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 49 / 52

41 Summary Figure 1: E, 0 t s < 192. w1-power torsion is shown in red, while black classes are free over F2[w1, w 4 2]. The w 4 2 multiples are not shown. E Figure: E, 0 t s < 192. w 1 -power torsion is shown in red, while black classes are free over F 2 [w 1, w2 4]. The w 2 4 multiples are not shown. Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 50 / 52

42 Summary Tusen takk Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 51 / 52

43 Summary Robert Bruner (WSU and UiO) A(2) Modules and their Cohomology Bergen 52 / 52