For the Ausoni-Rognes conjecture at n = 1, p > 3: a strongly convergent descent spectral sequence

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1 For the Ausoni-Rognes conjecture at n = 1, p > 3: a strongly convergent descent spectral sequence Daniel G. Davis University of Louisiana at Lafayette June 2nd, 2015

2 n 1 p, a prime E n is the Lubin-Tate spectrum, with π (E n ) = W (F p n) u 1,..., u n 1 [u ±1 ]. Here: W (F p n) is the ring of Witt vectors of the field F p n the complete power series ring is in degree zero u = 2 (this is the choice Ausoni makes in his Inventiones paper) G n = S n Gal(F p n/f p ), the extended Morava stabilizer group E n, a commutative S algebra G n acts on E n by maps of commutative S algebras. (The preceding two points are an application of the Goerss-Hopkins-Miller Theorem.)

3 ( A, a commutative S algebra ) ( K(A), the algebraic K theory spectrum of A, a commutative S algebra ) = K(E n ), a commutative S algebra By the functoriality of K( ), G n acts on K(E n ) by maps of commutative S algebras. L K(n) (S 0 ), the Bousfield localization of the sphere spectrum with respect to K(n), the nth Morava K theory spectrum. G n, a profinite group (more: a compact p-adic analytic group; finite v.c.d.) The K(n) local unit map of K(n)-local commut. S-algebras L K(n) (S 0 ) E n is a consistent profaithful K(n) local profinite G n Galois extension (due to Rognes, Behrens-D.).

4 V n, a finite p local complex of type n + 1 v : Σ d V n V n, a v n+1 self-map (d, some positive integer) Thus, v induces a sequence of maps of spectra. We set V n Σ d V n Σ 2d V n v 1 n+1 V n = colim j 0 Σ jd V n, the colimit of the above sequence, the mapping telescope associated to the v n+1 self-map v.

5 Conjecture (Ausoni, Rognes) The G n Galois extension L K(n) (S 0 ) E n induces a map K(L K(n) (S 0 )) v 1 n+1 V n (K(E n )) hgn v 1 n+1 V n that is a weak equivalence, and associated with the target of this weak equivalence is a homotopy fixed point spectral sequence that has the form E s,t 2 = (V n ) t s ((K(E n )) hgn )[v 1 n+1 ], with E s,t 2 = Hc s ( Gn ; (V n ) t (K(E n ))[vn+1 1 ]).

6 Remark We supplement the statement of the conjecture with the following comments: the E 2 -term E s,t 2 = Hc s ( Gn ; (V n ) t (K(E n ))[vn+1 1 ]) of its spectral sequence is given by continuous cohomology; its object (K(E n )) hgn is a continuous homotopy fixed point spectrum; and the conjecture is actually just a piece of the family of conjectures made by Ausoni and Rognes we only stated the part that we have been focusing on.

7 For every integer t, there is an isomorphism (V n ) t (K(E n ))[v 1 n+1 ] = π t (K(E n ) v 1 n+1 V n). = When the conjectured spectral sequence E s,t 2 = (V n ) t s ((K(E n )) hgn )[v 1 n+1 ], exists, since it is for homotopy fixed points, there should also be an equivalence (K(E n )) hgn v 1 n+1 V n (K(E n ) v 1 n+1 V n) hgn. Obtaining this equivalence and a homotopy fixed point spectral sequence E s,t 2 = H s c (G n ; π t (K(E n ) v 1 n+1 V n)) = π t s ( (K(En ) v 1 n+1 V n) hgn) immediately implies the existence of the conjectured spectral sequence.

8 To make progress on the conjecture, one obstacle that must be overcome is that there are no known constructions of the (continuous) homotopy fixed point spectra for any n and p. (K(E n )) hgn, (K(E n ) v 1 n+1 V n) hgn Also, there are no known constructions of the two homotopy fixed point spectral sequences that we have referred to.

9 In this talk, we are reporting on progress on this conjecture for n = 1, p 5, with V 1 = V (1), the type 2 Smith-Toda complex S 0 /(p, v 1 ). Thus, we have E 1 = KU p, p completed complex K theory, G 1 = Z p = Z p Z/(p 1)..B. Henceforth, we will use the term descent spectral sequence in place of homotopy fixed point spectral sequence.

10 Theorem 1 (D.) Let p 5. Given any closed subgroup K of Z p, there is a strongly convergent descent spectral sequence where E s,t 2 = π t s (( K(KUp ) v 1 2 V (1)) hk ), E s,t 2 = H s c (K; π t (K(KU p ) V (1))[v 1 2 ]), with E s,t 2 = 0, for all s 2 and any t Z. Also, there is an equivalence of spectra ( K(KUp ) v 1 2 V (1)) hk colim j 0 (K(KU p) Σ jd V (1)) hk.

11 Remarks about the theorem Each subgroup K is a profinite group. Both occurrences of ( ) hk in the theorem are for the application of the (continuous) K homotopy fixed points functor for the category of discrete K-spectra. The term spectrum means symmetric spectrum: the theorem, its proof, and the underlying theory are worked out in the setting of symmetric spectra of simplicial sets. Letting K = Z p gives some progress on the Ausoni-Rognes conjecture: for n = 1, p 5, and V 1 = V (1), we have obtained the desired descent spectral sequence H s c (G n ; π t (K(E n ) v 1 n+1 V n)) = π t s ( (K(En ) v 1 n+1 V n) hgn). The construction of K(KU p ) hz p remains open.

12 Theorem 2 (D.) For p 5, there is a canonical map of symmetric spectra η : K(L K(1) (S 0 )) v 1 2 V (1) ( K(KU p ) v 1 2 V (1)) hz p. Remark It is easy to see that if η is a weak equivalence and if (K(KU p ) v 1 2 V (1))hZ p K(KU p ) hz p v 1 2 V (1), then there would be an equivalence K(L K(1) (S 0 )) v 1 2 V (1) K(KU p) hz p v 1 2 V (1), which would be close to proving part of the Ausoni-Rognes conjecture in the n = 1, p 5 case.

13 For the next result... Let G be a profinite group, with = { α } α Λ an inverse system of open normal subgroups of G that satisfies (a) the maps in the diagram (indexed by the directed poset Λ) are given by inclusions (that is, α 1 α 2 in Λ if and only if α2 is a subgroup of α1 ), and (b) the intersection α Λ α is the trivial group {e}. Also, let X be a G spectrum such that the G module π t (X ) is a discrete G module, for every t Z.

14 Theorem (D.) Let G and X be as on the previous slide. Suppose that the map λ s π t(x ) : Hs c ( α ; π t (X )) H s ( α ; π t (X )) is an isomorphism for all s 0, every integer t, and each α Λ. If there exists a natural number r, such that for all integers t and every α Λ, H s c ( α ; π t (X )) = 0, for all s > r; or there exists some fixed integer l, such that π t (X ) = 0, for all t > l, then there is a zigzag of G equivariant maps X holim Sets(G +1, X f ) X dis that are weak equivalences in Sp Σ, with X dis ΣSp G.

15 For Theorem 1, we didn t need the full power of the preceding theorem; we only needed a simpler version...

16 Definition A spectrum X is an f spectrum if π (X ) is degreewise finite. Theorem (D.) Let p be any prime and let H be any finite discrete group. If X is a (Z p H) spectrum and an f spectrum, then there is a zigzag X X X dis of (Z p H) spectra and (Z p H) equivariant maps that are weak equivalences of symmetric spectra, and X dis is a discrete (Z p H) spectrum. X = X dis = X h(zp H) := (X dis ) h(zp H)

17 The construction of X dis is equal to colim m 0 holim [n] is elementary!: X dis ( Sets(Zp H,, Sets(Z p H, }{{} (n+1) times ) (p X f ) ) m Z p) {e}, }{{} (n+1) times where each (p m Z p ) {e} is an (open normal) subgroup of Z p H and p m Z p has its usual meaning.

18 ow we build on these tools for the situation of filtered colimits.

19 Let G be any profinite group and X any G spectrum. Definition If G, X, and (an inverse system of open normal subgroups of G) satisfy the hypotheses of either of the last two theorems, then we say that the triple (G, X, ) is suitably finite. Definition Let G be a profinite group with a fixed inverse system of open normal subgroups of G, and let {X } be a filtered diagram of G spectra such that for each, (G, X, ) is a suitably finite triple and X is a fibrant spectrum. We refer to (G, {X }, ) as a suitably filtered triple.

20 Let (G, {X }, ) be a suitably filtered triple. There is a zigzag of G equivariant maps colim X colim holim Sets(G +1, (X ) f ) colim(x ) dis that are weak equivalences in ΣSp. The composition colim π t (X ) = πt (colim (X ) dis ) = colim π t ((X ) dis ) consists of two isomorphisms in the category of discrete G modules (in particular, each of the above abelian groups is a discrete G module).

21 Definition Given a suitably filtered triple (G, {X }, ), we have seen that the G spectrum colim X can be identified with the discrete G spectrum colim (X ) dis. Thus, it is natural to define (colim X ) hg = ( colim(x ) dis ) hg. We can extend this definition to an arbitrary closed subgroup K in G: since the K spectrum colim X can be regarded as the discrete K spectrum colim (X ) dis, we define (colim X ) hk = ( colim(x ) dis ) hk.

22 We say that a profinite group G has finite virtual cohomological dimension ( finite v.c.d. ) if G contains an open subgroup that has finite c.d. Theorem (D.) Let G be a profinite group with finite v.c.d. If (G, {X }, ) is a suitably filtered triple and K is a closed subgroup of G, then there is a conditionally convergent descent spectral sequence Er, (K) that has the form E s,t 2 (K) = Hs c (K; π t (colim X )) = π t s ( (colim X ) hk ).

23 An almost complete sketch of the proof of this theorem... Let U be an open subgroup of G that has finite c.d. Then U K is an open subgroup of K, and since U has finite c.d. and U K is closed in U, there exists some r such that for any discrete (U K) module M, Hc s (U K; M) = Hc s (U; Coind U U K (M)) = 0, whenever s > r, by Shapiro s Lemma. This shows that K has finite v.c.d. Then, as a special case of a result due to [Behrens-D., D.], we obtain the conditionally convergent spectral sequence ( (colim E s,t 2 = Hc s (K; π t (colim(x ) dis )) = π t s (X ) dis ) ) hk, and this is the desired spectral sequence.

24 A little more useful detail... Since K has finite v.c.d., ( colim (X ) dis ) hk holim Γ K colim (X ) dis, and for each m 0, the m-cosimplices of the cosimplicial spectrum Γ K colim (X ) dis satisfy the isomorphism ( Γ K colim(x ) dis ) m = colim V ok m K m /V colim (X ) dis, where K m is the m-fold Cartesian product of K (K 0 is the trivial group {e}, equipped with the discrete topology).

25 Our spectral sequence is the homotopy spectral sequence for the spectrum holim Γ K colim(x ) dis. Based on [Behrens-D., D.], one might expect us to instead form the homotopy spectral sequence for holim Γ ( K colim (X ) dis ) fk. But since each (X ) dis is a fibrant spectrum, colim (X ) dis is already a fibrant spectrum, so that we do not need to apply ( ) fk to it (so that we are taking the homotopy limit of a cosimplicial fibrant spectrum). This completes our sketch-proof.

26 Theorem (D.) Let G be a profinite group with finite v.c.d., let (G, {X }, ) be a suitably filtered triple such that {} is a directed poset, and let K be a closed subgroup of G. If there exists a nonnegative integer r such that for all t Z and each, Hc s (K; π t (X )) = 0 whenever s > r, then descent spectral sequence Er, (K) is strongly convergent and there is an equivalence of spectra (colim X ) hk colim (X ) hk.

27 We give the complete proof of this result... For all t Z, when s > r, we have E s,t 2 (K) = Hs c (K; π t (colim X )) = colim H s c (K; π t (X )) = 0, so that the spectral sequence is strongly convergent, by [Thomason s Lemma 5.48].

28 If V is an open normal subgroup of K m, where m 0, then K m /V is finite, and hence, an earlier isomorphism implies that ( Γ K colim(x ) dis ) m = colim colim V ok m K m /V (X ) dis ( = colim Γ K (X ) dis ) m, so that there is an isomorphism of cosimplicial spectra. Γ K colim (X ) dis = colim Γ K (X ) dis

29 Therefore, we have ( colim (X ) dis which gives ) hk holim Γ K colim (X ) dis (colim X ) hk holim colim Γ K (X ) dis colim = holim colim Γ K (X ) dis, holim ( colim (X ) dis = colim (X ) hk, Γ K (X ) dis and the canonical colim/holim exchange map above is a weak equivalence if there exists a nonnegative integer r such that for every t and all, H s[ π t ( Γ K (X ) dis )] = 0, when s > r. ) hk

30 The proof is completed by noting that there are isomorphisms H s[ ( π t Γ K (X ) dis )] = H s c (K; π t ((X ) dis )) = Hc s (K; π t (X )), for all s 0.

31 Ausoni showed that K(ku p ) V (1) is an f spectrum. = K(KU p ) V (1) is an f spectrum. Then our tools give K(KU p ) v 1 2 (( V (1) = colim K(KUp ) Σ jd V (1) ) ) dis j 0 f ΣSp Z p and ( K(KUp ) v2 1 V (1)) hk (( = (colim K(KUp ) Σ jd V (1) ) ) ) dis hk j 0 f.

32 We have not constructed (K(KU p )) hz p for any p. evertheless, we have a draft result related to this conjectural object... A Recollection If G is any profinite group and X is a (naive) G spectrum, then G can be regarded as a discrete group and one can always form the discrete homotopy fixed point spectrum Draft theorem X hg = Map G (EG +, X ). When p 5, there is an equivalence of spectra ( K(KUp ) v 1 2 V (1)) hz p (K(KU p )) hz p v 1 2 V (1).

33 Does this work shed any light on the Ausoni-Rognes Conjecture for higher n? For any n and p: there exists K c G n, with G n /K = Z p. I believe it is reasonable to think that there exists an equivalence ( ) (K(E n ) vn+1 1 V n) hgn (K(E n ) vn+1 1 V n) hk hz p.

34 Also, if K(E n ) V n can be shown to be an f -spectrum, and (K(E n ) V n ) hk can be constructed as an f -spectrum, then I believe that the tools and techniques of this work will yield a construction of ( (K(E n ) v 1 n+1 V n) hk ) hz p.

Delta-discrete G spectra and iterated homotopy fixed points

Delta-discrete G spectra and iterated homotopy fixed points DANIEL G. DAVIS Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G spectrum. If H and K are closed