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A Note on the Cohomology of the Langlands Group

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A Note on the Cohomology of the Langlands Group TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 4, April 2015, Pages 2905–2920 S 0002-9947(2014)06230-2 Article electronically published on February 25, 2014 A NOTE ON THE COHOMOLOGY OF THE LANGLANDS GROUP EDWARD S.T. FAN, WITH AN APPENDIX BY M. FLACH Abstract. We begin with a comparison of various cohomology theories for topological groups. Using the continuity result for Moore cohomology, we establish a Hochschild-Serre spectral sequence for a slightly larger class of groups. We use these properties to compute the cohomology of the Langlands group of a totally imaginary field. The appendix answers a question raised by Flach concerning the cohomological dimension of the group R. 1. Introduction In the first half of this article, we try to collect the recent continuity result of Moore cohomology of compact groups defined via measurable cochains announced in [2] and compare it with the parallel result on the topological group cohomology defined via topos theory [5] with the aid of a series of comparison results [2,5,10,18] between various cohomology theories. We will recall the definitions of different cohomology theories of topological groups here and briefly discuss the relations between them. In the second half, we will give an application to compute the topological group cohomology of the Langlands group LF . As an extension of the i result in [5], we show that H (LF , Z) has infinite rank for all even i ≥ 4inthecase where F is a totally imaginary number field. This result gives a negative answer to the conjectural use of the Langlands group to modify the definition of a Weil- etale topos Y W for the spectrum of the ring of integers of a number field F ,whose i cohomologies H (YW , Z) are expected to vanish for i ≥ 4. The appendix answers a question raised in [5, 9.3]. Since the present article is in some sense a natural continuation of [5], we decided to include this answer here. 2. Basic definitions In this section, we will briefly recall the definition of various cohomology theories for topological groups with coefficients on their corresponding topological modules. 2.1. Topos theoretic cohomology of topological groups. Let Top be the (small) category of topological spaces with continuous maps, and Jopen the open covering topology on Top. Then (Top,Jopen)isasiteandsetT = Sh(Top,Jopen)to be the associated category of sheaves of sets on it. Given a topological group G and a topological G-module A (i.e. a topological abelian group with a linear continuous G- action), let y :Top→T be the Yoneda embedding, where y(X)=HomTop ( ,X). Then yG is a group object in T and set BG to be the classifying topos of G, i.e. the Received by the editors September 3, 2012 and, in revised form, June 12, 2013. 2010 Mathematics Subject Classification. Primary 11F75, 14F20; Secondary 20J06, 22A99. Key words and phrases. Cohomology, topological groups, Langlands group. c 2014 American Mathematical Society Reverts to public domain 28 years from publication 2905 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2906 EDWARD S.T. FAN category of yG-objects in T .Inparticular,yA is an abelian group object in BG. We define the topos theoretic cohomology of a topological group G with coefficients in a G-module A to be (2.1) Hi(G, A)= RiΓ (yA) , where Γ(A):=HomBG (∗, A) is the global section functor. (Here ∗ denotes the trivial yG-object, i.e. the constant sheaf of a point with trivial G-action.) 2.2. Continuous cochain cohomology of topological groups. As above, sup- pose G is a topological group and A is a topological G-module. Let Ci (G, A)be the set of continuous maps Gi → A for i ≥ 0 and 0 otherwise. Then we define the coboundary maps d : Ci(G, A) → Ci+1(G, A)by df (g1,...,gi+1):=g1f(g2,...,gi+1) i j + (−1) f(g1,g2,...,gj gj+1,...,gi+1) j=1 i+1 +(−1) f(g1,...,gi). Simple computation shows that d2 =0.Thus Ci(G, A),d form a cochain com- plex and the corresponding cohomology is defined to be the continuous cochain cohomology of G with coefficients in A, denoted Ker d : Ci(G, A) → Ci+1(G, A) (2.2) Hi (G, A):= . cts Im (d : Ci−1(G, A) → Ci(G, A)) 2.3. Moore measurable cochain cohomology for locally compact groups. Let G be a locally compact group and A a complete metric G-module. It is well known that there is a Haar measure on G so that both G and A are equipped with a Borel structure (i.e. a σ-field of subsets of G or A which contains all singletons and where the group operations are Borel functions) generated by open subsets [12]. i i → Let CM (G, A) be the set of Haar-a.e. equivalence classes of Borel maps G A for ≥ i → i i 0 and 0 otherwise. Define the coboundary maps δ : CM (G, A) CM (G, A)by δf(g1,...,gi+1):=g1f(g2,...,gi+1) i j + (−1) f(g1,g2,...,gj gj+1,...,gi+1) j=1 i+1 +(−1) f(g1,...,gi), where all equalities hold almost everywhere in corresponding Haar measures. It is 2 i easy to see that δ = 0, henceforth (CM (G, A),δ) form a cochain complex and its co- homology defines the Moore measurable cochain cohomology of G with coefficients in A, denoted Ker δ : Ci (G, A) → Ci+1(G, A) (2.3) Hi (G, A):= M . M i−1 → i Im δ : CM (G, A) C (G, A) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A NOTE ON THE COHOMOLOGY OF THE LANGLANDS GROUP 2907 2.4. Algebraic cohomology of topological groups. Let G be a topological group, MG the category of complete metric G-modules, and A a complete metric G-module. Set EXTn (Z,A)tobethesetofn-term exact sequences 0 → A → MG B1 →···→Bn → Z → 0inMG,whereZ is the group of integers with the discrete topology and trivial G-action and Extn (Z,A) is the quotient of EXTn (Z,A) MG MG by the equivalence relation generated by chain maps with identity on the first and last terms. Then the algebraic cohomology of G with coefficients in A is defined to be i i Z (2.4) Ha(G, A):=ExtMG ( ,A). 2.5. Ordinary cohomology of classifying spaces. Let G be a topological group and A be an abelian group with trivial G-action. The cohomology of G with coefficients in A defined via classifying space is given by i i (2.5) Hsp(G, A)=Hsing(BG, A), i where BG denotes the topological classifying space of G and Hsing is the ordinary ith singular cohomology. 3. Relations between various cohomology theories of topological groups This entire section is devoted to gathering up comparison results of various cohomology theories of topological groups appearing in the literature. From this hull of information, we conclude a few useful consequences which facilitate the calculation of the cohomology of the Langlands group in the next section. 3.1. Recollections of comparison results. We will summarize the comparison theorems between the five cohomology theories for topological groups mentioned above that appeared in [2, 5, 10, 18]. All the results here are not new, but by putting them together, we get a relatively complete picture of the relations among different cohomology theories. It turns out that the comparison theorems extend the availability of the Hochschild-Serre spectral sequences for infinite dimensional compact groups under some mild conditions. Theorem 3.1 ([5], Proposition 5.1). Let G be a topological group and A be a topo- q n i i logical G-module. If HT (G ,yA)=0for all n, q > 0,thenH (G, A)=Hcts(G, A). q n n Here HT (G ,yA) denotes the sheaf cohomology of the topological space G with coefficients in the sheaf represented by A. Theorem 3.2 ([5], Proposition 9.5). Let G be a paracompact, locally compact group i i and let A be a vector group with continuous G-action. Then H (G, A)=Hcts(G, A). Theorem 3.3 ([5], Proposition 5.2). Let G be a topological group and A be a discrete G-module with trivial G-action. Suppose that Gn is locally contractible for i i all n>0;thenH (G, A)=Hsp(G, A). Theorem 3.4 ([18]). Let G be a locally compact group and A be a complete metric i i G-module. If G is Hausdorff, then HM (G, A)=Ha(G, A). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2908 EDWARD S.T. FAN Theorem 3.5 ([18], Theorems 1 and 3). Let G be a finite dimensional paracompact, locally compact group. Then (1) If G is zero dimensional, then i i HM (G, A)=Hcts(G, A) for any locally compact G-module A. (2) If A is a vector group with continuous G-action, then i i HM (G, A)=Hcts(G, A). Theorem 3.6 ([5], Corollary 2, [10], Propositions 1.4, [18], Theorem 2). Let G be a finite dimensional paracompact, locally compact group. Suppose A is a com- plete metric G-module with the following property: For any short exact sequence of complete metric abelian topological groups 0 → A → B →τ C → 0, τ has the homotopy lifting property for every finite dimensional paracompact space.
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