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Local Class Field Theory Local Class Field Theory Pan Yan Summer 2015 These are notes for a reading course with D. Wright on Local Class Field Theory in Summer 2015. The notes are prepared and written by Pan Yan ([email protected]). If you notice any mistakes or have any comments, please let me know. Contents 1 Introduction 3 1.1 Outline of the course . .3 1.2 References . .3 1.3 Goals of the course . .3 2 Group Cohomology 4 2.1 Group rings . .4 2.2 G-modules . .4 2.3 Group cohomology via cochains . .6 2.4 Group cohomology via projective resolutions . 11 2.5 Homology . 14 2.6 Change of groups . 17 2.7 Tate cohomology . 23 2.8 Tate cohomology via complete resolutions . 29 2.9 Cup products . 29 2.10 Tate cohomology of cyclic groups . 32 2.11 Cohomological triviality . 36 2.12 Tate's Theorem . 40 3 Profinite Groups 45 3.1 Inverse systems and inverse limits . 45 3.2 Topological structure of profinite groups . 46 3.3 Examples of profinite groups . 47 3.4 Direct systems and direct limits . 49 1 3.5 Discrete G-modules . 50 3.6 Cohomology of profinite groups . 51 3.7 Galois cohomology . 52 4 Local Class Field Theory 55 4.1 Statements of the main theorems . 55 4.2 The fundamental class . 57 4.3 The local reciprocity map . 65 4.4 Lubin-Tate formal group law . 70 2 1 Introduction 1.1 Outline of the course Group Cohomology { Chapter IV of [CF67] (this will take about 2-3 weeks); Profinite Groups { Chapter V of [CF67] (this will take about 2 weeks); Local Class Field Theory { Chapter VI of [CF67] (this will take the rest of the summer semester). 1.2 References The main reference is Algebraic Number Theory by Cassels & Fr¨ohlich [CF67]. Other references include [ANT], [Cas86], [FT91], [LT65], [Mil13], [Neu86], [Ser80], [Sha]. Note: Errata for Cassels & Fr¨ohlich [CF67] can be found at http://wwwf.imperial. ac.uk/~buzzard/errata.pdf (maintained by Kevin Buzzard). 1.3 Goals of the course Class Field Theory is the study of abelian extensions of (local or global) fields. In the case of Local Class Field Theory, we are mainly interested in abelian extensions of nonar- chimedean local fields. Specifically, given a nonarchimedean local field K, we want to describe all finite abelian extensions of K totally in terms of the arithmetic of the base field K. To this end, we want to understand group cohomology (especially Tate cohomology for finite groups and Tate's Theorem), inverse limits and profinite groups, direct limits, cohomology of profinite groups and Galois cohomology, the statements and proofs of Local Reciprocity Law and Local Existence Theorem. 3 2 Group Cohomology 2.1 Group rings Definition 2.1.1. Let G be a group, R a commutative ring with identity. The group ring P R[G] is the set of all formal finite sums g2G agg with each ag 2 R, i.e., 8 9 <X = R[G] = agg j ag 2 R; almost all ag = 0 :g2G ; 8 9 <X = = agg j ag 2 R; only finitely many ag 6= 0 : :g2G ; The addition is defined as 0 1 0 1 X X X @ aggA + @ bggA = (ag + bg)g; g2G g2G g2G and the multiplication is the involution 0 1 0 1 X X X X @ aggA @ bggA = (agbhgh) = agh−1 bh g: g2G g2G g;h2G g2G In this course we are mainly interested in the integral group ring 8 9 <X = Z[G] = agg j ag 2 Z; almost all ag = 0 : :g2G ; 2.2 G-modules From now on G always means a group, unless otherwise indicated. Definition 2.2.1. A (left) G-module is an abelian group A together with a G-action on A (i.e., a map G × A ! A defined by (g; a) 7! g · a) such that (i) 1 · a = a; 8a 2 A; (ii) g1 · (g2 · a) = (g1g2) · a; 8a 2 A; g1; g2 2 G; (iii) g · (a1 + a2) = g · a1 + g · a2; 8a1; a2 2 A; g 2 G. Remark 2.2.2. (i) A right G-module is defined similarly by replacing the above G-action with A × G ! A defined by (a; g) 7! g−1a. (ii) We will assume all G-modules are left G-modules unless otherwise indicated. (iii) G-modules are the same as Z[G]-modules. 4 Definition 2.2.3. A homomorphism ' : A ! B of G-modules is a homomorphism of abelian groups such that '(ga) = g'(a) for all a 2 A; g 2 G (hence it is compatible with the G-action). The group of G-module homomorphisms is denoted as HomG(A; B) = HomZ[G](A; B). A G-module homomorphism ' : A ! B makes the following diagram commutative. G−action A > A ' ' _ G−action _ B > B If A; B are G-modules, we denote the group of all abelian group homomorphisms A ! B as Hom(A; B). Note that Hom(A; B) actually has a G-module structure: if ' 2 Hom(A; B), we can define the map G × Hom(A; B) ! Hom(A; B) A ! B ! g · (' : A ! B) 7! a 7! g'(g−1a) which makes Hom(A; B) as a G-module. Definition 2.2.4. A G-module A is trivial if g · a = a for all a 2 A; g 2 G. Definition 2.2.5. Let A be a G-module. The group of G-invariants of A, denoted as AG, is AG = fa 2 A j g · a = a; 8g 2 G; a 2 Ag: Remark 2.2.6. (i) AG is the maximal trivial submodule of A. Indeed, AG is trivial, by definition. Now suppose B ⊂ A is a trivial submodule of A. Let b 2 B, for any g 2 G, we have g · b = b. So b 2 AG. Hence B ⊂ AG. (ii) If A is a trivial G-module, then AG = A. If A; B are G-modules, then HomG(A; B) = f' : A ! B j '(ga) = g'(a); 8g 2 G; a 2 Ag; (Hom(A; B))G = f' : A ! B j '(a) = g · '(a) = g'(g−1a); 8g 2 G; a 2 Ag: Hence G HomG(A; B) = (Hom(A; B)) : So G ∼ G HomG(Z;A) = (Hom(Z;A)) = A : 5 Since the covariant functor Hom is left exact, it follows that AG is also a covariant left exact functor. More specifically, if 0 ! A ! B ! C is an exact sequence of G-modules, then 0 ! AG ! BG ! CG is an exact sequence of abelian groups. The i-th cohomology group Hi(G; A) of G with coefficients in A can be defined as the i-th derived functor on A of the functor of G-invariants. Alternatively, we give more specific definitions in the following two sections. 2.3 Group cohomology via cochains Definition 2.3.1. Let A be a G-module, and i ≥ 1. (i) The group of i-cochains of G with coefficients in A, denoted as Ci(G; A), is the set of functions from Gi to A, i.e., Ci(G; A) = f' : Gi ! Ag. i i i i+1 (ii) The i-th differential d = dA : C (G; A) ! C (G; A) is the map i i X j d (')(g0; g1; ··· ; gi) =g0'(g1; ··· ; gi) + (−1) '(g0; ··· ; gj−2; gj−1gj; ··· ; gi) j=1 i+1 + (−1) '(g0; ··· ; gi−1): Remark 2.3.2. C0(G; A) = f' : G0 ! Ag = f' : fpg ! Ag =∼ A. Lemma 2.3.3. For any i ≥ 0, we have di+1 ◦ di = 0. So (Ci(G; A); di) is a cochain complex d0 d1 0 > C0(G; A) > C1(G; A) > C2(G; A) > ··· : Definition 2.3.4. Let i ≥ 0. (i) The group Zi(G; A) = ker di is the group of i-cocycles of G with coefficients in A. (ii) The group B0(G; A) = 0;Bi(G; A) = imdi−1 (i ≥ 1) is the group of i-coboundaries of G with coefficients in A. Remark 2.3.5. Bi(G; A) ⊂ Zi(G; A) since di ◦ di−1 = 0. Definition 2.3.6. The i-th cohomology group of G with coefficients in A is defined as Zi(G; A) Hi(G; A) = : Bi(G; A) Remark 2.3.7. The cohomology groups measure how far the cochain complex (Ci(G; A); di) is from being exact. Here are some examples of cohomology groups of low degrees. 6 Lemma 2.3.8. (i) H0(G; A) = AG. (ii) Z1(G; A) = f' : G ! A j '(gh) = g'(h) + '(g); 8g; h 2 Gg: B1(G; A) = f' : G ! A j 9a 2 A such that '(g) = ga − a; 8g 2 Gg: (Elements in Z1(G; A) are called crossed homomorphism.) (iii) If A is a trivial G-module, then H1(G; A) = Hom(G; A). Proof. (i) H0(G; A) = Z0(G; A)=B0(G; A) = Z0(G; A) since B0(G; A) = 0 by definition. Z0(G; A) = ker d0, where d0 : C0(G; A) = A ! C1(G; A) is defined by d0(a)(g) = ga − a. So ker d0 = fa 2 A j ga − a = 0; 8g 2 Gg = AG. Hence H0(G; A) = AG. (ii) By definition, B1(G; A) = im d0 = f' : G ! A j 9a 2 A such that '(g) = ga − a; 8g 2 Gg: Z1(G; A) = ker d1, where d1 : C1(G; A) ! C2(G; A) is defined by d1(')(g; h) = g'(h) − '(gh) + '(g); where (g; h) 2 G2: Hence Z1(G; A) = f' : G ! A j g'(h) − '(gh) + '(g) = 0; 8g; h 2 Gg = f' : G ! A j '(gh) = g'(h) + '(g); 8g; h 2 Gg: (iii) If A is a trivial G-module, then Z1(G; A) = f' : G ! A j '(gh) = g'(h) + '(g); 8g; h 2 Gg = f' : G ! A j '(gh) = '(h) + '(g); 8g; h 2 Gg = Hom(G; A): On the other hand, B1(G; A) = 0 since ga − a = 0 for any g 2 G; a 2 A.
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