Cohomological interpretation of Brauer Groups 20172252 Seewoo Lee Abstract In this note, we introduce a centeral simple algebra and a Brauer group, which classifies central division algebra over a given field. We prove that there exists a canonical isomorphism between a relative Brauer group Br(L=k) and the second group cohomology H2(Gal(L=k);L×), which shows that the cohomology of group can be used to classify central division algebras over a given field. We also present some examples and computations. 1 Introduction Let k be a field. A central simple algebra A over k is a finite dimensional associative algebra which is simple and center is k. For example, the quoternions H is a 4-dimensional central simple algebra over R. Our basic question is the following: Classify the central simple algebras over a given field k. This seems very hard problem at first glance. However, we will show that the Artin-Wedderburn theorem claims that every central simple algebra over k is isomorphic to a matrix algebra over some central division algebra over k which is determined unique up to isomorphism. Hence the theorem reduces the problem as Classify the central division algebras over a given field k. However, this isn't an easy problem. To do this, we will define the Brauer group Br(k) of a given field, which is a group that classifies central division algebras over k. More precisely, each elements of Br(k) corresponds to the isomorphism class of the central division algebras over k. Computation of the Brauer group can be done by the computation of the second cohomology group, by the following main theorem we will prove: Br(L=k) ' H2(Gal(L=k);L×): Using this, we will compute Brauer groups over various fields such as R; Fp; Qp; Q. Also, we will show that there exists a noncommutative division algebra of any given positive characteristic p > 0. In Appendix, we will introduce another interpretation of the second cohomology group H2(G; A) as certain group extensions of G by A. Before we start, we will give some basic definitions. Definition 1. Let k be a field and A be a finite dimensional k-vector space. (1) A is k-algebra if A has a ring structure and k is contained in the center of A. (2) k-subalgebra of A is a subring of A containing k. (3) k-algebra homomorphism ' : A ! B is a ring homomorphism with '(a) = a for all a 2 k. (4) Oppsite Aopp is the k-algebra A with same addition but with reversed multiplication α·β := βα. l (5) For a basis e1; : : : ; en for A as a k-vector space, the constants faijg that satisfies n X l eiej = aijel l=1 are called the structure constants of A relative to the basis (ei)i. 2 Simple Algebras; Semisimple Modules In this section, we define CSA, central simple algebras over a field k, and study some basic properties. At last, we will show that classification of CSA's can be reduced to the classification of central division algebras over k (Artin-Wedderburn theorem), and the second one can be done by computing so called Brauer groups. 2.1 Semisimple Modules In this article, A-modules are finitely generated left A-modules. Let V be an A-module. If V is a such A-module and A is finite dimensional k-vector space, then V is also a finite dimensional k-vector space with a k-algebra homomorphism A ! Endk(V ) (representation of A on V ). Definition 2. (1) V is a faithful A-module if the map A ! Endk(V ) is injective. (2) V is a simple A-module if V 6= 0 and the only A-submodule of V are 0 and B itself. (3) V is a semisimple A-module if it is isomorphic to a direct sum of simple A-modules. (4) V is indecomposable if it can not be written as a direct sum of two nonzero A-modules. Theorem 1. Every A-module V admits a filtration V = V0 ⊃ · · · ⊃ Vr = 0 whose quotients Vi=Vi+1 are simple A-modules. If V = W0 ⊃ · · · ⊃ Ws = 0 is a second such filtration, then r = s and there is a permutation σ of f0; : : : ; r − 1g such that Vi=Vi+1 ' Wσ(i)=Wσ(i)+1 for all i. Proof. Existence follows from finite dimensionality of V . Uniqueness can be proved as in the Jordan- H¨oldertheorem. Corollary 1. If V ' V1 ⊕ · · · ⊕ Vr ' W1 ⊕ · · · ⊕ Ws with all A-modules Vi;Wj's are simple, then r = s and there exists σ 2 Sr such that Vi ' Wσ(i) for all i. P Proposition 1. Let V be an A-module. Assume that V = i2I Si for some simple submodules Si L of V . Then for every submodule W of V , there exists a subset J if I such that V = W ⊕ i2J Si. Proof. Use Zorn's lemma; choose maximal J among the subsets of I such that the sum SJ = P j2J Sj is direct and W [ SJ = 0. Then we can prove that W ⊕ SJ = V . Corollary 2. TFAE: (a) V is semisimple; (b) V is a sum of simple submodules; (c) every submodule of V has a complement. Corollary 3. Sums, submodules and quotient modules of semisimple modules are semisimple. 2 2.2 Simple k-algebras Definition 3. A k-algebra A is simple if it contains no proper two-sided ideals other than 0. Example 1. 1. Mn(k), the ring of n × n matrices over k, is a simple k-algebra. 2. A is a division algebra (skew field) if every nonzero element has an inverse. Every division algebra is simple. Also, matrix ring over a division algebra is also simple. 3. For a; b 2 k×, define a quaternion algebra H(a; b) be the k-algebra with basis 1; i; j; ij (as a k-vector space) and with the multiplication determined by i2 = a; j2 = b; ; ij = −ji: One can show that H(a; b) is either a division algebra or isomorphic to M2(k), hence simple. 2.3 Centralizers Definition 4. Let A be a k-subalgebra of B. Then the centralizer of A in B is CB(A) = fb 2 B : ba = ab 8a 2 Ag which is also a k-subalgebra of B. Example 2. 1. We have CMn(k)(k · In) = Mn(k) and CMn(k)(Mn(k)) = k · In. 2. If A ⊂ Mn(k) is a set of diagonal matrices in Mn(k), then CMn(k)(A) = A. For both cases, we have C(C(A)) = A. This can be generalized for any faithful semisimple A- modules. Theorem 2 (Double centralizer theorem). Let A be a k-algebra and V be a faithful semisimple A-module. Then C(C(A)) = A (centralizers taken in Endk(V )). Proof. Let D = B(A) and B = C(D). Clearly A ⊂ B and the reverse inclusion follows from the next lemma when we take v1; : : : ; vn to generate V as a k-vector space. Lemma 1. For any v1; : : : ; vn 2 B and b 2 B, there exists a 2 A s.t. avi = bvi holds for 1 ≤ i ≤ n. Proof. We will prove this for n = 1 first. Since Av1 is an A-submodule of V , by the Proposition 1, we can find W ≤ V s.t. V = Av1⊕W . Consider the projection π : V ! V defined as (av1; w) 7! (av1; 0). Since π is A-linear, π 2 D and π(v) = v if and only if v 2 Av1. Now π(bv1) = b(πv1) = bv1 so bv1 2 Av1. For general n, let W be the direct sum of n copies of V with A acting diagonally, i.e. a(v1; : : : ; vn) = (av1; : : : ; avn); a 2 A; vi 2 V: Then W is a semisimple A-module and we can check that CEndk(W )(A) = Mn(D) = Mn(CEndk(V )(A)) and CEndk(W )(Mn(D)) = B · In. Now apply the case n = 1 to A; W; b and (v1; : : : ; vn) 2 W . 3 2.4 Classification of simple k-algebras In this subsection, we prove the Artin-Wedderburn theorem, which claims that the classification of central simple k-algebras can be reduced to the classification of central division algebras. Lemma 2 (Schur's lemma). If the A-module S is simple, then EndA(S) is a division algebra. Proof. Let γ : S ! S be an A-linear map. Then ker(γ) is an A-submodule of a simple A-module S, so it is either S or 0. In the first case, γ = 0, and in the second case it is an isomorphism, i.e., it has an inverse that is also A-linear. Theorem 3 (Artin-Wedderburn). Every simple k-algebra A is isomorphic to Mn(D) for some n and some division k-algebra D. Proof. Choose any simple A-module S, for example, any minimal left ideal of A. Then A acts faithfully on S, and let D = CEndk(S)(A). By the double centralizer theorem, CEndk(S)(D) = A, so A ' EndD(S). By Schur's lemma, D ' EndA(S) is a division algebra, so S is a free D-module and n opp EndD(S) ' EndD(D ) ' Mn(D ). 2.5 Modules over simple k-algebras Proposition 2. For every simple k-algebra A, the simple submodules of AA are the minimal left ideals in A; any two such ideals are isomorphic as left A-modules, and A is a direct sum of its minimal left ideals.