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Over a global field any finite number of central simple algebras of exponent dividing m is split by a common cyclic field extension of degree m. We show that the same property holds for function fields of 2-dimensional excellent schemes over a henselian local domain of dimension one or two with algebraically closed residue field.

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

The homology groups h1(l/k), h2(l/k), and h3(l/k) of the Brauer complex for a triquadratic field extension l=k(a,b,c) are studied. In particular, given D∈APTARAMOPREFIX2 Br (k(a,b,c)/k), we find equivalent conditions for the image of D in h2(l/k) to be zero. We consider as well the second divided power operation γ2:APTARAMOPREFIX2 Br (l/k)→H4(k,Z/2Z), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h2⊗h1→H3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F/k such that the cohomology group h1(F(a,b,c)/F) of the Brauer complex contains the images of prescribed elements of k∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L/F with nontrivial h3(L/F) are given. As a consequence we show that the homology group h3(L/F) can be arbitrarily big.

Let F be a field, char F ≠ 2 , L / F a quartic field extension. Define by G L / F the group of elements r ∈ F ⁎ such that D ∪ ( r ) = 0 for any regular field extension K / F and any D ∈ Br 2 ( K L / K ) . We show that G L / F = F ⁎ 2 N L / F L ⁎ . As a consequence we prove that the Hasse norm theorem modulo squares holds for L / F .

Let T/A be an integral extension of noetherian integrally closed integral domains whose quotient field extension is a finite cyclic Galois extension. Let S/R be a localization of this extension which is unramified. Using a generalized cyclic crossed product construction it is shown that certain reflexive fractional ideals of T with trivial norm give rise to Azumaya R-algebras that are split by S. Sufficient conditions on T/A are derived under which this construction can be reversed and the relative Brauer group of S/R is shown to fit into the exact sequence of Galois cohomology associated to the ramified covering T/A. Many examples of affine algebraic varieties are exhibited for which all of the computations are carried out.

Let [math] be a prime integer and [math] a field of characteristic different from [math] . We prove that the essential [math] -dimension ed [math] of the class [math] of central simple algebras of degree [math] is at least [math] . The integer [math] measures complexity of the class of central simple algebras of degree [math] over field extensions of [math] .

Let F be a field of characteristic not 2, a 1 , … , a n ∈ F ∗ , M = F ( a 1 , … , a n ) . Let further G be the subgroup of F ∗ / F ∗ 2 generated by a 1 , … , a n . For any I ⊂ { 1 , … , n } put a I = ∏ i ∈ I a i ( a ∅ = 1 ) . Consider the complex ∏ I ≠ ∅ K I ∗ / K I ∗ 2 → φ G ⊗ F ∗ / F ∗ 2 → ψ Br 2 F → res Br 2 M → ∏ N M / E ∏ F ⊂ E ⊂ M , [ M : E ] = 2 Br 2 E , where K I = F ( a I ) are all the quadratic extensions of F , containing in M , φ ( { w I } ) = ∑ I a I ⊗ N K I / F ( w I ) , ψ ( a I ⊗ z ) = ( a I , z ) . The homology group of this complex at i + 1 -th term from the left is denoted by h i ( M / F ) . Independently of char F we give examples of these complexes with nontrivial h 3 ( M / F ) and prescribed n ⩾ 3 and a 1 , … , a n − 1 . Also similar examples concerning the group h 1 ( M / F ) are constructed.

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements \(\bar a,\overline {b_1 } , \ldots ,\overline {b_n } \in k^* /k^{*2} \) are linearly independent over ℤ/2ℤ. We construct a field extension K/k and a quaternion algebra D = (u, v) over K such that (1) the field K has no proper extension of odd degree (2) the u-invariant of K equals 4 (3) the multiquadratic extension \(K(\sqrt {b_1 } , \ldots \sqrt {b_n } )/K\) is not 4-excellent, and the quadratic form 〈uv,-u,-v, a〉 provides a relevant counterexample (4) the central division algebra A = D ⊗E (a, t0) ⊗E (b1, t1) ⋯ ⊗E (bn, tn) does not decompose into a tensor product of two nontrivial central simple algebras over E, where E = K ((t0))((t1)) … ((tn)) is the Laurent series field in the variables t0, t1, …, tn (5) ind A = 2n+1.

The two-dimensional local field K = Fq((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have xp − x = ut−p =: h, then the condition that (y, f | h]K = 0 for any y e K is equivalent to the condition f e Im(Nm(L*)).

Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

Let [Formula: see text] be a field, [Formula: see text], [Formula: see text][Formula: see text], [Formula: see text] a quartic field extension. We investigate the divided power operation [Formula: see text] on the group [Formula: see text]. In particular, we show that any element of [Formula: see text] is a symbol [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is the quadratic trace form associated with the extension [Formula: see text]. As an application, we obtain certain results on the Stifel–Whitney maps [Formula: see text].

Let K K be a field and B r ( K ) Br(K) its Brauer group. If L / K L/K is a field extension, then the relative Brauer group B r ( L / K ) Br(L/K) is the kernel of the restriction map r e s L / K : B r ( K ) → B r ( L ) res_{L/K}:Br(K)\rightarrow Br(L) . A subgroup of B r ( K ) Br(K) is called an algebraic relative Brauer group if it is of the form B r ( L / K ) Br(L/K) for some algebraic extension L / K L/K . In this paper, we consider the m m -torsion subgroup B r m ( K ) Br_{m}(K) consisting of the elements of B r ( K ) Br(K) killed by m m , where m m is a positive integer, and ask whether it is an algebraic relative Brauer group. The case K = Q K=\mathbb {Q} is already interesting: the answer is yes for m m squarefree, and we do not know the answer for m m arbitrary. A counterexample is given with a two-dimensional local field K = k ( ( t ) ) K=k((t)) and m = 2 m=2 .

If D is a tame central division algebra over a Henselian valued field F, then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF. After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map [D] ↦ [GD]g yields an index-preserving isomorphism from the tame part of the Brauer group of F to the graded Brauer group of GF. This isomorphism is shown to be functorial with respect to field extensions and corestrictions, and using this it is shown that there is a correspondence between F-subalgebras of D (with center tame over F) and graded GF-subalgebras of GD.

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty $, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0 \le b _{p} \le a _{p} \le \infty $, equals the sequence abrd$_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero., Comment: LaTeX, 14 pages: published in Journal of Algebra {\bf 428} (2015), 190-204; the abstract in the Metadata updated to fit the one of the paper

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$, and reduce the problem to the case where $G$ is simple. For simple groups we show that the answer is positive if $G$ is split of type ${\sf A}_{n}$ or ${\sf C}_n$, and negative for groups of other types, except possibly ${\sf G}_{2}$. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of $G$ on itself. The results and methods of this paper have played an important part in recent A. Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\sf A}_n$, ${\sf C}_n$, and ${\sf G}_2$., Final version, 37 pages. To appear in Compositio Mathematica.

In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H^1(K, S) -> H^1(K, G) is surjective for every field extension K/k. We give several applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. In particular, we prove that for every z in H^1(K, G) there exists an abelian field extension L/K such that z_L \in H^1(L, G) is represented by a G-torsor over a projective variety. From this we deduce that z_L has trivial point obstruction. We also show that a (strong) variant of the algebraic form of Hilbert's 13th problem implies that the maximal abelian extension of K has cohomological dimension =< 1. The last assertion, if true, would prove conjectures of Bogomolov and Koenigsmann, answer a question of Tits and establish an important case of Serre's Conjecture II for the group E_8., New material added on the no-name lemma in Section 4 and on Hilbert's 13th problem in Section 9. A mistake in the proof of Proposition 2.3 is corrected

Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic. Given a commutative ring extension C over A of prime characteristic p, there are three variants for the Brauer group B(C/A) of central separable A-algebras split by C: the Amitsur cohomology group H2(C/A, Gm), the Chase-Rosenberg group PV(C/A), and Hochschild's group 4(C, g) of regular restricted Lie algebra extensions of C by the Lie algebra g of all A-derivations on C. In this paper we show that if C is finitely generated projective as an A-module and C [g] = EndA (C), then H2(C/A, Gm) , C) ,l(C/A). As a corollary we show that Hi(C/A, Gm) is zero for all i > 2. When C is a field, these are the results of Berkson, Hochschild and Rosenberg and Zelinsky [4], [11], [12]. As in [11] we show that the Lie algebra extensions which arise from central separable algebras are trivial extensions when regarded as ordinary extensions so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. We also show that if R is a commutative C-algebra which is finitely generated, projective as a C-module, then the Brauer group B(R/A) is mapped onto the Brauer group B(R/C). The last result is also due to Hochschild when C is a field [10]. ?1 contains the background on projective modules which came into the picture. Due to their peculiar behavior all relevant automorphisms turn out to be inner which explains why instead of some exact sequences we get two isomorphism theorems. In ?2 the isomorphism of e(g, C) with 91(C/A) is proved. ?3 and ?4 provide the preliminary materials for ?5. ?3 contains an exposition on the theory of differentials in rings of prime characteristic. Its application to Amitsur cohomology is given in ?4. The main results are given in ?5. Throughout this paper C over A always denotes a commutative ring extension of prime characteristic p such that C is finitely generated projective as an A-module Received by the editors October 30, 1969. AMS 1970 subject classifications. Primary 13A20.

and Fk* the group of units of Fk. Define ring homomorphisms ei (i = 1, , k+ 1) of Fk to Fk+l by(2'3) ei(fl? * ?fk) =fl? * fi?1 fi ... ?fk and define a multiplicative homomorphism Ak: Fk* -> F(k+l)* by Ak(X) = el(x) (E2(X)) * ... (ek+l(x))+1. Then the groups Fk* and mappings Ak form the Amitsur complex; the kth homology group Ker Ak+l/Im Ak we denote by Hk(F). In case F is a finite dimensional extension field, Amitsur showed that H2(F) is isomorphic to the Brauer group of central simple C-algebras split by F. He also showed that in case F is a normal separable extension, Hn(F) is isomorphic to Hn(G, F*) the nth cohomology group of the Galois group G of F over C with coefficients in the group of nonzero elements of F. In this paper, we extend and simplify Amitsur's results. We begin by showing (?2) that in case F is a separable field extension of C, Hn(F) --'Hn([G: H], K*), where K is a normal closure of F with Galois group G, H is the subgroup corresponding to F, and the cohomology group on the right side is the relative cohomology group as introduced in [I]. Next, we study Hn(F) when C is not necessarily a field but when n = 2. In ?3, under weak hypotheses on F, we exhibit a homomorphism of H2(F) to the (generalized) Brauer group of central separable algebra classes split by F [5]. This homomorphism is an isomorphism under stronger hypotheses on F and C. These hypotheses are slightly weaker than assuming all projective C, F, and F2 modules are free and include the cases (10) C is semilocal (not necessarily Noetherian) and F is a C-algebra which is a-finitely generated projective C-module containing C 1 as a direct summand and (20) C=K[x], F=L[x], with K a field and L a finite dimensional commutative K-algebra. Hochschild in [13] has given a description of the Brauer group in case F is a purely inseparable extension field of C of exponent 1. In [3, ?7], Amitsur

In this chapter we set the stage for a systematic study of the Brauer group. The details of the program will be worked out in the next chapter, using two main tools: the cohomology theory that was introduced in Chapter 11, and the properties of subfields of central simple algebras which will be established in the first five sections of this chapter. The last section of the chapter gives appUcations of the theory, including Wedderbum’s Theorem on Finite Division Algebras, and the Cartan-Brauer-Hua Theorem.

LetKbe a field and(K) the Brauer group ofK. It consists of the similarity classes of finite central simple algebras overK. For any field extensionF/Kthere is a natural mapping(K) →(F) which is obtained by assigning to each central simple algebraA/Kthe tensor productwhich is a central simple algebra overF. The kernel of this map is the relative Brauer group(F/K), consisting of thoseA ∈(K) which are split byF.

In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (=CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras., Comment: 30 pages. Revised version. A strong relation between the birationality of Severi-Brauer varieties and NC motives was added

A field is stable if for every division algebra A in its Brauer group order of A = index of A. Index and order in the BraLer group of a field F with discrete valuation and perfect residue class field K are calculated. Division algebras with specified order and index are constructed. For F complete, necessary and sufficient conditions for the stability of F are given in terms of the Brauer group of K. These results follow. A finite extension of a stable field need not be stable. The power series field K((x)) is stable for K a local field. K((x)) and K(x) are not stable for K a global field. Introduction. A field F is stable if for every division algebra A in the Brauer group Br(F), order eF(A) = index SF(A). In general order divides index and they have the same prime factors. Any local or global field is stable by class field theory. Many results concerning subfields and subalgebras of division algebras depend on stability. See, for example, [Sc] and [R]. For a profinite group G, the character group G = continuous Hom(G, Q/Z). For a field F, G(F) = Gal(FS/F) with Fs a separable algebraic closure of F. If L is a finite Galois extension of F and f is a character on Gal (L/F), then f inflates to a character on G(F). These topics are exposed in [S]. Lemma 1. Let F be a field and G = G(F). If f e G, H = Ker f, and L = fixed field of H, then L is a finite cyclic extension of F with Gal (LIF) = G/H and order f = index [ G: H] = [ L: F]. If L is a finite Galois extension of F, then L is cyclic over F iff there is a character on Gal (L/F) of order [L: F]. Proof. Immediate from Galois theory and the fact that every finite subgroup of Q/Z is cyclic. For M a field extension of F and f e G(F), fM = the restriction of f to G(M). This definition assumes Fs C Ms. If fM = ?, M is said to split f. For Received by the editors March 26, 1974 and, in revised form, May 3, 1974. AMS (MOS) subject classifications (1970). Primary 16A40, 12B20, 12G05, 12J10; Secondary 13A20, lOMIO. Copyright ? 1975, American Mathematical Society 33 This content downloaded from 157.55.39.186 on Sun, 09 Oct 2016 04:23:44 UTC All use subject to http://about.jstor.org/terms