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We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting A 1 ≤ A 2 if deg A 1 = deg A 2 and every étale subalgebra of A 1 is isomorphic to a subalgebra of A 2 , and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras. [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra with bijective antipode. In this paper we prove that, if A is a Hopf brace with some projection on H, then there exists a compatible braided Hopf brace R such that A is isomorphic to R ♯ H as Hopf braces, where R ♯ H is some Radford's biproduct Hopf algebra. This should be viewed as the brace version of well-known Radford's theorem about Hopf algebras with a projection. This provides a new method to construct Hopf braces by some braided Hopf algebras. [ABSTRACT FROM AUTHOR]

Let F be a field, n ≥ 2 a positive integer not divisible by. Suppose the group of roots of unity of degree n is contained in F ∗ . Let l, m be positive integers, m ≥ 2 , n is divisible by m 2 , n = m k . Assume that ξ n ∈ F if n is odd, and ξ 2 n ∈ F if n is even. Applying the Bloch–Kato conjecture and the divided power operations on K 2 (F) / n , we construct an invariant in H 2 l + 2 (F , μ n) for elements A ∈ Br (F) such that mA is the sum of l symbol algebras of degree k. Namely, if A = ∑ i = 1 l D i + α = ∑ i = 1 l D ~ i + α ~ , where D i , D ~ i are symbol algebras of degree n, and , then k D 1 ∪ ⋯ ∪ k D l ∪ α = k D ~ 1 ∪ ⋯ ∪ k D ~ l ∪ α ~ ∈ H 2 l + 2 (F , μ m). When ξ 2 n ∉ F , but ξ n ∈ F we construct the above invariant in the case where l = 1 , m = 2 , and n ≥ 4 is 2-primary, using Suslin's theorem on the torsion in K 2 (F) and the divided power operation γ 2 : K 2 (F) / 2 → K 4 (F) / 2 . In the second part of the paper we consider one problem on the symbol length for K q (F) / n . For n , q ≥ 2 and u ∈ K q (F) / n denote by L (n) (u) the minimal number k such that u is the sum of k symbols in K q (F) / n . For any m ≥ 1 we give a sufficient condition on u for the equality L (m n) (m u) = L (n) (u) . Using the index reduction formula for central simple algebras, we also construct examples with L (m n) (m u) < L (n) (u) for any noncoprime m and n. In the case we give an upper bound for L (n) (u) L (m n) (m u) . [ABSTRACT FROM AUTHOR]

In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$ -categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$ , we find that the 'algebraic' Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $\pi _0(E)$ , recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$ -algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an 'exotic' $KO$ -algebra with the same coefficient ring as $\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$ , previously studied by Mathew and Stojanoska. [ABSTRACT FROM AUTHOR]

Let F be a field, A an F-csa of degree n. Saltman proves that there is a Zariski dense subset of elements d ∈ A such that the Severi-Brauer variety of A is birationally equivalent to the affine variety defined by restricting the reduced norm to subsets of the form K + d where K is a commutative separable maximal F subalgebra of A. In this work, we show that Saltman's result actually implies that there is a non-empty dense subset of subspaces V ≤ A of dimension n + 1 such that the Severi-Brauer variety of A is birationally equivalent to the projective variety defined by restricting the reduced norm of A to V. We then show that for symbol algebras, a standard n-Kummer subspace induces the birational equivalence as above and use it to reprove Amitsur's conjecture for the case of symbol algebras. [ABSTRACT FROM AUTHOR]

Let H be a weak Hopf algebra with an idempotent element p. Denote by Hp some twisted weak Hopf algebra associated with a generalized Drinfeld twist. Let be a bicovariant differential calculus on H. We give some method to construct a bicovariant differential calculus on Hp by a generalized Drinfeld twist, where is some subspace of Γ determined by. As an application, we present some examples of bicovariant differential calculi on some face algebra and its quantum double. [ABSTRACT FROM AUTHOR]

The standard period-index conjecture for Brauer groups of p-adic surfaces S predicts that ind (α) | per (α) 3 for every α ∈ Br (Q p (S)) . Using Gabber's theory of prime-to- ℓ alterations and the deformation theory of twisted sheaves, we prove that ind (α) | per (α) 4 for α of period prime to 6p, giving the first uniform period-index bounds over such fields. [ABSTRACT FROM AUTHOR]

Let S be a commutative ring and Q a group that acts on S by ring automorphisms. Given an S-algebra endowed with an outer action of Q, we study the associated Teichmüller class in the appropriate third group cohomology group. We extend the classical results to this general setting. Somewhat more specifically, let R denote the subring of S that is fixed under Q. A Q-normalS-algebra consists of a central S-algebra A and a homomorphism σ:Q→Out(A) into the group Out(A) of outer automorphisms of A that lifts the action of Q on S. With respect to the abelian group U(S) of invertible elements of S, endowed with the Q-module structure coming from the Q-action on S, we associate to a Q-normal S-algebra (A,σ) a crossed 2-fold extension e(A,σ) starting at U(S) and ending at Q, the Teichmüller complex of (A,σ), and this complex, in turn, represents a class, the Teichmüller class of (A,σ), in the third group cohomology group H3(Q,U(S)) of Q with coefficients in U(S). We extend some of the classical results to this general setting. Among others, we relate the Teichmüller cocycle map with the generalized Deuring embedding problem and develop a seven term exact sequence involving suitable generalized Brauer groups and the generalized Teichmüller cocycle map. We also relate the generalized Teichmüller cocycle map with a suitably defined abelian group of classes of representations of Q in the Q-graded Brauer category of S with respect to the given action of Q on S. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a del Pezzo surface of degree 5 defined over a field F. A theorem of Manin and Swinnerton‐Dyer asserts that every Del Pezzo surface of degree 5 is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree 5 over a field F are precisely the F‐forms of the moduli space M ¯ 0 , 5 of stable curves of genus 0 with five marked points. Suppose n ⩾ 5 is an integer, and F is an infinite field of characteristic ≠2. It is easy to see that every twisted F‐form of M ¯ 0 , n is unirational over F. We show that (a)if n is odd, then every twisted F‐form of M ¯ 0 , n is rational over F; (b)if n is even, there exists a field extension F / k and a twisted F‐form X of M ¯ 0 , n such that X is not retract rational over F. [ABSTRACT FROM AUTHOR]

Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a fieldk. We first construct a weak Hopf algebra [Δ(1)(H⊗H)Δ(1)]R, which is based on the subalgebra of the tensor product algebraH⊗H. Next we verify that ifHis factorizable, then the Drinfeld’s quantum double ofHis isomorphic to [Δ(1)(H⊗H)Δ(1)]R. [ABSTRACT FROM AUTHOR]

In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable if it is adjoint to a bilinear Pfister form over all splitting fields of the algebra. A stronger result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra. [ABSTRACT FROM AUTHOR]

Let H be a coquasi-triangular Hopf algebra. We first show that the group of braided autoequivalences of the category of H -comodules is isomorphic to the group of braided-commutative bi-Galois objects. Next, by investigating the latter, we obtain that the group of braided autoequivalences of the representation category of Lusztig’s quantum group u q ( s l ( 2 ) ) ′ is isomorphic to the projective special linear group P S L ( 2 ) , where q is a root of unity of odd order N > 1 . [ABSTRACT FROM AUTHOR]

We study the Brauer group of an affine double planeπ:X→𝔸2defined by an equation of the formz2 = fin two separate cases. In the first case,fis a product ofnlinear forms ink[x,y] andXis birational to a ruled surfaceℙ1×C, whereCis rational ifnis odd and hyperelliptic ifnis even. In the second case,f = y2−p(x) is the equation of an affine hyperelliptic curve. Forπas well as the unramified part ofπ, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

By Merkurjev's Theorem every central simple algebra of exponent two is Brauer equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over a given field is split, then there exists no central simple algebra of exponent two over this field. This note provides an independent elementary proof for the latter fact. [ABSTRACT FROM AUTHOR]

LetAbe a central simple algebra over a fieldF. Letk1,…,krbe cyclic extensions ofFsuch thatk1 ⊗F… ⊗Fkris a field. We investigate conditions under whichAis a tensor product of symbol algebras where each fieldkilies in a symbolF-algebra factor of the same degree askioverF. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4. [ABSTRACT FROM PUBLISHER]

In this paper, we study cyclic twists of an elliptic curve of period two or three, and apply a recent result of Ciperiani and Krashen to describe explicitly their relative Brauer groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group acting on the rational function field C ( x g : g ∈ G ) by C -automorphisms h ( x g ) = x h g for any g , h ∈ G . Noether's problem asks whether the invariant field C ( G ) = k ( x g : g ∈ G ) G is rational (i.e. purely transcendental) over C . By Fischer's theorem, C ( G ) is rational over C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G of order p 9 and of order p 6 such that C ( G ) is not rational over C by showing the non-vanishing of the unramified Brauer group: Br nr ( C ( G ) ) ≠ 0 , which is an avatar of the birational invariant H 3 ( X , Z ) tors given by Artin and Mumford where X is a smooth projective complex variety whose function field is C ( G ) . For p = 2 , Chu, Hu, Kang and Prokhorov proved that if G is a 2-group of order ≤32, then C ( G ) is rational over C . Chu, Hu, Kang and Kunyavskii showed that if G is of order 64, then C ( G ) is rational over C except for the groups G belonging to the two isoclinism families Φ 13 with Br nr ( C ( G ) ) = 0 and Φ 16 with Br nr ( C ( G ) ) ≃ C 2 . Bogomolov and Böhning's theorem claims that if G 1 and G 2 belong to the same isoclinism family, then C ( G 1 ) and C ( G 2 ) are stably C -isomorphic. We investigate the birational classification of C ( G ) for groups G of order 128 with Br nr ( C ( G ) ) ≠ 0 . Moravec showed that there exist exactly 220 groups G of order 128 with Br nr ( C ( G ) ) ≠ 0 forming 11 isoclinism families Φ j . We show that if G 1 and G 2 belong to Φ 16 , Φ 31 , Φ 37 , Φ 39 , Φ 43 , Φ 58 , Φ 60 or Φ 80 (resp. Φ 106 or Φ 114 ), then C ( G 1 ) and C ( G 2 ) are stably C -isomorphic with Br nr ( C ( G i ) ) ≃ C 2 . Explicit structures of non-rational fields C ( G ) are given for each cases including also the case Φ 30 with Br nr ( C ( G ) ) ≃ C 2 × C 2 . [ABSTRACT FROM AUTHOR]

Let ( H , R ) be a quasitriangular weak Hopf algebra over a field k . We show that there is a braided monoidal isomorphism between the Yetter–Drinfeld module category YD H H over H and the category of comodules over some braided Hopf algebra H R in the category M H . Based on this isomorphism, we prove that every braided bi-Galois object A over the braided Hopf algebra H R defines a braided autoequivalence of the category YD H H if and only if A is quantum commutative. In case H is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of YD H H trivializable on M H is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in M H form a group measuring the Brauer group of ( H , R ) as studied in [21] in the Hopf algebra case. [ABSTRACT FROM AUTHOR]

We investigate the Brauer groupBRM(k,H) of a cocommutative Hopf algebraH(Hcan be nonfinitely generated).BRM(k,H) is the bigger Brauer group ofH-module Taylor–Azumaya algebras (which do not necessarily contain a unit). An exact sequenceis obtained. Furthermore, we prove the surjectivity ofunder certain circumstances. This generalizes Beattie's exact sequence to the “infinite” case. [ABSTRACT FROM PUBLISHER]

We study algebra classes and divisor classes on a normal affine surface of the formz2 = f(x,y). The affine coordinate ring isT = k[x,y,z]/(z2 − f), and ifR = k[x,y][f−1] andS = R[z]/(z2 − f), thenSis a quadratic Galois extension ofR. If the Galois group isG, we show that the natural map H1(G, Cl(T)) → H1(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H1(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples. [ABSTRACT FROM PUBLISHER]