Brauer group of projective spaces tetsuya uematsu abstract. One of the main theorems on the schur subgroup of the brauer group is the brauer witt theorem. Brauer group of a moduli space of parabolic vector bundles over a curve article pdf available in journal of ktheory ktheory and its applications to algebra geometry and topology 803 may. Several notions, describing how close xis to projective space, have been developed. Brauer algebras and the brauer group noah schweber abstract.
Brauer group of moduli spaces of pglrbundles over a curve. In this paper we further generalize this concept and define the brauer group of a ringed space x, % where cu is a sheaf of commutative rings with unit. Rational, unirational and stably rational varieties. Every projective schur algebra is brauer equivalent to a radical. Galois theory and projective geometry 5 projective space subject to pappus axiom is a projectivization of a vector space over a. Brauer severi varieties are twisted forms of the projective space.
Such a space always exists, is a cw complex, kill higher homotopy groups via postnikov towers and is. This interpretation will enable us to give another, geometric construction of the brauer group. We show that the brauer group of any moduli space of stable pairs with fixed determinant over a curve is zero. Just as ktheory classi es vector bundles on a space, the brauer group will help classify projective bundles. Let x be a smooth complex projective curve with genusx2, and let m be the moduli space parametrizing isomorphism classes of stable vector bundles e over x of rank r with r e. Let xc be a connected smooth projective curve of genus g x. For a group g and a group a with gaction, there is a rst cohomology set h1g. Brauer groups and etale cohomology in derived algebraic. The brauer group of kummer surfaces and torsion of. The varieties x arising in this way are called severi brauer varieties. Manin published examples of rational surfaces that is, surfaces birational over. Related content the cohomological brauer group of a real algebraic variety vyacheslav a krasnovon the brauer group of an arithmetic scheme sergei g tankeev. Witt and hasse discovered the connection between csas and. The brauer group of a eld f, classifying central simple algebras over f, plays a critical role in class eld theory.
Going back to our description of the brauer group in terms of h1. The brauer group of kummer surfaces and torsion of elliptic curves alexei n. On the brauer group of a surface stanford university. In the proceedings of the international congress of mathematicians, amsterdam, 1954, r. In particular, we will take r kg, the group ring of a nite group g. The above description of the brauer group in terms of cohomology has a manifestation over schemes, but the story is not as clean. Every schur algebra is brauer equivalent to a cyclotomic algebra. In mathematics, the brauer group of a field k is an abelian group whose elements are morita equivalence classes of central simple algebras over k, with addition given by the tensor product of algebras. Skorobogatov may 28, 2019 to the memory of sir peter swinnertondyer abstract we present a method for calculating the brauer group of a surface given by a diagonal equation in projective space.
Cohomology and the brauer groups of diagonal surfaces dami an gvirtz and alexei n. Classifying projective bundles modulo line bundles this talk is about the brauer group. We establish various fundamental facts about brauer groups in this setting, and we provide a computational tool, which we use to compute the brauer group in several. Our results extend those of mukai on semihomogeneous vector bundles, and yield a geometric view of the brauer group of abelian varieties. Homogeneous projective bundles over abelian varieties. Brauer group of the moduli spaces of stable vector bundles. We compute the brauer group of the moduli stack of stable pgl rbundles on a curve x over an algebraically closed field of characteristic zero. A recent theorem of hoffmann shows that the picard group of the moduli space does not depend on the base field. The brauergrothendieck group imperial college london. Several notions, describing how close xis to projective space. Division algebras, the brauer group, and galois cohomology.
Brauer severi schemes are defined as etale locally trivial projective. The severibrauer variety associated to a central simple. Every projective schur algebra is brauer equivalent to a. The naive height on the projective space over 15 2. It is a famous consequence of tsens theorem that a smooth curve over an algebraically closed field has trivial brauer group. Introduction our main motivation for considering the chow group of 0cycles and the brauer group is the rationality problem in algebraic geometry. We show a wellknown result which states that the brauer group of a projective space over a eld is isomorphic to that of the base eld. Brauer group of moduli spaces of pglrbundles over a. An algebra is a vector space v over a eld k together with a kbilinear product of vectors under which v is a ring. The brauer group of the universal moduli space of vector bundles. Zarhin introduction in this paper we are interested in computing the brauer group of k3 surfaces. As with the projective bundles, there is a class of azumaya algebras which are. If y is the projective space pn, then the exactness of e for x implies its.
On the brauer group of a projective curve springerlink. Cohomology and the brauer groups of diagonal surfaces. This paper presents a direct, torsiontheoretic description of the brauer group o f a projective schemex. Let nr be the moduli stack of stable pglr,k bundles on x. Using the torsiontheoretic description of the brauer group o f a projective variety, an elegant, direct description of the brauer group of an arbitrary projective curve is established. This is not the rst place brauer groups appeared historically, but will be a convenient place to begin. A certain class of algebras, called brauer algebras algebras which split over a nite galois extension. It was defined by the algebraist richard brauer the brauer group arose out of attempts to classify division algebras over a field. Universal triviality of the chow group of 0cycles and the. We also show that this brauer group of such a moduli stack coincides with the brauer group of the smooth locus of the corresponding coarse moduli space of stable pgl rbundles on x. Classifying projective bundles modulo line bundles this talk is about the brauer group, an object of startling use and ubiquity.
Is every complex algebraic variety birationally equivalent to a complex projective manifold. It is given in terms of reflexive modules over the normalized curve and the brauer groups of. It can also be defined in terms of galois cohomology. Mcfaddin the severi brauer variety associated to a. Over a number eld k, the geometric picard group picx of a projective k3 surface xkis a free zmodule of rank 1. We first study the brauer group of a topological space, which can be. Brauerseveri varieties are twisted forms of the projective space. Brauerseveri schemes are defined as etale locally trivial projective. He focuses on comparing the cohomological brauer group and the brauer group in terms of azumaya algebras. A group cohomological interpretation of these examples was given by the.
To this end, we define and develop in section ii the properties of locally projective sheaves of. Brauer groups and galois cohomology of commutative ring. Our aim is to study the brauer group of nrand that of the smooth locus of nr. In each of these cases, the \ brauer group of a smooth projective model is a natural group attached to the variety, if it is nonzero, it tells us that the variety is not rational. The brauer group associated to the eld k, denoted brk, is the set of isomorphism classes. If xhas a krational point, then is the zero map, i. Division algebras, the brauer group, and galois cohomology robert hines january 5, 2015 introduction classifying or even nding noncommutative division algebras is a di cult task. The brauer group and the cohomological brauer group 101 7. Brauer groups, tamagawa measures, and rational points on. There are two corresponding streams of definitions and theorems allowing one to study the brauer group.
But what about curves over non algebraically closed fields. In particular, we nd an explicit formula for the brauer group of fourfolds bered in quadrics of dimension 2 over a rational surface. The brauer group brxof a smooth projective variety xis a fundamental object of study. Pdf brauer group of a moduli space of parabolic vector. Both groups are stable birational invariants of smooth projective varieties that have been used to great success.
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