Topological methods for complex-analytic Brauer groups

Schröer, Stefan

doi:10.1016/j.top.2005.02.005

Cited by 28 publications

(37 citation statements)

References 51 publications

(63 reference statements)

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“…The Brauer groups exist in complex analytic topological spaces and the equality between Brauer groups in view of geometry is evaluated in [5]. If the complex analytic topological space X = Y an is compact, then the canonical map between Brauer groups f : Br(Y) → Br(X) is invertible.…”

Section: Topology Of Complex Analytic Spacesmentioning

confidence: 99%

On the Topological Structure and Properties of Multidimensional (C, R) Space

Bagchi

2020

Symmetry

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Generally, the linear topological spaces successfully generate Tychonoff product topology in lower dimensions. This paper proposes the construction and analysis of a multidimensional topological space based on the Cartesian product of complex and real spaces in continua. The geometry of the resulting space includes a real plane with planar rotational symmetry. The basis of topological space contains cylindrical open sets. The projection of a cylindrically symmetric continuous function in the topological space onto a complex planar subspace maintains surjectivity. The proposed construction shows that there are two projective topological subspaces admitting non-uniform scaling, where the complex subspace scales at a higher order than the real subspace generating a quasinormed space. Furthermore, the space can be equipped with commutative and finite translations on complex and real subspaces. The complex subspace containing the origin of real subspace supports associativity under finite translation and multiplication operations in a combination. The analysis of the formation of a multidimensional topological group in the space requires first-order translation in complex subspace, where the identity element is located on real plane in the space. Moreover, the complex translation of identity element is restricted within the corresponding real plane. The topological projections support additive group structures in real one-dimensional as well as two-dimensional complex subspaces. Furthermore, a multiplicative group is formed in the real projective space. The topological properties, such as the compactness and homeomorphism of subspaces under various combinations of projections and translations, are analyzed. It is considered that the complex subspace is holomorphic in nature.

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Section: Topology Of Complex Analytic Spacesmentioning

confidence: 99%

On the Topological Structure and Properties of Multidimensional (C, R) Space

Bagchi

2020

Symmetry

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“…It follows from [28,Proposition 1.3] that Br ′ (X) is isomorphic to the analytic cohomological Brauer group Br ′ (X an ), the torsion part of H 2 (X an , O × Xan ). The exponential sequence 0 → Z → O Xan → O × Xan → 1 induces the following exact sequence:…”

Section: A Characterization In Terms Of Divisor Class Groups and Braumentioning

confidence: 99%

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

Srinivas

Takagi

2017

Advances in Mathematics

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Abstract. An F -nilpotent local ring is a local ring (R, m) of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules H i m (R). A singularity in characteristic zero is said to be of F -nilpotent type if its modulo p reduction is F -nilpotent for almost all p. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of F -nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups.

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“…• The best result for complex spaces is the result of Schröer [47] which gives a purely topological condition that ensures Br(X an ) = Br ′ (X an ). This condition applies to complex Lie groups, Hopf manifolds, and all compact complex surfaces except for a class which conjecturally does not exist, namely the class VII surfaces that are not Hopf surfaces, Inoue surfaces, or surfaces containing a global spherical shell.…”

Section: A Problem Of Grothendieckmentioning

confidence: 99%

The period-index problem for twisted topologicalK–theory

2014

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We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d , we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K -theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/ℓ, 2), where ℓ is a prime, we construct a sequence of spaces with an order ℓ class in the Brauer group, but whose indices tend to infinity.19L50, 16K50, 57T10, 55Q10, 55S35 IntroductionThis paper gives a solution to a period-index problem for twisted topological K -theory. The solution should be viewed as an existence theorem for twisted vector bundles.Let X be a connected CW-complex. An Azumaya algebra A of degree n on X is a noncommutative algebra over the sheaf C of complex-valued functions on X such that A is a vector bundle of rank n 2 and the stalks are finite dimensional complex matrix algebras M n (C). Examples of Azumaya algebras include the sheaves of endomorphisms of complex vector bundles and the complex Clifford bundles Cl(E) of oriented evendimensional real vector bundles E . The Brauer group Br(X) classifies topological Azumaya algebras on X up to the usual Brauer equivalence: A 0 and A 1 are Brauer equivalent if there exist vector bundles E 0 and E 1 and an isomorphismof sheaves of C -algebras. Define Br(X) to be the free abelian group on isomorphism classes of Azumaya algebras modulo Brauer equivalence.The period-index problem for twisted topological K -theory 13 Proposition 2.5 If A is an Azumaya algebra of degree n with class α, then A ∼ = End(E) for some α-twisted vector bundle E of rank n. The α-twisted sheaf is unique up to tensoring with untwisted line bundles. Proof See [16, Theorem 1.3.5] or [35, Section 3]. Proposition 2.6 Tensoring with E * , the dual of E , induces an equivalence of categories Vect α → Vect A . Proof See [16, Theorem 1.3.7] or [35, Section 3]. Proposition 2.7 If A and B are Brauer-equivalent Azumaya algebras over X , then Vect A and Vect B are equivalent categories.Proof This follows from the previous two propositions.If X is a finite CW-complex, the twisted topological K -group KU 0 (X) α may be identified with the Grothendieck group of left A-modules.Proposition 2.8 If X is compact and Hausdorff, and if A is an Azumaya algebra on X with class α (so that α is torsion), then K A 0 (X) ∼ = KU 0 (X) α . This isomorphism is uniquely defined up to the natural action of H 2 (X, Z) on the left.Proof This follows from [9, Section 3.1]. In fact, they show that KU ...

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Topological methods for complex-analytic Brauer groups

Cited by 28 publications

References 51 publications

On the Topological Structure and Properties of Multidimensional (C, R) Space

On the Topological Structure and Properties of Multidimensional (C, R) Space

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

The period-index problem for twisted topologicalK–theory

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