Unramified division algebras do not always contain Azumaya maximal orders

Antieau, Benjamin; Williams, Ben

doi:10.1007/s00222-013-0479-7

Cited by 17 publications

(44 citation statements)

References 14 publications

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“…This paper is one of a sequence, [1,3,4], in which classical homotopy theory is used to derive statements about and derive intuition for the study of division algebras, quadratic forms and the Brauer group of rings and schemes. We begin with an overview of the context of the paper, the reader is referred to [1] for further details.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…from which it follows that the extension K(Z, 5) → BP c (n, rn) [4] → BP c (n, rn) [3] is classified by r times a generator of H 5 (K(Z/n, 2), Z). We deduce, from a Serre spectral sequence, for instance, that H i (BP c (n, rn), Z) = 0 for i = 1, 2, that H 3 (BP c (n, rn), Z) ∼ = H 2 (BP c (n, rn), Z/n) = H 2 (BP c (n, rn) [3], Z/n) = H 2 (K(Z/n, 2), Z/n) = Z/n · ξ, and also that H 5 (BP c (n, rn), Z) = H 5 (BP c (n, rn) [4], Z) = Z/r · Q(ξ),…”

Section: Smooth Complex Varietiesmentioning

confidence: 99%

“…Bh [4] ( ( P P P P P P P P P P P P BPUrn [4] K(Z/n, 2) = BPUn [3] Bh [3] ( ( P P P P P P P P P P P P where the horizontal maps are classifying maps for the vertical maps, which are K(Z, 4)bundle maps. The map indicated by ×r is the map obtained by applying the functor K(·, 5) to π 4 (BPU n ) → π 4 (BPU rn ), which is an inclusion of Z in Z given by 1 → r, and consequently the map in the diagram represents the operation ×r on H 5 (·, Z).…”

Section: Bpun[4]mentioning

confidence: 99%

“…Associated to the fibration BPU n [4] → K(Z/n, 2), we have a Serre spectral sequence for integral cohomology, part of the E 2 -page of which is illustrated in Figure 2. Therefore, H 5 (BPU n [4], Z) = (Z/ (n))/ im(f n ).…”

Section: Bpun[4]mentioning

confidence: 99%

“…In the case where n > 2, because π 5 (BPU n ) = 0, it follows that BPU n [4] BPU n [5], and so H 5 (BPU n [4], Z) = H 5 (BPU n , Z) = 0, and f n is consequently surjective. In the case where n = 2, we discover another extension problem, associated to the principal fibration K(Z/2, 5) → BPU 2 [5] → BPU 2 [4]. Since H 5 (BPU 2 [5], Z) = H 5 (BPU 2 , Z) = 0, there is an exact sequence 0 → Z/4/ im f 2 → 0 → Z/2 → * .…”

Section: Bpun[4]mentioning

confidence: 99%

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The topological period-index problem over 6-complexes

Antieau

Williams

2013

Journal of Topology

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By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

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Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Section: Smooth Complex Varietiesmentioning

confidence: 99%

Section: Bpun[4]mentioning

confidence: 99%

Section: Bpun[4]mentioning

confidence: 99%

Section: Bpun[4]mentioning

confidence: 99%

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The topological period-index problem over 6-complexes

Antieau

Williams

2013

Journal of Topology

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Period-index bounds for arithmetic threefolds

Antieau

Auel

Ingalls³

et al. 2019

Invent. math.

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The standard period-index conjecture for Brauer groups of p-adic surfaces S predicts that ind(α)| per(α) 3 for every α ∈ Br(Qp(S)). Using Gabber's theory of prime-toℓ 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.If α has period prime to 6p, then ind(α) | per(α) 4 .Recall from [38] that k is a semi-finite field if it is perfect and if for every prime ℓ, the maximal prime-to-ℓ extension of k is pseudo-algebraically closed with Galois group Z ℓ . Finite fields and pseudo-finite fields are semi-finite. As a special case, we obtain the following result.Corollary 1.2. Let S be a geometrically integral surface over a p-adic field K. If α ∈ Br(K(S)) has period relatively prime to 6p, then ind(α) | per(α) 4 . 1 that for a surface over C((t)) one has ind(α)| per(α) 2 , while for a surface over a p-adic field one has ind(α)| per(α) 3 . The nonoptimality of our results is undoubtedly due to our overly generous splitting of ramification. The approach taken in [38] improves these kinds of bounds at the expense of a layer of stacky complexity.Outline. In Section 2, we generalize work of Pirutka [43] on splitting the ramification of Brauer classes. Section 3 considers Gabber's refined theory of ℓ ′ -alterations in the context of splitting ramification. Sections 4 and 5 discuss the existence and deformation theory of twisted sheaves on proper models of the function field we consider. Theorems 1.1 and 1.4 are proved in Section 6. Starting in Section 4, we freely use the theory of twisted sheaves. An introduction to the use of twisted sheaves to study questions about the Brauer group can be found in [35] and [36].Notation. If X is a scheme and R is a commutative ring, we denote by H i (X, µ n ), H i (R, G m ), and so on the correspondingétale cohomology groups, and by Br(X) and Br(R) the respective Brauer groups of Azumaya algebras. Given a locally noetherian scheme X and a G-gerbe π : X → X for some closed subgroup G ֒→ G m , we will write Coh (1) (X) for the category of coherent X-twisted sheaves. Similarly, given a locally noetherian scheme X, we will write Coh(X) for the usual categories of coherent sheaves on X. When F is an X-twisted sheaf and M is an O X -module, for simplicity we write F ⊗ M for the X-twisted sheaf F ⊗ O X π * M .

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An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

First

2022

manuscripta math.

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Unramified division algebras do not always contain Azumaya maximal orders

Cited by 17 publications

References 14 publications

The topological period-index problem over 6-complexes

The topological period-index problem over 6-complexes

Period-index bounds for arithmetic threefolds

An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

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