Period-index bounds for arithmetic threefolds

Abstract

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 .

The period-index problem for real surfaces

“…We collect here a few facts that will be used in §3. 3 A local computation at points x ∈ S 0 (C) \ S * (C) shows that j 0…”

“…Two outstanding results are de Jong and Lieblich's theorems on function fields of surfaces over algebraically closed ([25], see also [44,Theorem 4.2.2.3]) or finite fields [45, Theorem 1.1] (see [3] for results on function fields of p-adic surfaces).…”

Splitting Brauer classes using the universal Albanese

“…Antieau and Auel's proof of Theorem 1.0.2. A different proof of Theorem 1.0.2, due to Benjamin Antieau and Asher Auel [AA18], uses results on the stable birational geometry of symmetric powers of Brauer-Severi varieties to show that an appropriate symmetric power of a curve splitting a Brauer class also splits the class. Here is a sketch of their proof; more details may appear elsewhere in the future.…”