Twisted sheaves and the period-index problem
By Max Lieblich. Compositio Mathematica, 2005
158 citations recorded by the source corpus.
Abstract We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic zero, which (among other things) extends de Jong’s result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.
Subfields: Geometry and Topology, Mathematical Physics, Pharmacology
View Paper: https://www.semanticscholar.org/paper/f9ad94ccf2215383c7ebcf13c20943056e3bb62f
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In Geometry and Topology
- D. Huybrechts
- Max Lieblich
- Martin Olsson
- Benjamin Antieau
- B. Williams
- Emanuele Macrì
- P. Stellari
- Bumsig Kim
In Mathematical Physics
- D. Huybrechts
- Max Lieblich
- Martin Olsson
- Benjamin Antieau
- B. Williams
- Emanuele Macrì
- P. Stellari
- Yukinobu Toda
In Pharmacology
Citations in the corpus
- The topological period–index problem over 6‐complexes (2012, background): “For a given Brauer class α, there is a notion of α-twisted coherent sheaf, see [21] for particulars.”
- Lectures on K3 Surfaces (2016, methodology): “For function fields of surfaces and for surfaces over finite fields this has been addressed by de Jong [135] and Lieblich [364].”
- Period and index in the Brauer group of an arithmetic surface (2007, methodology): “In this section, we will construct a division algebra over K with period p and index pr.”
- Fourier-Mukai partners of K3 surfaces in positive characteristic (2011, background): “of M is ismorphic to Y ×Bµr. Since r is relatively prime to p, we have that the Brauer class associated to the gerbe M → Mis trivial. In particular, there is an invertible M-twisted sheaf L on M (see [L2] for basic results on twisted sheaves). Now let V be the universal twisted sheaf on M ×R Xand V ”