The bigger Brauer group and twisted sheaves

AbstractWe 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.

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The Equivariant Brauer Group of a Cocommutative Hopf Algebra

Communications in Algebra ◽

10.1080/00927872.2012.692417 ◽

2013 ◽

Vol 41 (11) ◽

pp. 4161-4194

Author(s):

J. Dello ◽

Y. H. Zhang

Keyword(s):

Hopf Algebra ◽

Brauer Group ◽

Cocommutative Hopf Algebra

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NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.43 ◽

2020 ◽

pp. 1-25

Author(s):

CHIARA CAMERE ◽

ALBERTO CATTANEO ◽

ANDREA CATTANEO

Keyword(s):

Moduli Spaces ◽

Quadratic Forms ◽

Symplectic Manifolds ◽

Hilbert Schemes ◽

Small Rank ◽

Twisted Sheaves ◽

Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

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