Benjamin Antieau
Galois theory of difference equations with periodic parameters.
Wed, 09/08/2010 - 15:01 — SomNambulAuthors: Benjamin Antieau, Alexey Ovchinnikov, Dmitry Trushin
Subjects: Commutative Algebra
AbstractWe develop a Galois theory for systems of linear difference equations with
periodic parameters, for which we also introduce linear difference algebraic
groups. We then apply this to constructively test if solutions of linear
q-difference equations, with complex q, not a root of unity, satisfy any
polynomial q'-difference equations with q' being a root of unity. In
particular, we provide a detailed analysis of such relations satisfied by
Jacobi's theta-function.- 3 comments
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Cech approximation to the Brown-Gersten spectral sequence.
Mon, 12/21/2009 - 16:01 — SomNambulAuthors: Benjamin Antieau
Subjects: K-Theory and Homology
AbstractIn this paper, it is established that the spectral index of a cohomological
Brauer class is divisible by the its period. This relies on the establishment
of a theorem on Cech approximation to the Brown-Gersten spectral sequence via
Cech cohomology of presheaves. This theorem is most likely known, but the
author knows of no published proof.Cohomological obstruction theory for Brauer classes and the period-index problem.
Tue, 09/15/2009 - 11:27 — SomNambulAuthors: Benjamin Antieau
Subjects: Algebraic Geometry
AbstractLet $U$ be a noetherian, quasi-compact, and connected scheme. Let $\alpha$ be
a class in $H^2(U_{et},G_m)$. For each positive integer $m$, we use the
$K$-theory of $\alpha$-twisted sheaves to identify obstructions to $\alpha$
being representable by an Azumaya algebra of rank $m^2$. We define the spectral
index of $\alpha$, denoted $spi(\alpha)$, to be the least positive integer such
that all of the associated obstructions vanish. Let $per(\alpha)$ be the order
of $\alpha$ in $H^2(U_{et},G_m)$.
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