Equivariant Vector Bundles and Rigid Cohomology on Drinfeld's Upper Half Space over a Finite Field
Kuschkowitz
Mark
Kuschkowitz, Mark
aut
2016-07-12
2016-08-22
en
<p>Fix a finite field k and let X be Drinfeld's Upper Half Space over k of dimension n. Let S be the polynomial ring over k in n+1 variables, let G be the algebraic k-group scheme associated with the general linear group of degree n+1 and let Y=Proj S. Fix an algebraic action of G on Y.
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In the first part of this thesis, the global sections F(X) on X of a G-equivariant vector bundle F on Y are considered as a G(k)-representation. Due to a theorem of Orlik (2008), the computation of F(X) essentially reduces to the computation of the local cohomologies of F on Y with support in a k-rational closed subvariety Z. This local cohomology is considered as a representation of a certain parabolic subgroup P of G which stabilizes Z resp. of its Levi subgroup L. Three types of descriptions of these local cohomology modules are given:
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In the case where F arises from a representation of the stabilizer of a chosen base point of Y, a very general result on the structure of the local cohomology of F on Y with support in Z as an L-module is translated from Orlik's results.
For general F, an even coarser strucutre result is proved by using an affine projection of Y onto Y-Z. In particular, when F equals the structure sheaf or a sheaf of differential forms on Y (resp. a Serre twist of either sheaf), this result is made very precise.<br>
In the case where the graded S-module associated with F is generated in degrees <2, a higher divided power version of the distribution algebra of G is used to for a more conceptual approach to the description of the local cohomologies mentioned in terms of the unipotent radical of P.
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In the second part of this thesis, the rigid cohomology of X is computed in two ways.
<br><br> The first method proceeds by computation of the rigid cohomology of the complement
of X in Y (which is projective itself, thus its rigid cohomology is simply the de Rham cohomology of an associated rigid-analytic tube). Then application of the associated long exact sequence for rigid cohomology with proper supports yields the rigid cohomology of X.
<br><br> The second method proceeds by direct computation of the direct limit of the de Rham cohomologies of a certain cofinal family of strict open neighborhoods of the tube of X in the ambient rigid-analytic projective space.
<br><br> The resulting cohomology formula has been known since 2007, when Große-Klönne proved that it is the same as the one obtained from l-adic cohomology using different methods.
urn:nbn:de:hbz:468-20160822-115624-2
2016-08-22T09:56:20.114Z
2016-08-22T10:18:15.795Z
published
Diss
fbc/mathematik/diss2016/kuschkowitz