* Principal bundles
@ 2013-07-21 9:47 Johannes Huebschmann
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From: Johannes Huebschmann @ 2013-07-21 9:47 UTC (permalink / raw)
To: categories
Dear Colleagues
Here is a question.
An answer results perhaps from combining Gelfand and Tannaka duality.
Many thanks in advance
Best wishes
Johannes
Let $X$ be an affine variety
with coordinate ring $B$
endowed with an action
$X\times G \to X$ of an affine group $G$,? and let $Y$
denote the quotient, with coordinate ring $A=B^G$.
By a theorem of Oberst \cite{MR0444680},
the projection $p\colon X \to Y$ is an affine principal fiber bundle
if and only if the functor
from $(G,B)$-modules to $A$-modules
which assigns to a? $(G,B)$-module $M$ the invariant subspace $M^G$
is an equivalence of categories.
Is there a similar characterization of a topological
or smooth principal bundle with structure group
a (presumably compact?)? Lie group
in the literature?
?
@article {MR0444680,
AUTHOR = {Oberst, Ulrich},
TITLE = {Affine {Q}uotientenschemata nach affinen, algebraischen
{G}ruppen und induzierte {D}arstellungen},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {44},
YEAR = {1977},
NUMBER = {2},
PAGES = {503--538},
ISSN = {0021-8693},
MRCLASS = {14L99},
MRNUMBER = {0444680 (56 \#3030)},
MRREVIEWER = {T. Kambayashi},
}
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