Fibers in Ore Extensions |
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Authors: | S Paul Smith James J Zhang |
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Institution: | (1) Department of Mathematics, University of Washington, Seattle, WA, 98195, U.S.A |
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Abstract: | Let R be a finitely generated commutative algebra over an algebraically closed field k and let A=Rt;,] be the Ore extension with respect to an automorphism and a -derivation . We view A as the coordinate ring of an affine noncommutative space X. The inclusion RA gives an affine map : XSpecR, and X is a noncommutative analogue of A
1×SpecR. We define the fiber X
p
of over a closed point pSpecR as a certain full subcategory ModX
p
of ModA. The category ModX
p
has the following structure. If p has infinite -orbit, then ModX
p
is equivalent to the category of graded modules over the polynomial ring kx] with degx=1. If p is not fixed by , but has finite -orbit, say of size n, then ModX
p
is equivalent to the representations of the quiver Ã
n–1 with the arrows all going in the same direction. If p is fixed by , then ModX
p
is equivalent to either Modk or Modkx]. It is also shown that X is the disjoint union of the fibers X
p
in a certain sense. |
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Keywords: | Ore extension fibers noncommutative algebraic geometry |
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