Fibers in Ore Extensions

Let R be a finitely generated commutative algebra over an algebraically closed field k and let A=R[t;,] 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 A¹×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 k[x] 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 Modk[x]. It is also shown that X is the disjoint union of the fibers X_p in a certain sense.