The Geometry of Noncommutative Plane Curves

摘要

We consider (noncommutative) affine algebras over algebraically closed fields. A 1-dimensional representations of such an algebra, A, corresponds to a point on an algebraic set in an affine space. Different points of (nonisomorphic 1-dimensional simple A-modules) can have nonzero “Ext”-groups. We will show these groups will in many cases give information back to the algebra.In this note, we mainly focus on algebras of the form k x, y /(f).We show that if such an algebra has a representation x → a and y → b for all points P = (a, b) in the plane and nonzero “Ext” for all such, then f  ([x, y])2.Another case we can solve completely. If an algebra A = k x, y /(f), where f  ([x, y]) has for all points in the plane, then f = [x, y] + f ₀, where f ₀  ([x, y])2.We show by examples that the 1-dimensional representations and their corresponding “Ext”-groups can be the same, but the higher dimensional simple representations are in general quite different. We also prove that each curve has a model with an n-dimensional simple representation for any n  1.View full textDownload full textKey WordsExtensions, Ext-relations, Modules, Plane curves2000 Mathematics Subject Classification14A22, 14H50, 14R, 16D60, 16G30Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00927870802107892