Curves on quasi-schemes
Let \(X\) be a non-commutative scheme. A Grothendieck category of quasi-coherent sheaves on \(X\) is called a quasi-scheme and in a certain sense it can be identified with \(X\). Suppose that there is a regularly embedded hypersurface \(Y\subset X\). The authors investigate curves on \(X\) which are in general position with respect to \(Y\). In particular it is shown that the category of quasi-coherent sheaves on such a curve \(C\) is isomorphic to a certain quotient category of graded modules over the \(n\)-dimensional commutative polynomial ring graded by \(\mathbb{Z}^n\), where \(n\) is the number of points of the intersection \(C\cap Y\). The results obtained are applied to some interesting examples of non-commutative schemes such as the enveloping algebra of the two-dimensional non-Abelian Lie algebra, the quantum affine and projective planes, the quantum projective space of \textit{M. Vancliff} [J. Algebra 165, No. 1, 63-90 (1994; Zbl 0837.16023)], and others.
scientific article; zbMATH DE number 1287957