Curves on Quasi-Schemes

Abstract

This paper concerns curves on noncommutative schemes, hereafter called quasi-schemes. Aquasi-scheme X is identified with the category $$Mod{\text{ }}X$$ ofquasi-coherent sheaves on it. Let X be a quasi-scheme having a regularly embeddedhypersurface Y. Let C be a curve on X which is in ‘good position’ withrespect to Y (see Definition 5.1) – this definition includes a requirement that Xbe far from commutative in a certain sense. Then C is isomorphic to $$\mathbb{V}_n^1 $$ , where n is the number of points of intersection of Cwith Y. Here $$\mathbb{V}_n^1 $$ , or rather $$Mod{\text{ }}\mathbb{V}_n^1 $$ , is the quotient category $$GrModk[x_1 , \ldots ,x_n ]/\{ {\text{K}}\dim \leqslant n - 2\} {\text{ of }}\mathbb{Z}^n $$ -graded modules over the commutative polynomial ring, modulo the subcategory ofmodules having Krull dimension ≤ n − 2. This is a hereditary category whichbehaves rather like $$Mod\mathbb{P}^1 $$ , the category of quasi-coherentsheaves on $$\mathbb{P}^1 $$ .