Curves on Quasi-schemes

This paper concerns curves on non-commutative schemes, hereafter called quasi-schemes. A quasi-scheme X is identiied with the category ModX of quasi-coherent sheaves on it. Let X be a quasi-scheme having a regularly embedded hypersurface Y. Let C be a curve on X which is in \good posi-tion" with respect to Y (see Deenition 5.1)|this deenition includes a requirement that X be far from commutative in a certain sense. Then C is isomorphic to V 1 n , where n is the number of points of intersectionof C with Y. Here V 1 n , or rather ModV 1 n , is the quotient category GrModkx 1 ; : : : ; xn ]=fKdim n?2g of Z n-graded modules over the commutative polynomialring, modulo the subcat-egory of modules having Krull dimension n?2. This is a hereditarycategory which behaves rather like ModP 1 , the category of quasi-coherent sheaves on P 1. 0. Introduction There are several motivations for this paper, one of which is suggested by the title. Non-commutative algebraic geometry has not yet developed a conceptual framework which allows one to say, for example, that a curve in one space (quasi-scheme) is isomorphic to a curve in another space. This gap is particularly apparent when one considers line modules. One wishes to think of line modules as lines, and then to decide whether the lines in one quantum projective space are isomorphic to those in another. There are at least two reasons this has not been done: rst, it is unclear how to deene a morphism between schemes; second a line module is a single module whereas a space (quasi-scheme) is an abelian category (Deenition 1.1). This paper is concerned with the second issue. We can avoid the rst issue since our concern is to show that certain pairs of quasi-schemes are isomorphic, and it is clear that isomorphism should be deened in terms of equivalence of categories. The following simple question illustrates the problems surrounding the second issue. Let R be a ring with a right ideal I. How should we associate to R=I a full abelian subcategory of ModR, which is a substitute for ModR=I which only exists when I is a two-sided ideal? If one has a sound recipe for producing such a category, one can then compare R=I with S=J, where J is a right ideal in another ring S, by comparing the categories \ModR=I" and …