Na¨ive Noncommutative Blowups at Zero-dimensional Schemes

In an earlier paper [KRS] we defined and investigated the properties of the na¨ıve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one na¨ıvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in qgr-A is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper [RS1] to prove that a large class of noncommutative surfaces can be written as na¨ıve blowups.