We study the cohomology and structure of the category of sheaves for the quasi-finite, flat Grothendieck topology over a prescheme X. (Definitions given in ? 1.) Of particular interest is the surprising result (Theorem 1) that the flat cohomological dimension is infinite in general. However, by obtaining a complete structure theorem (? 4) for the category of sheaves over Spec k, k a field, we show that the sheaves causing this unpleasant phenomenon can never be group schemes of finite type (? 5). In fact, the p-dimension of a field restricted to such group schemes is bounded by 2.