In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that ...

Via the BGG-correspondence a simplicial complex ∆ on [n] is transformed into a complex of coherent sheaves L̃(∆) on the projective space P. In general we compute the support of each of its cohomology sheaves. When the Alexander dual ∆ is Cohen-Macaulay there is only one such non-zero cohomology sheaf. By considering when this sheaf can be an a’th syzygy sheaf in a locally free resolution, we get...

We define a notion of flexible functor from a category X to the category of topological spaces (or other similar categories). This is a generalization of the notion of functor, where the condition of compatibility with composition is replaced by the data of homotopies, homotopies between the homotopies, and so on. When X is a site, we define a sheaf condition and show that a flexible functor (t...

1.1. Crash course in blowing up. Last time I began to talk about blowing up. Let X be a scheme, and I ⊂ OX a sheaf of ideals on X. (Technical requirement automatically satisfied in our situation: I should be a coherent sheaf, i.e. finitely generated.) Here is the “universal property” definition of blowing-up. Then the blow-up of OX along I is a morphism π : X̃ → X satisfying the following univer...