Grothendieck topology

Artin, M. Grothendieck Topologies (mimeographed notes, Harvard, 1962). 

In these notes, a site (i.e., a category with a Grothendieck topology, in the sense of [42]) is required to be a small category whose topology is defined by a pretopology (see [42]). 

Recall that a site is a category with a Grothendieck topology, see [13, 11.1.1.5]. All the sites we consider in this paper are essentially small (equivalent to a small category) and, to simplify the exposition, we always assume they have enough points (see [13]). 

These lectures were intended for graduate students who have basic knowledge on algebraic geometry and ordinary topology. The only results which will be used but not proved in this note are Grothendieck s construction of the Hilbert scheme (Theorem 1.1) and results on intersection cohomology ( 6.1). I recommend to the reader to accept these results when he/she is not familiar with them. 

Un foncteur contravariant defini sur Sch/S, A valeurs dans les ensembles, est aussi appeltf un pr faisceau sur S. Nous aurons besoin de r sultats elemen-taires sur les topologies de Grothendieck (cf. SGA 3 IV et SGA 4). Nous utilise-rons les trois topologies suivantes ... 

There are a number of treatments of Grothendieck duality for the Zariski topology (not to mention other contexts, see e.g., [Gfi], [De], [LO]), for example, Neeman s approach via Brown representability [N], Hashimoto s treatment of duality for diagrams of schemes (in particular, schemes with group actions) [Hsh], duality for formal schemes [AJL ], as well as various substantial enhancements of material in Hartshorne s classic [H], such as [C], [S], [LNS] and [YZ]. Still, some basic results in these notes, such as Theorem (3.10.3) and Theorem (4.4.1) are difficult, if not impossible, to find elsewhere, at least in the present generality and detail. And, as indicated below, there are in these notes some significant differences in emphasis. 

In this chapter we review and elaborate on—with proofs and/or references— some basic abstract features of Grothendieck Duality for schemes with Zariski topology, a theory initially developed by Grothendieck [Gr ], [H], [C], Dehgne [De ], and Verdier [V ]. The principal actor in this Chapter is the twisted inverse image pseudofunctor, described in the Introduction. The basic facts about this pseudofunctor—which may be seen as the main results in these Notes—are existence and flat base change, Theorems (4.8.1) and (4.8.3). 

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