The Stacks project

35.1 Introduction

In the chapter on topologies on schemes (see Topologies, Section 34.1) we introduced Zariski, ├ętale, fppf, smooth, syntomic and fpqc coverings of schemes. In this chapter we discuss what kind of structures over schemes can be descended through such coverings. See for example [Gr-I], [Gr-II], [Gr-III], [Gr-IV], [Gr-V], and [Gr-VI]. This is also meant to introduce the notions of descent, descent data, effective descent data, in the less formal setting of descent questions for quasi-coherent sheaves, schemes, etc. The formal notion, that of a stack over a site, is discussed in the chapter on stacks (see Stacks, Section 8.1).


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