# Categories and Grothendieck Topologies [Lecture notes] by Jörg Zintl

By Jörg Zintl

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Second, by specifying a topology on the full subcategory of affine schemes 30 Aff . 2]. Let C be a category and C be a full subcategory of C, both of them admitting fibre products which are chosen to be compatible with respect to the inclusion functor. Suppose that we are given a pretopology P on C and a pretopology P on C . ). Suppose furthermore that the following two conditions are satisfied. (T1) For all objects X ∈ Ob (C) there exists a covering family {fα : Xα → X}α∈A ∈ Cov (X), such that Xα ∈ Ob (C ) for all α ∈ A .

G ~~ γ Proof. 34. 40. By property (T2) from above there exists a family {hδ : Zδ → X}δ∈D ∈ Cov (X) such that Zδ QQQ QQQ QQQhδ QQQ QQQ  fα,β fαQQ( /X / Xα,β Xα > DD ~ ~ DD ~ ~ DD ~~ g DD ~~ γ " Yγ commutes, and this is doing the job. 3 Standard topologies Let C = Sch, and let C = Aff be the full subcategory of affine schemes. Let the pretopology on C be given by the Zariski topology. In this section we shall describe the most frequently used topologies on the category of schemes. 42 Definition.

Let X ∈ Ob (C ) be an affine scheme. We define the following sets of families of morphisms. (i) Cov etf (X) := {fα : Xα → X}α∈A : fα ´etale, finite, Xα → X surjective α∈A (ii) Cov et (X) := {fα : Xα → X}α∈A : fα ´etale, Xα → X surjective α∈A    {fα : Xα → X}α∈A : fα flat, quasi-finite, of finite presentation, (iii) Cov f ppf (X) :=  Xα → X surjective  α∈A (iv) Cov f pqc (X) := {fα : Xα → X}α∈A : fα flat, Xα → X surjective      . e. only finite sets A of indices, are permitted. 43 Remark.