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**Additional info for Categories and Grothendieck Topologies [Lecture notes]**

**Sample text**

First, by applying coverings by affine schemes only. 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 .

1]. 41 Corollary. If X ∈ Ob (C ), then a family U = {gγ : Yγ → X}γ∈C is a J-covering family of X if and only if there is a family {fα : Zα → X}α∈A ∈ Cov (X) such that for all α ∈ A there exists an index γ ∈ C and a commutative diagram fα Zα A AA AA AA A Yγ /X ~? ~ ~~ ~~ 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.

In particular E is a scheme together with a morphism π : E → X such that there exists a Zariski open cover {Uα }α∈A of X with π −1 (Uα ) ∼ = Uα ×Ar −1 and πα := π|π (Uα ) is just the projection pr1 to the first factor. Clearly, {iα : Uα → X}α∈A is a covering family with respect to PZar . Locally there are zero-sections sα : Uα → Uα × Ar ∼ = π −1 (Uα ) ⊂ E. Now we have a commutative diagram Uα B sα BB BB B iα BB ! X. /E } } }} }} π } }~ This shows that U := {π : E → X} is a JZar -covering family of X.