By José Bertin (auth.), Pierre Dèbes, Michel Emsalem, Matthieu Romagny, A. Muhammed Uludağ (eds.)

This Lecture Notes quantity is the fruit of 2 research-level summer season colleges together prepared through the GTEM node at Lille collage and the workforce of Galatasaray college (Istanbul): "Geometry and mathematics of Moduli areas of Coverings (2008)" and "Geometry and mathematics round Galois conception (2009)". the quantity makes a speciality of geometric equipment in Galois concept. the alternative of the editors is to supply a whole and complete account of contemporary issues of view on Galois concept and similar moduli difficulties, utilizing stacks, gerbes and groupoids. It comprises lecture notes on étale primary team and primary crew scheme, and moduli stacks of curves and covers. examine articles entire the collection.

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**Example text**

Our aim now is to illustrate the descent formalism in two very special cases. The ﬁrst one, elementary, is the descent of schemes along a Zariski cover, and the second one is the so-called Galois descent of schemes. These two examples can help to understand the deﬁnition of stacks (coming soon). Let ???? ∈ Sch a scheme. Its functor of points ℎ???? : Schop → Set, ???? → ℎ???? (????) = HomSch (????, ????), is clearly a Zariski sheaf, indeed an fppf sheaf. In other words if ???? = ∪???? ???????? is an open cover of ???? ∈ Aﬀ, then the following diagram with obvious arrows is exact ∏ GG ∏ ℎ (???? ∩ ???? ).

51 that we get the inclusions: ´ Schemes ⊂ Algebraic spaces ⊂ Etale sheaves. 18 Finally a last comment. For a ﬁeld ????, call ????-point of an algebraic space ???? a point of ???? (????). An ´etale neighborhood of a ????-point ???? is an ´etale map (????, ★) → ???? of a pointed aﬃne scheme ????, ★ ∈ ???? (????), and ★ → ????. Now taking the inductive limit of the local rings ????????,★ , we can deﬁne the local ring of ???? at ???? (see [4] or [62] Deﬁnition 18 For simplicity, we write ???? (????) instead of ???? (Spec ????). Algebraic Stacks with a View Toward Moduli Stacks of Covers 39 04KG).

47. ∑ Let ???? be a commutative ring, and (????1 , . . , ???????? ) a family ∏???? of elements of ???? ???????? . Prove that the canonical morphism ???? → such that ???? = ???? ???? ????=1 ????=1 ???????????? is faithfully ﬂat. 48. 45, let (???????? ) be an open covering of ????. Denote ????????′ , ????????′ the corresponding open subschemes of ???? ′ , ???? ′ . Then a descent datum ???? : ∼ ????∗1 (???? ′ ) → ????∗2 (???? ′ ) is eﬀective if and only if for each ????, the induced descent datum ???????? on ′ ???????? relatively to ???????? : ????????′ → ???????? is eﬀective. 49.