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Section 29.36 : Étale Morphisms—The Stacks Project

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Then set U = ∐i∈IVi with induced morphism U → V → X. This is étale and surjective as a composition of étale and surjective representable transformations of functors (via the general Second, we can imagine several generalizations of this notion to morphisms of higher relative dimension (for example, one can ask for morphisms which are étale locally compositions of at

Section 66.6 : Special coverings—The Stacks project

35.32 Properties of morphisms étale local on source-and-target Let $\mathcal {P}$ be a property of morphisms of schemes. There is an intuitive meaning to the phrase “$\mathcal {P}$ is étale

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Lemma 35.25.2. A universally injective étale morphism is an open immersion. First proof. Let f: X → Y be an étale morphism which is universally injective. Then f is open (Morphisms, Lemma Since $t$ is separated and étale, and in particular separated and locally quasi-finite (by Morphisms, Lemmas 29.35.10 and 29.36.16) we see that the restriction of $G$ to each $T_ i$

This section really belongs in the chapter on morphisms of algebraic spaces, but we need the notion of an algebraic space étale over another in order to define the small étale site of an Section 35.30: Properties of morphisms local in the smooth topology on the source Section 35.31: Properties of morphisms local in the étale topology on the source

Let P be a property of morphisms of schemes which is étale local on source-and-target. Let Q be the associated property of morphisms of germs, see Lemma 35.33.2. Table of contents Part 2: Schemes Chapter 41: Étale Morphisms of Schemes Section 41.14: Topological properties of étale morphisms Theorem 41.14.1 (cite) Flat locally finitely presented, smooth, and étale morphisms are universally open, see Morphisms, Lemmas 29.25.10, 29.34.10, and 29.36.13. The case of surjective, quasi-compact, flat

Table of contents Part 2: Schemes Chapter 41: Étale Morphisms of Schemes Section 41.13: Étale and smooth morphisms (cite) Then V → Y is étale as a base change of U → X (Morphisms, Lemma 29.36.4) and the pullback D×X V is a strict normal crossings divisor on V by Lemma 41.21.3.

By Morphisms, Lemma 29.51.1 for every η ∈ U which is the generic point of an irreducible component of U, there exists an open neighbourhood V ⊂ U of η such that s−1(V) → V is finite. Table of contents Part 3: Topics in Scheme Theory Chapter 59: Étale Cohomology Section 59.26: Étale morphisms Proposition 59.26.2 (cite) The reader is encouraged to read up on local complete intersection morphisms of schemes in that section first. The property “being a local complete intersection morphism” of morphisms of

Recall that étale morphisms are open, see Morphisms, Lemma 29.36.13. It follows (from the construction of pullback on sheaves) that \epsilon ^ {-1}\mathcal {O}_ {Zar} is the sheafification The morphism $V \to T’$ is ind-quasi-affine by More on Morphisms, Lemma 37.66.8 (because étale morphisms are locally quasi-finite, see Morphisms, Lemma 29.36.6). By More on Then f is étale at x and g is étale at y. Proof. After replacing X by an open neighbourhood of x we may assume g ∘ f is étale. Then we find f is unramified by Morphisms,

Table of contents Part 2: Schemes Chapter 37: More on Morphisms Section 37.17: Closed immersions between smooth schemes (cite) 37.38 Slicing smooth morphisms In this section we explain a result that roughly states that smooth coverings of a scheme S can be refined by étale coverings. The technique to prove this relies

[1] Actually we use here also Schemes, Lemma 26.11.1 (soberness schemes), Morphisms, Lemmas 29.36.12 and 29.25.9 (generalizations lift along étale morphisms), Lemma 66.4.5

Suppose given a set I and algebraic spaces Fi, i ∈ I. Then F =∐i∈IFi is an algebraic space provided I, and the Fi are not too “large”: for example if we can choose surjective étale

Of course, since the composition of étale morphisms is étale (Morphisms, Lemma 29.36.3) we see that conversely any étale neighbourhood $ (V, v)$ of $ (U, u)$ is an étale neighbourhood is fully faithful by Étale Morphisms, Lemma 41.20.3. Thus the claim implies the theorem. Proof of the claim. Recall that a universal homeomorphism is the same thing as an integral, universally

We are going to use that an étale morphism is flat, syntomic, and a local complete intersection morphism (Morphisms, Lemma 29.36.10 and 29.36.12 and More on Morphisms,

Table of contents Part 4: Algebraic Spaces Chapter 67: Morphisms of Algebraic Spaces Section 67.50: Separated, locally quasi-finite morphisms (cite) Proof. We are going to use Lemma 35.26.4. By Morphisms, Lemma 29.35.3 the property of being unramified (resp. G-unramified) is local for Zariski on source and target. By Morphisms,

35.14 Descent of finiteness and smoothness properties of morphisms In this section we show that several properties of morphisms (being smooth, locally of finite presentation, and so on) 101.35 Étale morphisms An étale morphism of algebraic stacks should not be defined as a smooth morphism of relative dimension $0$. Namely, the morphism

Table of contents Part 4: Algebraic Spaces Chapter 67: Morphisms of Algebraic Spaces Section 67.50: Separated, locally quasi-finite morphisms Proposition 67.50.2 (cite) where $\pi $ is a homeomorphism onto an open subset. Smooth morphisms of schemes are the analogue of these maps in the category of schemes. See Lemma 29.34.11 and Lemma

Table of contents Part 4: Algebraic Spaces Chapter 66: Properties of Algebraic Spaces Section 66.19: Points of the small étale site (cite) We will prove a very precise result relating weakly étale morphisms to étale morphisms later (see Pro-étale Cohomology, Section 61.9). In this section we stick with the basics. 67.27 Quasi-finite morphisms The property “locally quasi-finite” of morphisms of schemes is étale local on the source-and-target, see Descent, Remark 35.32.7. It is also stable under base

Table of contents Part 7: Algebraic Stacks Chapter 101: Morphisms of Algebraic Stacks Section 101.21: Special presentations of algebraic stacks (cite)