In algebraic geometry, the **étale topology** is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use.

For any scheme *X*, let Ét(*X*) be the category of all étale morphisms from a scheme to *X*. This is the analog of the category of open subsets of *X* (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of *X*. The intersection of two objects corresponds to their fiber product over *X*. Ét(*X*) is a large category, meaning that its objects do not form a set.

An **étale presheaf** on *X* is a contravariant functor from Ét(*X*) to the category of sets. A presheaf *F* is called an **étale sheaf** if it satisfies the analog of the usual gluing condition for sheaves on topological spaces. That is, *F* is an étale sheaf if and only if the following condition is true. Suppose that *U*→*X* is an object of Ét(*X*) and that *U*_{i}→*U* is a jointly surjective family of étale morphisms over *X*. For each *i*, choose a section *x*_{i} of *F* over *U*_{i}. The projection map *U*_{i}×*U*_{j}→*U*_{i}, which is loosely speaking the inclusion of the intersection of *U*_{i} and *U*_{j} in *U*_{i}, induces a restriction map *F*(*U*_{i}) →*F*(*U*_{i}×*U*_{j}). If for all *i* and *j* the restrictions of *x*_{i} and *x*_{j} to *U*_{i}×*U*_{j} are equal, then there must exist a unique section *x* of *F* over *U* which restricts to *x*_{i} for all *i*.

Suppose that *X* is a Noetherian scheme. An abelian étale sheaf *F* on *X* is called **finite locally constant** if it is a representable functor which can be represented by an étale cover of *X*. It is called **constructible** if *X* can be covered by a finite family of subschemes on each of which the restriction of *F* is finite locally constant. It is called **torsion** if *F*(*U*) is a torsion group for all étale covers *U* of *X*. Finite locally constant sheaves are constructible, and constructible sheaves are torsion. Every torsion sheaf is a filtered inductive limit of constructible sheaves.

Grothendieck originally introduced the machinery of Grothendieck topologies and topoi to define the étale topology. In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The **small étale site of ***X* is the category *O*(*X*_{ét}) whose objects are schemes *U* with a fixed étale morphism *U* → *X*. The morphisms are morphisms of schemes compatible with the fixed maps to *X*. The **big étale site of ***X* is the category *Ét/X*, that is, the category of schemes with a fixed map to *X*, considered with the étale topology.

The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology. Consequently, to define an étale cover of a scheme *X*, it suffices to first cover *X* by open affine subschemes, that is, to take a Zariski cover, and then to define an étale cover of an affine scheme. An étale cover of an affine scheme *X* can be defined as a jointly surjective family {*u*_{α} : *X*_{α} → *X*} such that the set of all α is finite, each *X*_{α} is affine, and each *u*_{α} is étale. Then an étale cover of *X* is a family {*u*_{α} : *X*_{α} → *X*} which becomes an étale cover after base changing to any open affine subscheme of *X*.

Let *X* be a scheme with its étale topology, and fix a point *x* of *X*. In the Zariski topology, the stalk of *X* at *x* is computed by taking a direct limit of the sections of the structure sheaf over all the Zariski open neighborhoods of *x*. In the étale topology, there are strictly more open neighborhoods of *x*, so the correct analog of the local ring at *x* is formed by taking the limit over a strictly larger family. The correct analog of the local ring at *x* for the étale topology turns out to be the strict henselization of the local ring .^{[ citation needed ]} It is usually denoted .

- For each étale morphism , let . Then is a presheaf on
*X*; it is a sheaf since it can be represented by the scheme .

In category theory, a branch of mathematics, a **Grothendieck topology** is a structure on a category *C* that makes the objects of *C* act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a **site**.

In mathematics, a **sheaf** is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.

In mathematics, a **scheme** is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

In mathematics, the **étale cohomology groups** of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct **ℓ-adic cohomology**, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaves** are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

**Motivic cohomology** is an invariant of algebraic varieties and of more general schemes. It includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.

In algebraic geometry, a branch of mathematics, a morphism *f* : *X* → *Y* of schemes is **quasi-finite** if it is of finite type and satisfies any of the following equivalent conditions:

In mathematics, **algebraic spaces** form a generalization of the schemes of algebraic geometry, introduced by Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.

In algebraic geometry, an **étale morphism** is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

In mathematics, the **flat topology** is a Grothendieck topology used in algebraic geometry. It is used to define the theory of **flat cohomology**; it also plays a fundamental role in the theory of descent. The term *flat* here comes from flat modules.

In mathematics a **stack** or **2-sheaf** is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

In algebraic geometry, the **Nisnevich topology**, sometimes called the **completely decomposed topology**, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.

In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the **exceptional inverse image functor** is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.

In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four **image functors for sheaves** that belong together in various senses.

In algebraic topology, a branch of mathematics, the **Čech-to-derived functor spectral sequence** is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, **coherent sheaf cohomology** is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

In mathematics, a **constructible sheaf** is a sheaf of abelian groups over some topological space *X*, such that *X* is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It is a generalization of constructible topology in classical algebraic geometry.

In algebraic geometry, **dévissage** is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent.

This is a **glossary of algebraic geometry**.

- Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie".
*Publications Mathématiques de l'IHÉS*.**20**. doi:10.1007/bf02684747. MR 0173675. - Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie".
*Publications Mathématiques de l'IHÉS*.**32**. doi:10.1007/bf02732123. MR 0238860. - Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.).
*Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 2*. Lecture notes in mathematics (in French).**270**. Berlin; New York: Springer-Verlag. pp. iv+418. doi:10.1007/BFb0061319. ISBN 978-3-540-06012-3. - Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.).
*Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3*. Lecture notes in mathematics (in French).**305**. Berlin; New York: Springer-Verlag. pp. vi+640. doi:10.1007/BFb0070714. ISBN 978-3-540-06118-2. - Deligne, Pierre (1977).
*Séminaire de Géométrie Algébrique du Bois Marie - Cohomologie étale - (SGA 4½)*. Lecture notes in mathematics (in French).**569**. Berlin; New York: Springer-Verlag. pp. iv+312. doi:10.1007/BFb0091516. ISBN 978-3-540-08066-4. - J. S. Milne (1980),
*Étale cohomology*, Princeton, N.J: Princeton University Press, ISBN 0-691-08238-3 - J. S. Milne (2008).
*Lectures on Étale Cohomology*

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