Making statements based on opinion; back them up with references or personal experience.

Paper author has not included all suggestions in peer review. Artin showed that proper algebraic spaces over the complex numbers are more or less the same as Moishezon spaces. Etale morphisms play an important role in étale cohomology theory (cf. It derives a natural topology from the projection operator. the $ k ( y) $- Not every singular algebraic surface is a scheme. scheme $ f ^ { - 1 } ( y) = X \otimes _ {Y} k ( y) $

A far-reaching generalization of algebraic spaces is given by the algebraic stacks. Z What are some (edit:) specific benefits of viewing a sheaf in this sense? It derives a natural topology from the projection operator.

The dimension of X at x is then just defined to be d. A morphism f: Y → X of algebraic spaces is said to be étale at y ∈ Y (where x = f(y)) if the induced map on stalks.

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Isn't this a special case of the more general phenomenon that it's best to know as many ways as possible of thinking about a concept, not least because some questions become trivial when thought about using one picture and they're less clear with another. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Responding to the Lavender Letter and commitments moving forward, Extracting the Sheaf and espace étalé condition from an abstractly given equivalence between these two spaces. Are environmentalists responsible for Californian forest fires? One can always assume that R and U are affine schemes. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor. It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of C by the corresponding lattice are not isomorphic as algebraic spaces but the corresponding analytic spaces are isomorphic. @Eivin Dahl: no, there are many more etale maps that coverings (typically, $Y$ is very non-Hausdorff in the etale maps corresponding to sheaves). Etale morphisms play an important role in étale cohomology theory (cf. I find this construction completely unmotivated without going through étalé spaces. X A point on an algebraic space is said to be smooth if ÕX, x ≅ k{z1, …, zd} for some indeterminates z1, …, zd. I know that the older view of a sheaf on $X$ was to consider it as a triple a local homeomorphism.

If R is the trivial equivalence relation over each connected component of U (i.e.

Commutative-group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes. Thanks for contributing an answer to MathOverflow! As an exercise I recently proved the equivalence of categories between ètalè spaces over X and sheaves of sets over X, which essentially also entails proving the sheafification functor and its properties.

How to align decimal point of table entries having units and no units.

The structure sheaf OX on the algebraic space X is defined by associating the ring of functions O(V) on V (defined by étale maps from V to the affine line A1 in the sense just defined) to any algebraic space V which is étale over X. If $G=(G_0,G_1)$ is a groupoid, a $G$-sheaf is an étalé space $p:X\to G_0$ over $G_0$ together with an action map $G_1\times_{d,p} X\to X$ satisfying obvious axioms.

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. , $$ COHOMOLOGY OF ALGEBRAIC SPACES 6 07U6 Lemma5.2.

This article was adapted from an original article by V.I. for all x, y belonging to the same connected component of U, we have xRy if and only if x=y), then the algebraic space will be a scheme in the usual sense. 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. rev 2020.10.9.37784, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. This page was last edited on 17 August 2020, at 12:09. The #1 tool for creating Demonstrations and anything technical. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. can be defined equivalently as a locally finitely-presentable flat morphism such that for any point $ y \in Y $ Can you multiply p-values if you perform the same test multiple times? I guess it's more geometric, I think it was the original definition, it's similar to sections of bundles. Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom(Y, X) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence.

While fairly abstract at the outset, this seems to be (to me) an intuitive view; in particular, all of the manipulations and constructions with sheaves fit nicely into this schema. a ringed space Specm.A/(topological space endowed with a sheaf of k-algebras), and an affine variety over kis a ringed space isomorphic to one of this form. IHES, 32 (1967) [2] If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the étalé space construction without specifically telling you, and to me it makes the construction somewhat unmotivated. I think my attempts to locally trivialize an arbitrary étale map rely on tacit assumptions. These two definitions are essentially equivalent. Thanks for contributing an answer to Mathematics Stack Exchange!

What is the connection between direct/inverse image of set maps and direct/inverse image functors of sheaves? This gives a set-valued functor on $\mathrm{Open}(Y)$; it is easily shown to be a sheaf too. $$ This is the pullback sheaf $f^* F$. denote the ring of algebraic functions in x over k, and let X = {R ⊂ U × U} be an algebraic space.

What are the benefits of viewing a sheaf from the “espace étalé” perspective?

So, this means that if $F \in Sh(X)$, the local homeomorphism $E(F) \to X$ which corresponds to $F,$ viewed as a map of topoi is nothing but the etale geometric morphism $Sh(X)/F \to Sh(X)$. For smooth varieties the fact that $ f : X \rightarrow Y $

What is the point in yard signs in presidential elections? e.g. "Etale Space." Proper algebraic spaces over a field of dimension one (curves) are schemes. Etale spaces are examples of space that are not T2.

schemes is étale. 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. References [1] A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ.

To me the obvious answer involves sheafification of a presheaf. Additional additions From the étalé space point of view it is clear that covering spaces are indeed elements of the topos $Sh(X)$ of sheaves on $X$ and that the fundamental group of $Sh(X)$ (in the sense of Barr and Diaconescu) is the usual fundamental group of $X$ if $X$ is locally simply connected. or "I understand linear maps so what is the point of matrices?" The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa.

Is there a key for reporting or killing in Among Us? Let me expand on Yosemite Sam's comment. I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets.

Use MathJax to format equations. So covering map $\Leftrightarrow$ étale map?

X The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf.

Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Etale_morphism&oldid=46854, A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique", A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique: Etude locale des schémas et de morphismes de schémas". ÉTALE COHOMOLOGY 5 03N5 A family of morphismsDefinition 4.1. A topological groupoid or Lie groupoid C C is called an étale groupoid if the source-map s: Mor C → Obj C s : Mor C \to Obj C is a local homeomorphism or local diffeomorphism, respectively, and hence exhibits the space of morphisms as an étale space over the space of objects. How do I keep my character's gender a complete mystery? is a local isomorphism.). 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. is étale means that $ f $ {ϕ i: U i →X} i∈I is called an étale coveringS if each ϕ i is an étale morphism and their images cover X, i.e., X = i∈I ϕ i(U i). There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes.

Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. How does Stockfish know if the king is in check? is an affine $ Y $-

On the Wikipedia page Sheaf, under the section "The etale space of a sheaf," the author claims that the etale space of the sheaf of (continuous) sections of a continuous map $Y \to X$ is (homeomorphic to) $Y$ if and only if $Y \to X$ is a covering map.

Use MathJax to format equations. An étale morphism of schemes $ f : X \rightarrow Y $ (A locally finitely-presentable morphism $ f : X \rightarrow Y $ A topology arising from a sheaf of continuous functions. Fair enough, I wasn't really quite as specific in my phrasing: I want to know specific benefits. To learn more, see our tips on writing great answers.

How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?

Practice online or make a printable study sheet. The point set underlying the algebraic space X is then given by |U| / |R| as a set of equivalence classes. for linear maps instead of matrices, we get motivation for the formula for matrix multiplication, while for matrices we are given an efficient computation method. In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Artin for use in deformation theory. then in both cases I'm sure you can see a good answer.

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