F is injective. + To sections define the same germ if they agree on an open neighborhood of a point p. So they will also agree on a smaller neighborhood. Making statements based on opinion; back them up with references or personal experience. Stalk is a local object of a sheaf 2 $i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. By abuse of notation we will often denote $(U, s)$, $s_ x$, or even $s$ the corresponding element in $\mathcal{F}_ x$. S has the same set or group as stalks at every point: for any point x, pick an open connected neighbourhood. I have been thinking about this problem a little bit lately, but I have not been able to come up with a "canonical" solution. F Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Le dictionnaire des synonymes est surtout dérivé du dictionnaire intégral (TID). | Dernières modifications. From what the germ tells us, the bump could be infinitely wide, that is, f could equal the constant function with value 1. C1-functions and the sheaf of continuous functions. De nition-Lemma 4.6. Making statements based on opinion; back them up with references or personal experience. This is indeed true: Both statements are false for presheaves. For every point we have canonical bijections $A = (A_ p)_ x = \underline{A}_ x$, where the second map is induced by functoriality from the map $A_ p \to \underline{A}$. Example 6.11.3. A regular function is a morphism X! site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The same property holds for any point x if the topological space in question is a T1 space, since every point of a T1 space is closed. LA fenêtre fournit des explications et des traductions contextuelles, c'est-à-dire sans obliger votre visiteur à quitter votre page web ! In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. We cannot even reconstruct f on a small open neighborhood U containing the origin, because we cannot tell whether the bump of f fits entirely in U or whether it is so large that f is identically one in U. Such an element is given by a $\mathcal{C}^\infty $-function $f$ whose domain contains $x$. Then the map $i_p : * \to X$ factors as $i_U \circ j_p$ where $j_p : * \to U$ has image $\{p\}$ and $i_U : U \to X$ is the inclusion. by This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on X. Germs are more useful for some sheaves than for others. Bas Edixhoven ○ Lettris Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. But then $U = \bigcup _{x \in U} V^ x$ is an open covering. Let me phrase the question a little bit more formally. _ Note that when we have a sheaf of rings, the stalk is often a local ring. {\displaystyle {\mathcal {F}}} Example 6.11.5. \[ \mathcal{F}(U) \longrightarrow \prod \nolimits _{x \in U} \mathcal{F}_ x \] The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point. How to model two variables to NOT to belong to the same set partition using Constraint Programming. Let C 1 Let $\alpha_p : \mathscr{F}_p \to \mathscr{G}_p$ be a homomorphism for each $p \in X$. {\displaystyle {\mathcal {F}}} How should someone choose a PhD topic so that she doesn't fail? e (If you actually have a sheaf of groups, then you get a group object over $X$. Then the stalk is the same as the inverse image sheaf . This cirterion is sometimes useful, for example in the construction of fibered products of locally ringed spaces (German: http://maddin.110mb.com/pdf/faserprodukte.pdf). I have been thinking about this problem a little bit lately, but I have not been able to come up with a "canonical" solution. The constant sheaf Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes? dictionnaire et traducteur pour sites web. Even if we know in advance that f is a bump function, the germ does not tell us how large its bump is. This can also be stated as continuity of the $\alpha_p$ (see the answer of David Roberts). However, stalks of sheaves and presheaves are tightly linked: See the References in the article on sheaves. 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. Conceptually speaking, we do this by looking at small neighborhoods of the point. Suppose that we want to reconstruct f from its germ. sheaf definition: 1. a number of things, especially pieces of paper or plant stems, that are held or tied together…. Une fenêtre (pop-into) d'information (contenu principal de Sensagent) est invoquée un double-clic sur n'importe quel mot de votre page web. MathJax reference. Even if we know in advance that f is a bump function, the germ does not tell us how large its bump is. − S Think about it! Il est aussi possible de jouer avec la grille de 25 cases. How do we think of an element in the stalk $\mathcal{C}^\infty _{\mathbf{R}^ n, x}$? Then to any $y\in U$. See Categories, Section 4.21 for notation and terminology regarding (co)limits over systems. By functoriality of the inverse image construction, we have that $i_p^{-1} = (i_U \circ j_p)^{-1} = j_p^{-1} \circ i_U^{-1}$. 5 And a pair of such functions $f$, $g$ determine the same element of the stalk if they agree in a neighbourhood of $x$. Suppose that $s, s' \in \mathcal{F}(U)$ map to the same element in every stalk $\mathcal{F}_ x$ for all $x \in U$. 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