In coordinates [x: y: z] on the rst P2 and [a: b: c] on the second P2, Y is given by the bihomogeneous equation ax+ by+ cz= 0: Consider projection ˇof Y down to the rst P2. 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. Namely, for any open subset

Hence, by Categories, Section 4.19 the colimit $\mathcal{F}^{+}(U)$ may be described in the following straightforward manner. Then we conclude using Lemma 7.10.1. Hence by Categories, Lemma 4.19.2 we see that $\mathcal{F} \to \mathcal{F}^+$ commutes with finite limits (as a functor from presheaves to presheaves). Definition 7.10.9. Then $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} \mathcal{F}$ exists and is the sheafification of the colimit in the category of presheaves. Let $\mathcal{C}$ be a site. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. This last equality means that there exists some covering $\{ W_{jk} \to W_ j\} $ such that $s_ j|_{W_{jk}} = s'_ j|_{W_{jk}}$. The tag you filled in for the captcha is wrong. It is clear that $\mathcal{F} \to \mathcal{F}^+$ is injective because all the maps $\mathcal{F}(U) \to H^0(\mathcal{U}, \mathcal{F})$ are injective. we need to study some examples from other mathematical areas. But for the fpqc topology this is not possible.
With $\mathcal{F}$ as above Even though sites that people work with can be large (say $Aff$) 'nice' Grothendieck pretopologies are given by a set of covering families for each object, or at the very least have a cofinal set of covering families (this means there is a coverage given by a set of covering families, and this is enough to define sheaves, though generally weaker than a pretopology). The presheaf $\mathcal{F}^+$ is separated. With $\mathcal{F}$ as above. Denote $\theta ^2 : \mathcal{F} \to \mathcal{F}^\# $ the canonical map of $\mathcal{F}$ into its sheafification. The proof of Lemma 1.28 is divided in steps.. {\displaystyle {\mathcal {O}}_ {X}= {\widetilde {A}}} . A classic example of sheafification is the sheafification of the presheaf of holomorphic functions admitting a square root on with the classical topology. ) manifold

In the case of large sites without the condition WISC, the appropriate thing to consider is small sheaves, namely sheaves that are small colimits of representable sheaves. A particular example is the limit over the empty diagram. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$, and let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. A first example is a presheaf which satisfies the ``locality'' of sheaf axiom, but which fails to obey ``gluing lemma''. We leave it to the reader to see this element has the required property that $s_ j = s|_{V_ j}$. Let $\mathcal{F} : \mathcal{I} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a diagram. ExcitedAlgebraicGeometer As $f$ and $g$ induce the same map $U\to V$, the diagram, is commutative for every $i\in I$. There is a canonical map $\mathcal{F}(U) \to H^0(\mathcal{U}, \mathcal{F})$. We point out that later we will see that $\mathcal{F}^{+}(U)$ is the zeroth Čech cohomology of $\mathcal{F}$ over $U$. so, is the category of fpqc-sheaves not reflective then? There exists a covering $\{ U_ i \to U\} $ and sections $s_ i \in \mathcal{F}(U_ i)$ such that. Last revised on January 8, 2019 at 21:44:07. for an example. MathJax reference. for every $i, j$ there exists a covering $\{ U_{ijk} \to U_ i \times _ U U_ j\} $ of $\mathcal{C}$ such that the pullback of $s_ i$ and $s_ j$ to each $U_{ijk}$ agree. According to the remarks above the construction $\mathcal{U} \mapsto H^0(\mathcal{U}, \mathcal{F})$ is a contravariant functor on $\mathcal{J}_ U$.

In particular this lemma shows that if $\mathcal{U}$ is a refinement of $\mathcal{V}$, and if $\mathcal{V}$ is a refinement of $\mathcal{U}$, then there is a canonical identification $H^0(\mathcal{U}, \mathcal{F}) = H^0(\mathcal{V}, \mathcal{F})$.

We continue Example 4.2. Also, in Milne's book on etale cohomology, he mentions an example (I think he attributes it to Waterhouse but I don't have the book at hand to check) of a presheaf with no associated sheaf, but it's unclear to me what topology he's talking about. See the history of this page for a list of all contributions to it. Moreover, M ~. This process has two main nice features: it preserves the stalks, and any map from a presheaf to a sheaf factors through the sheafification. A small remark is that we can define $H^0(\mathcal{U}, \mathcal{F})$ as soon as all the morphisms $U_ i \to U$ are representable, i.e., $\mathcal{U}$ need not be a covering of the site. Johan Let $U$ be an object of $\mathcal{C}$. . We will use this notion to prove the following simple lemma about limits of sheaves.

defined as follows. Since the trivial covering $\{ \text{id}_ U\} $ is an object of $\mathcal{J}_ U$ we get a canonical map $\mathcal{F}(U) \to \mathcal{F}^+(U)$. On the other hand, by Lemmas 7.10.5 and Lemma 7.10.6 the colimits in the construction of $\mathcal{F}^+$ are really over the directed set $\mathop{\mathrm{Ob}}\nolimits (\mathcal{J}_ U)$ where $\mathcal{U} \geq \mathcal{U}'$ if and only if $\mathcal{U}$ is a refinement of $\mathcal{U}'$.

This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made). The morphism $f$ consists of a map $U\to V$, a map $\alpha : I \to J$ and maps $f_ i : U_ i \to V_{\alpha (i)}$.
In order to define the sheafification we study the zeroth Čech cohomology group of a covering and its functoriality properties. The bre over a point Lemma 7.10.4. Let $\mathcal{C}$ be a site and let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. Let $s \in \mathcal{F}^\# (U)$. $\square$. The constructions above define a presheaf $\mathcal{F}^+$ together with a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$. It is the presheaf that associates to each object $U$ of $\mathcal{C}$ a singleton set, with unique restriction mappings and moreover this presheaf is a sheaf. sheafification 0 points 1 point 2 points 6 years ago Factor has similar capability, with the addition of being able to choose how much of the runtime gets included to save on file size. Comment #4332 Then since $\{ W_{jk} \to U\} $ is a covering we see that $s, s'$ map to the same element of $H^0(\{ W_{jk} \to U\} , \mathcal{F})$ as desired. Let $\phi : i \to i'$ be a morphism of the index category. \[ H^0(\mathcal{U}, \mathcal{F}) = \{ (s_ i)_{i\in I} \in \prod \nolimits _ i \mathcal{F}(U_ i) \mid s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j} \ \forall i, j \in I \} .


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