Articles of sheaf theory

About the sheafification

Look at the following definition of the shefification (the source is the Stacks Project): I don’t understand what is the projection $\prod_{u\in U}\mathcal F_u\longrightarrow\prod_{v\in V}\mathcal F_v$. In particular, if $u\in U\setminus V$ what is the image of $s_u$ under this map? Thanks in advance.

Sheafyness and relative chinese remainder theorem

The relative chinese remainder theorem says that for any ring $R$ with two ideals $I,J$ we have an iso $R/(I\cap J)\cong R/I\times_{R/(I+J)}R/J$. Let’s take $R=\Bbbk [x_1,\dots ,x_n]$ for $\Bbbk $ algebraically closed. If $I,J$ are radical, the standard dictionary tells us $R(I\cap J)$ is the coordinate ring of the variety $\mathbf V(I)\cup \mathbf V(J)$. Furthermore, […]

A sheaf whose stalks are zero outside a closed subset

Let $X$ be a topological space. Let $Y$ be a closed subset of $X$. Let $f\colon Y \rightarrow X$ be the canonical injection. Let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Suppose $\mathcal{F}_x = 0$ for every $x \in X – Y$. Is the canonical morphism $\Gamma(X, \mathcal{F}) \rightarrow \Gamma(Y, f^{-1}(\mathcal{F}))$ an isomorphism? […]

Some question of sheaf generated by sections

Let $\mathcal{F}$ be a sheaf of abelian groups on a topological space $X$. Let $B= \cup_{U\subseteq X} \mathcal{F}(U)$ and let $A$ be the set of all finite subset of $B$. For each $\alpha \in A$, let $\mathcal{F}_\alpha$ be subsheaf of $\mathcal{F}$ generated by the section in $\alpha$ (over a various open sets). But I don’t […]

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal P(X)$$ where the powerset is a poset category. The action on objects is given by $$\exists (f)(A)=f[A],\; \mathsf I(f)(B)=f^\leftarrow (B),\; \forall (f)(A)=f[A^c]^c.$$ and one can prove $$\exists (f)\dashv \mathsf I(f) \dashv \forall (f).$$ […]

Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It’s well known that $Pic(\mathbb{P}^n)\simeq\mathbb{Z}$, with generator $O_{\mathbb{P}^n}(1)$. First question : it is right that if I show that $\phi^*O_{\mathbb{P}^m}(1)$ is generated by less than n+1 global sections, then must be $\phi^*O_{\mathbb{P}^m}(1)\simeq O_{\mathbb{P}^n}$ and so […]

If $U$ is connected, any two sections $U \to \mathfrak S$ either coincide or have disjoint images (Is my proof correct?)

I tried proving the following statement by Ahlfors, page 287: If $U$ is connected and $\varphi,\psi: U \to \Gamma(U,\mathfrak S)$, then either $\varphi$ and $\psi$ are identical, or the images $\varphi(U)$ and $\psi(U)$ are disjoint. Indeed, the sets with $\varphi – \psi = 0$ and $\varphi- \psi \neq 0$ are both open. Here $\mathfrak S$ […]

Extension by zero not Quasi-coherent.

Hartshorne’s Example 5.2.3 in Chapter 2 states that if $X$ is an integral scheme, and $U$ is an open subscheme with $i:U \rightarrow X$ the inclusion, then if $V$ is any open affine not contained in $U$, $i_{!}(\mathcal{O}_U) \mid_{V}$ will have no sections over $V$. But it will have non-zero stalks, and so cannot come […]

Sheafification: Show that $\tilde{\mathscr{F}_x}=\mathscr{F}_x$.

My today’s question is about a proof of this book. More precisely we are talking about the proof of Prop. 2.24 on page 52. The book says that we have $\tilde{\mathscr{F}_x}=\mathscr{F}_x$ for all $x\in X$. I tried to verify that but unfolding the definition of elements of $\tilde{\mathscr{F}_x}$ is very technical. So I got stuck. […]

Stalks of Skyscraper Sheaf

Here is Rotman’s definition of the skyscraper sheaf: Let $A$ be an abelian group, $X$ a topological space, and $x \in X$. Define a presheaf by $x_*A(U) = \begin{cases} A & \text{if } x \in U,\\ \{0\} & \text{otherwise.} \end{cases}$ If $U \subseteq V$, then the restriction map $\rho_U^V$ is either $1_A$ or $0$. He […]