# Showing the support of a sheaf may not be closed (Liu 2.5)

This is question 2.5 of Qing Liu.

I am new in algebraic geometry and really stuck on it and can’t do anything to solve it.

The question:
Let $$F$$ be a sheaf on $$X$$. Let $$operatorname{Supp} F={xin X:F_xneq 0}$$. We want to show that in general, $$operatorname{Supp} F$$ is not a closed subset of $$X$$. Let us fix a sheaf $$G$$ on $$X$$ and a closed point $$x_0in X$$. Let us define a pre-sheaf $$F$$ by $$F(U)=G(U)$$ if $$x_0notin U$$ and $$F(U)= {sin G(U):s_{x_0}=0}$$ otherwise. Show that $$F$$ is a sheaf and that $$operatorname{Supp} F = operatorname{Supp}Gsetminus {x_0}$$.

I don’t know how to solve this question:
To show a pre-sheaf is a sheaf I need to check the "uniqueness" and "gluing local sections".

For the uniqueness: Let $$U$$ be an open subset of $$X$$ , $$sin F(U)$$, if $$x_0notin U$$ , then since $$G$$ is a sheaf, I don’t see a problem for $$F$$ to be a sheaf.

If $$sin F(U)$$ and $$x_0 in U$$ and $${U_i}_i$$ be an open covering of $$U$$,then there exists an $$i_0$$ such that $$x_0in U_{i_0}$$. the image of $$s$$ in the stalk $$F_{x_0}$$ is $$s_{x_0}$$. $$F(U_{i_0})={sin G(U):s_{x_0}=0}$$ by definition. I don’t know what to do now? (so sorry and I know this is an easy question…)

Mathematics Asked by user468730 on November 14, 2021

Side note: a presheaf $$F$$ has the same stalks as its sheafification $$F^#$$. Thus, $$operatorname{supp}(F) = operatorname{supp}(F^#)$$ and it suffices to find a presheaf $$F$$ such that $$operatorname{supp}(F)$$ is not closed, if all you want to know is that the support of a sheaf may or may not be closed. This doesn't matter here because the problem tells you specifically to prove $$F$$ is a sheaf, but it's good to know this trick.

$$DeclareMathOperator{res}{res}$$ The first thing we need to do when checking that $$F$$ is a sheaf is to keep in mind the entire datum of the presheaf $$F$$; not just the objects $$F(U)$$ for $$U$$ an open subset of $$X$$, but also the restriction morphisms $$res_{U, V} : F(U) to F(V)$$ when $$V subseteq U$$. In this case, $$res_{U,V}$$ is just the restriction (haha) of the restriction morphism $$G(U) to G(V)$$ (which we check is well-defined). In this situation (where $$F(U)$$ is always a subset of $$G(U)$$ and the restriction morphisms of $$F$$ are induced by those of $$G$$), $$F$$ is called a subpresheaf of $$G$$. Nothing crazy going on here, but it's important to fully understand the object you're working with. For convenience, I will write $$sigma|_V$$ to mean $$res_{U,V}(sigma)$$, since $$U$$ can always be determined from context.

Next, as you said, we should check "uniqueness". Remember, the uniqueness axioms says that for all open covers $${U_i}_{i in I}$$ of an open set $$U subseteq X$$ and all $$sigma in F(U)$$, if $$sigma|_{U_i} = 0$$ for all $$i in I$$, then $$sigma = 0$$. We should try to prove this in the most straightforward possible way:

Let $${U_i}_{i in I}$$ be an open cover of an open set $$U subseteq X$$. Let $$sigma in F(U)$$ be arbitrary. Suppose that $$sigma|_{U_i} = 0$$ for all $$i in I$$. Since $$F$$ is a subpresheaf of $$G$$, we have in particular that $$sigma in G(U)$$ and $$sigma|_{U_i} = 0 in G(U_i)$$ for all $$i in I$$. Since $$G$$ is assumed to be a sheaf, we must have $$sigma = 0$$, as desired.

Indeed, this very simple argument shows that any subpresheaf of a sheaf is separated (a.k.a. satisfies the uniqueness axiom): no need to mention anything about $$x_0$$!

Now you just need to check gluing, which I'll omit because this proof appears in other answers.

Answered by diracdeltafunk on November 14, 2021

To see uniqueness, let $$Usubset X$$ be an open subset and $${U_i}$$ an open cover of $$U$$. Let $$s,tin F(U)$$ and let $$s_i,t_iin F(U_i)$$ be their restrictions. Then the condition that $$s_i=t_i$$ in $$F(U_i)$$ means that $$s_i=t_iin G(U_i)$$, which means that $$s=t$$ in $$G(U)$$ and as $$F(U)subset G(U)$$, we have that $$s=t$$ in $$F(U)$$.

To check gluing, let $$s_i$$ be a collection of sections of $$F(U_i)$$ so that $$s_i|_{U_icap U_j}=s_j|_{U_icap U_j}$$ as elements of $$F(U_icap U_j)$$. Then this equality is also true in $$G(U_icap U_j)$$, and by the assumption that $$G$$ is a sheaf, there is a section $$sin G(U)$$ so that $$s|_{U_i}=s_i$$. This implies that $$s_{x_0}=0$$ (if $$x_0in U$$ - if not, we have nothing to worry about), as the maps $$G(U)to G(U_i)to G_{x_0}$$ commute, so $$sin F(U)$$ as well and thus $$F$$ satisfies gluing.

Answered by KReiser on November 14, 2021

This kind of a statement follows from "unpacking definitions". You unpack the sheaf properties of $$G$$ together with their connection with $$F$$ to get that $$F$$ is a sheaf. In principle this is easy and there is nothing happening. In practice you can get lost, in particular if you are not familiar with the general concepts at play.

Since $$G$$ is a sheaf and $$F(U)subseteq G(U)$$ for any $$U$$ you get uniqueness of any gluings automatically, since they are also gluings in $$G$$. But lets be explicit, if $$U_alpha$$ is a collection of open sets and $$s,s' in F(bigcup_alpha U_alpha)$$ with $$slvert_alpha = s'lvert_alpha$$ for all $$alpha$$ we want to see that $$s=s'$$ must follow. Since $$F(bigcup U_alpha)subseteq G(bigcup U_alpha)$$ you get that $$s,s'$$ are both in $$G(bigcup U_alpha)$$ with $$slvert_alpha = s'lvert_alpha$$, since $$G$$ is a sheaf you get $$s=s'$$.

So whats left is to show is the existence of a gluing.

So suppose $$U_alpha$$ is a collection of open sets and $$s_alphain F(U_alpha)$$ for all $$alpha$$ with $$s_alphalvert_{U_alphacap U_beta} = s_betalvert_{U_alphacap U_beta}$$ for any $$alpha,beta$$, ie the compatibility conditions of gluing are satisfied. We want to show that there is an $$sin F(bigcup_alpha U_alpha)$$ with $$slvert_alpha = s_alpha$$ for every $$alpha$$. Now since $$G$$ is a sheaf and $$s_alphain G(U_alpha)$$ you can glue them to in $$G$$ get an $$sin G(bigcup_alpha U_alpha)$$ for which $$slvert_alpha =s_alpha$$, we just need to check that $$sin F(bigcup U_alpha)$$, ie that $$s_{x_0}=0$$ if $$x_0inbigcup_alpha U_alpha$$ (if $$x_0notin bigcup_alpha U_alpha$$ there is nothing to check). Suppose $$x_0in U_gamma$$ for a fixed $$gamma$$, since $$s_gamma in F(U_gamma)$$ this means that $$(s_gamma)_{x_0}=0$$, ie there is some open set $$Usubseteq U_gamma$$ containing $$x_0$$ for which you have that $$s_gamma lvert_U =0$$. But then: $$slvert_U=(slvert_{U_gamma})lvert_{U}=s_gammalvert_U=0$$ implying that $$s_{x_0}=0$$, ie $$sin F(bigcup_alpha U_alpha)$$.

Answered by s.harp on November 14, 2021

## Related Questions

### Isometric mapping of two subsets in a metric space

1  Asked on December 28, 2020 by user856180

### How to show $A$ is compact in $Bbb{R}$ with standard topology?

4  Asked on December 28, 2020 by happy-turn

### Existence of a complex sequence with given property

1  Asked on December 28, 2020 by praveen

### For i.i.d random variables $X$ and $Y$, is $E[X mid sigma(X+Y)] = frac{X+Y}{2}$?

1  Asked on December 27, 2020

### Rewrite rational function as the sum of a polynomial and partial fraction?

1  Asked on December 27, 2020 by sabrinapat

### Understanding the proof of $cl(cl(A))=cl(A)$

1  Asked on December 27, 2020 by averroes2

### For $a,binmathbb{R}$, there is an integer within $|{a} – {b}|$ from $|a-b|.$

1  Asked on December 26, 2020 by mathgeek

### Proof that the coefficients of formal power series must match

2  Asked on December 26, 2020 by maths-wizzard

### To prove an apparently obvious statement: if $A_1subseteq A_2 subseteq … subseteq A_n$, then $bigcup_{i=1}^n A_i = A_n$

2  Asked on December 26, 2020

### Is it hopeless to try and solve this equation analytically?

1  Asked on December 26, 2020 by user694069

### Show that $I(y)=int_0^1 frac{x-y}{(x+y)^3}dx$ exists

1  Asked on December 26, 2020

### Find the coordinates of the point P on the line d : 2x − y − 5 = 0, for which the sum AP + PB attains its minimum, when A(−7; 1) and B(−5; 5).

2  Asked on December 26, 2020 by hestiacranel

### The Range of Taxi Fares

1  Asked on December 25, 2020 by cameron-chandler

### Schreier transversal and a basis for commutator subgroup of $F_3$

2  Asked on December 25, 2020 by makenzie

### Can someone help me with this other limit?

2  Asked on December 25, 2020 by toni-rivera-vargas

### Can you prove why the recurrence $3P_{n} – 2P_{n + 1} = P_{n – 1}$ holds for some (a subset of the) prime numbers?

1  Asked on December 25, 2020 by joebloggs

### Probabilities involving people randomly getting off floors on an elevator

0  Asked on December 25, 2020 by user747916

### Backward-Euler implicit integration for multiple variables

1  Asked on December 25, 2020 by user541686

### What are the continuous functions $mathbb{N} to mathbb{R}$ when $mathbb{N}$ is given the coprime topology?

1  Asked on December 25, 2020 by thedaybeforedawn