# Questions tagged [schemes]

The concept of scheme mimics the concept of manifold obtained by gluing pieces isomorphic to open balls, but with different "basic" gluing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, the spectrum of a commutative ring with unit.

2,195 questions
Filter by
Sorted by
Tagged with
27 views

9 views

27 views

### Can a closed subset of a scheme $X$ be a support?

Suppose $X$ is a scheme, and $T$ its closet subset. I want to ask that is there a canonical way to obtain a quasi-coherent $\mathscr O_X$-ideal $I$ such that $T=supp \mathscr O_X/I$? This is the ...
36 views

### Basics about action on a Scheme by finite Group

Let $Y$ be a scheme and a finite group $G$ acts on $Y$ in the sense that $G$ embeds in $Aut_{Sch}(Y)$ in category of schemes. I have two question concerning general properties dealing with action by ...
55 views

### Diagonal action by $\mathbb{Z}/2$ on $\mathbb{P}^1 \times \mathbb{P}^1$

I have a basic question about a certain action on product of projective lines $\mathbb{P}^1 \times \mathbb{P}^1$ by group $\mathbb{Z}/2$ which Sasha introduced in this MO discussion. How does it here ...
40 views

### Surjectivity of proper morphism

I am trying to prove that, given a proper morphism of schemes $f \colon X \rightarrow Y$, if $Y$ and all fibers of $f$ are connected then $X$ is connected. It seems to me that it would help knowing if ...
20 views

### open subscheme and étale fundamental group

Let $X$ be an irreducible scheme, $U\subset X$ be a nonempty open subscheme. Is $\pi^{ét}_1(U)\to \pi^{ét}_1(X)$ surjective? For example, if $X$ is normal integral scheme, then this follows from ...
12 views

### Universal cover with respect to étale topology of scheme

Let $X$ be a connected quasi-compact quasi-separated scheme. I tried to define a universal cover as follows (thanks to the help of one friend). Consider $I$ the directed set of all open normal ...
46 views

### Subset of a scheme where a section vanishes

Suppose $(X,\mathscr{O})$ is a scheme, $s\in \mathscr{O}(U)$. I am trying to prove that $\{x\in U: s_x=0\in \mathscr{O}_x(\text{the stalk})\}$ is open in $U$ but not necessarily closed. I can see why ...
50 views

70 views

35 views

### Blowup of $\Bbb{A}^2$ at the intersection of a cubic with a triple line

I'm familiar with blowups in classical algebraic geometry but I'm still learning about blowup of schemes. For now I'm trying to be as concrete as possible, because the formal definition is ...
### Proof that a set is open using closed points on a $k$-scheme
Let $S \subset X$, where $X$ is an affine scheme of finite type over $k$ and $S(k)$ is not empty. Let us suppose that the set of points of $S(k)$ is inside of the interior $\text{int } S$, that is, ...