# Tagged Questions

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

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### Examples of affine schemes

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional ...
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### Why do the Zariski distinguished open subsets form a base?

Let $R$ be commutative ring with unit. I have to prove that the distinguished open sets form a base for the Zariski topology i.e. any non-empty open set is a union of distinguished ones. We have that ...
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### Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
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### What is Galois theory for schemes?

I have heard about "Galois theory for schemes" in this note. I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, ...
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### Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
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### Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
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### Stalk map on fiber products of schemes

Let $X$ and $Y$ be schemes and $x$ a point on $X$. Let $f:X\rightarrow Y$ be a morphism. Recall this induces a map $f_x:\mathcal O_{Y,f(x)}\rightarrow \mathcal O_{X,x}$ on the level of sheaves. ...
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### A particular closed subscheme

Look at the following definition: Let $(X,\mathscr O_X)$ be a scheme. A closed subscheme of $(X, \mathscr O_X)$ is a scheme $(Z, \mathscr O_Z)$ such that: $Z$ is a closed subset of $X$ ...
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### Dominant morphism, equal dimensions: always finite?

Let $f:X\to Y$ be a dominant morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that dim $X$ = dim $Y$. Question: must f be finite? It seems ...