# 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.

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### About definition of finite presentation morphism

Simple question: why finite type morphisms of scheme are required to be only quasi-compact while finitly presented morphisms are asked to by quasi-compacts and quasi-separated?
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### Equality of morphisms $f,g:K\rightarrow X$ of schemes, where $K$ is a reduced scheme.

Suppose that $f,g:K\rightarrow X$ are morphisms of schemes, where $K$ is a reduced scheme. I want to show that $f=g$ if and only if for all $x\in K$, $f(x)\equiv g(x)$. Here $f(x)\equiv g(x)$ means ...
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### Motivation for separated and proper schemes

Hartshorne mention at the beginning of section 4 in chapter 2 that the definition of separated is similliar to hausdorff. We all can see that. That is also what I found in google. Again - we all can ...
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### Nillradical is prime ideal, then the ring is not a product ring.

Let $A$ be a commutative ring and $nill（A）$　is not a prime ideal. This is just a characterization of $SpecA$ to be irreducible. Then, according to the argument of general topology, irreducible ...
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### Affine scheme which is irreducible but not connected.

Could you let me know the example of affine scheme which is irreducible but not connected? I know affinescheme SpecA is irreducible only if nillradical of A is prime ideal. But I have trouble with ...
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### Vanishing of cohomology of affine scheme

In EGA I 5.1, more specifically the proof of 5.1.9, which states that $X$ is affine iff the closed subscheme defined by a quasi-coherent sheaf of ideals $\mathscr{I}$ such that $\mathscr{I}^n = 0$ for ...
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### Morphism $f:X\rightarrow\text{Spec}(B)$ is quasi - affine, if $X$ is quasi - affine.

I have a question regarding a proof from Bosch's Algebraic Geometry book, namely section 9.5, Proposition 3, part (ii): Let $f:X\rightarrow Y=\text{Spec}(B)$ be a morphism of schemes. $f$ is quasi-...
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### What is the $k^{al}$-points of a variety over $k$?

Let $X=\mathrm{Spec}(\mathbb{Q}[x,y]/(y^2-x^3-x-1))$. So $X$ is an elliptic curve over $\mathbb{Q}$ given by function $y^2=x^3+x+1$. Now I wonder what is meaning of a $\mathbb{C}$ points of $X$? I ...
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### When is relative Hilbert scheme $\text{Hilb}_{X/S}^{p(t)}$ flat over $S$?

Let $S$ be a scheme of finite type over the complex numbers $\mathbb{C}$ and let $X\subset S\times\mathbb{P}^r$ be a projective family over $S$. In the book "Geometry of Algebraic Curves Volume II" ...
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### Concerning the spectrum of a quasi coherent $\mathcal{O}_{X}$ algebra

I am reading chapter 11 (on vector bundles) of algebraic geometry I by Wedhorn/Görtz. There they have the following proposition 11.1: I understand what they write in equation 11.2.1. However, I do ...
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### The quotient of the constant sheaf by the structure sheaf is flasque.

Let $X$ be an integral scheme of finite type over $k$ with $\dim(X)=1$. Let $\mathcal{K}_{X}$ be the constant sheaf on $X$ with value $K(X)$, where $K(X)$ is the function field of $X$. I want to ...
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### Geometric points in fibre of finite étale morphism $\phi : Y \rightarrow X$ is independent of fibre

I am reading the following notes http://www.math.toronto.edu/~jacobt/Lecture6.pdf and trying to understand the conclusion of Lemma 2.1. We are trying to show that the number of geometric points above ...
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### The affine scheme $\text{Spec}(\mathbb{C}\times\mathbb{C})$ is a projective $\mathbb{C}$-scheme.
I want to show that $X:=\text{Spec}(\mathbb{C}\times \mathbb{C})$ is a projective scheme. So we have to find a closed immersion from $X$ to $\mathbb{P}^{1}_{\mathbb{C}}$. Brief explanation of my ...