Questions tagged [affine-schemes]

The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.

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Open set of the cuspidal curve is not principal

I'm struggling with this problem: Let $k$ a field and $X=\operatorname{Spec} k[x,y]\big/(x^3-y^2)$ and $U=X\setminus\{(x-1,y-1)\}$. Show that $U$ is not a principal open set in $X$. I tried to show ...
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Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S =$ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
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Where is the error in this proof that all morphisms of schemes are quasi-compact?

Lemma 1. All preimages of affine open subschemes are affine (Inverse of open affine subscheme is affine). Lemma 2. All affine schemes are quasi-compact (Proof that an affine scheme is quasi compact). ...
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Defining morphism of sheaves [duplicate]

Suppose we have sheaves $\mathcal{F},\mathcal{G}$ on a topological space $X$ where $\mathcal{U}$ is a base of $X$. Then to define a morphism $\varphi:\mathcal{F}\rightarrow \mathcal{G}$, is it enough ...
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Easy question regarding open subschemes of closed subschemes

I am trying to follow the proof of the theorem 3.42 of Wedhorn's and Görtz book on algebraic geometry. The step I can't follow is like this: suppose we have $Z \subseteq X = \operatorname{Spec}A$ a ...
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Question about a proof that all étale morphisms are locally standard étale

In chapter 1 of Milne's Étale cohomology book from 1980, Theorem 3.14 states that : If $f: Y\longrightarrow X$ is étale in some open neighbourhood of a point $y\in Y$, then there are affine open ...
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Could you provide an example for a nonreduced local ring with reducible Spectrum?

As the title says, could you provide a local ring which is nonreduced and its $Spec$ is reducible? It's one of the exercise in the Wedhorn's book. To get a local ring I only need to do localization at ...
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