# 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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### image of Segre embedding is cut out (scheme-theoretically) by vanishing minors

I am doing exercise 10.6.B of Vakil's FOAG (July 31, 2023 version). Basically, I need to show that the image of the Segre embedding is cut out by equations so that the matrix \begin{bmatrix} a_{00}&...
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### Morphism of schemes and closed points [duplicate]

I apologize for the vagueness of my question. In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the &...
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### Gluing in the sheaf of meromorphic functions on an affine scheme

Let $X = \operatorname{Spec}(R)$ be an affine Noetherian scheme, and $U_i = D(f_i)$ (with $f_i \in R$) be (finitely many) principal open sets with $X = \bigcup_i U_i$. We have the sheaf of meromorphic ...
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### Cokernel of canonical skyscraper map not generated by global sections.

I am trying to solve the following exercise (Görtz and Wedhorn Exercise 7.7): Let $A$ be an integral domain with field of fractions $K$, set $X=\operatorname{Spec} A$ and let $\eta \in X$ be the ...
### does natural structure morphism from Proj $S$ to Spec $A$ require $S$ a finitely generated graded ring?
The question is already asked here natural structure morphism from Proj $S$ to Spec $A$ . And It’s exercise 7.3.I in Vakil’s FOAG: 7.3.I. EASY EXERCISE. If $S$ is a finitely generated graded A-...