# 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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### Is every affine variety an affine scheme?

I have seen that the notion of affine scheme is a generalization of the notion of affine varieties where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski ...
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### Gluing step in the construction of the fiber product of $S$-schemes

I am reading through the construction of $X \times_S Y$, where $X$ and $Y$ are $S$-schemes in Liu's Algebraic Geometry and Arithmetic Curves (Proposition 3.1.2), and I am somewhat stuck justifying a ...
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### Shortest path from undergrad to the (co)tangent complex?

After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being ...
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### For a family of extensions of sheaves, does nonsplit over generic fibre imply nonsplit over special fibre?

Let $A\subseteq K$ be discrete valuation ring in its field of fractions. Let $\kappa:=A/\mathfrak m$ be the residue field. Let $X$ be a scheme, flat and projective over $A$. Consider a short exact ...
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### There is birational and integral morphism between glued curve and original curve

Let $C_0:$ be a projective closure of affine curve $y^{2}=x^4-7$. $C_0$ has singular point $[0:1:0]$ at infinity. Let another affine curve be $C_1: v^{2}=u^{4}(1/u^4-7)=1-7u^4$. To make smooth ...
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### A complete variety is proper over $k$: why does it suffice to check universally closed on varieties instead of all schemes?

A complete variety $X$ over $k$ is defined to be one where the projection map $X\times_{\operatorname{spec} k} Y\to Y$ is closed for every variety $Y$. It is stated on Wikipedia and various places ...
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### Degree of the image of a curve given by canonical divisor

Suppose $C$ is a curve of genus $g$. Let $K$ be a canonical divisor on $C$, let $n\geq 3$ be an integer, and let $C\to\mathbb{P}^N$ be the map determined by the linear system $|nK|$, where $N$ can be ...
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### Equivalent definition of geometric vector bundles over schemes

The following is Hartshonre's definition of geometric vector bundles: Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\to Y$ together with ...
1 vote
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### Degree of image of the canonical map [duplicate]

Suppose $X$ is a nonhyperelliptic curve of genus $g\geq 3$. There is an embedding $X\to\mathbb{P}^{g-1}$ determined by the canonical linear system. Hartshorne states that the image of this map is a ...
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### fibre product with a scheme in terms of its glueing data

So heres the situation. Let S be a scheme. Imagine you have two S-schemes X,Y with restrictions U,V respectively that are glued via a homeomorphism $f:U\cong V$. lets call the sheaf we obtain by this ...
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### If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$.

I am trying to understand the following statement: If $X$ is a not a rational curve, then $\dim |P|=0$ for all $P$ in $X$. This is stated by Hartshorne while proving Lemma 5.1, that a canonical linear ...
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### For a proper scheme $X,$ $X(\mathbb{Z})=X(\mathbb{Q})?$

Let $X$ be an integral model of a $\mathbb{Q}$-variety. If $X$ is proper, I found in a paper (page 1 in https://arxiv.org/pdf/2010.11763.pdf) that says $X(\mathbb{Z})=X(\mathbb{Q})$ from the valuation ...
1 vote
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### flatness of structural sheaf of scheme over ring of global sections

Let $X$ be an affine scheme. $U \subseteq X$ open. then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$. But I want to prove it only by knowing the definition of structure sheaf ...
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### Görtz-Wedhorn Proposition 7.46. Coherence in sheaves over locally noetherian schemes.

I am trying to undestand the following proposition in Görtz-Wedhorn Algebraic Geometry I. What I do not understand is the reduction to the affine case he does in order to prove $(iii) \Rightarrow (i)$....
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### Is there a notion of inverse image for schemes?

Let $f: X \to Y$ be a morphism of schemes, and let $Z$ be a subscheme of $Y$. Is there any reasonable way to talk about an "inverse image" of $Z$?
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### Sheaf morphism from closed subscheme is a closed immersion

For $K=\bar{K}$ a field consider $X=\mathbb P^1_K$, $Z=\{P_1,\dots,P_n\}\subseteq X$ closed points. Give $Z$ the reduced induced closed subscheme structure and write $\iota:Z\to X$ the closed ...
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### Monomorphism of schemes which is not a an open or closed immersion

I have showed that every open/closed immersion of schemes is a monomorphism of schemes, but I know that the converse is not true, i.e., not every monomorphism of schemes is an open/closed immersion. ...
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### How do I interpret the intersection of a variety with a "non-closed hyperplane?"

I am trying to understand Vakil's statement and proof of Bertini's theorem, which has been updated since many of the questions related to it were posted on this website (for what it's worth, I'm not ...
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### detail in proof that existence of ample invertible sheaf implies separatedness

Let $X$ be a scheme and $\mathcal{L}$ be an ample invertible sheaf on $X$, then lemma 28.26.7 in stacks project says that $X$ is separated. The proof uses the valuative criterion and goes as follows. ...
I was doing Professor Vakil's FOAG in exercise 9.1 F needs to prove: that quasicoherent sheaf of ideals on $Y$ produce an closed subscheme on $Y$. Here is my attempt, to construct it locally is not ...