# Questions tagged [projective-schemes]

This tag is for questions relating to "projective scheme".

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### injective sheaves on a projective scheme cannot be coherent [duplicate]

Let $X$ be a projective scheme. If it helps (e.g. gives way to a short/elegant answer) fix a base field $k$ and assume smoothness. It is often said that the category of coherent sheaves over $X$ does ...
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### If $R_0 \not\cong S_0$ but $\exists n \in \mathbb N$ with $R_d \cong S_d$ for all $d \ge n$, then $\operatorname{Proj}R \cong \operatorname{Proj}S$? [duplicate]

Let $R=R_0 \oplus R_1 \oplus \cdots$ and $S=S_0 \oplus S_1 \oplus \cdots$ be finitely generated graded rings with $R_0 \not\cong S_0$. If there exists a positive integer $n$ such that $R_d \cong S_d$ ...
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### Definition of projectively normal in Harshorne Ex II.5.14

In Hartshorne, Ex II.5.14, he defines that a closed subscheme $X \in \mathbb{P}^r_A$ is called projectively normal if the homogeneous coordinate ring $S(X)$ is integrally closed. This is also given by ...
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### Projective scheme over a ring

I’m reading Qing Liu’s Algebraic Geometry and Arithmetic Curves. I’m confusing my double definitions of projective morphism over a ring $A$. In definition 2.3.42, projective scheme over $A$ is an $A$-...
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### Weighted projective space and projective space are isomorphic

From Vakil's book: Exercise 8.2.N Show that the weighted projective space $\mathbb{P}(m, n) = Proj(k[x, y])$ (where $x$ and $y$ have degrees $m$ and $n$ respectively) is isomorphic to $\mathbb{P}^1$. ...
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### Explicit proof that $\mathbb{P}_B^n\cong\mathbb{P}_A^n\times_{\operatorname{Spec}A}\operatorname{Spec} B$

On page 103 of Hartshore, just before the definition of projective morphisms, he states that if $\varphi:A\to B$ is a ring morphism and $(f,f^{\#}):\operatorname{Spec}B\to \operatorname{Spec}A$ is the ...
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### Are these rings the same? A detail in the Proj construction

The following detail is needed in the affine-by-affine gluing construction of the Proj of a graded rings. Every single reference that I know of takes the claim for granted. Let $A$ be a nonnegatively ...
### Map from $B$-scheme to projective space
I am trying to understand exercise 6.3.M(a) in Vakil's algebraic geometry notes. It goes as follows: Suppose $B$ is a ring. If $X$ is a $B$-scheme, and $f_0,...,f_n$ are $n+1$ functions on $X$ with no ...