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2

I figured it out. Let's write $\mathrm{QCoh}(\mathcal O_S)$ for the category of quasicoherent $\mathcal O_S$-modules and $\mathrm{Mod}(\mathcal O_S)$ for the category of all $\mathcal O_S$-modules. It turns out that the inclusion $\iota:\mathrm{QCoh}(\mathcal O_S) \to \mathrm{Mod}(\mathcal O_S)$ always has right adjoint, which we will denote by $Q:\mathrm{...


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Let $P \in K[x_1,\ldots,x_n]$ be nonzero on $V_K(f_1,\ldots,f_m)$. We want to show that there exists a point $z \in k^n$ such that $P(z) \neq 0$ and every $f_i(z)$ is zero. Let $z_0 \in K^n$ be a common root of the $f_i$ with $P(z_0) \neq 0$. Let $A$ be the finitely generated sub-$k$-algebra of $K$ (hence an integral domain) generated by the coefficients of $...


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Question: "So the degree of a curve could be the degree of its image as a closed subscheme of $\mathbb{P}^n$ as defined in (I,7) (with the Hilbert polynomial) is it correct? It seems that with this definition, the degree of a curve depends on the embedding one can consider." Answer: Yes, the degree of a projective variety/scheme depends on the ...


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As he says in the text, Vakil is describing a (non-full) subcategory of the category of open sets of the scheme $X$. Namely, he is considering the category whose objects are open affine subsets $U$ and where there is a single morphism $U \to V$ if and only if $U$ is a distinguished open affine of $V$ The reason this is not a base in the usual topological ...


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Probably trace, but this is the first time I've seen this construction called that. Yes, that's correct. No, there's no $k$-map $k(X)\to k(Y)$ - any map of fields must be an injection, but the target has smaller transcendence degree than the source (assuming $Y$ is a strict subvariety) which is impossible. Instead, locally on $U\subset X$ write $f=g/h$ ...


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I think this is true. So as $S$ is quasi-compact, some power $\mathcal L^{\otimes n}$ will be relatively very ample, inducing a closed immersion $$j: X \hookrightarrow \mathbf P(E).$$ Then we may descend both $X$ and $\mathbf P(E)$ to a closed immersion $X_i \subset \mathbf P(E_i)$. We may also descend the isomorphism $\mathcal L^{\otimes n} \cong j^* \...


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