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How are the definitions of Weil divisors with subsets and subscheme equivalent
They are equivalent because of the following. Given any closed subset $Z\subseteq X$ of a scheme there is NOT a unique scheme structure on $Z$. But, there is a unique minimal scheme structure the one coming from the reduced structure on $Z$. Explicitly, find an ideal sheaf $\mathcal{I}$ fake Cartier love bracelet gold and diamond of $X$ which cuts out $Z$ and take the scheme structure on $Z$ given by $\mathcal{O}_Z/\sqrt{\mathcal{I}}$. This is called the reduced structure on $Z$
Your answer should be pretty clear once you recall that a scheme $Z$ is integral if and only if it's reduced and irreducible. So, to a closed irreducible subset $Z\subseteq X$ of comdimension $1$, there is a unique closed integral subscheme having replica cartier love bracelet $Z$ as its underlying space.
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They are equivalent because of the following. Given any closed subset $Z\subseteq X$ of a scheme there is NOT a unique scheme structure on $Z$. But, there is a unique minimal scheme structure the one coming from the reduced structure on $Z$. Explicitly, find an ideal sheaf $\mathcal{I}$ fake Cartier love bracelet gold and diamond of $X$ which cuts out $Z$ and take the scheme structure on $Z$ given by $\mathcal{O}_Z/\sqrt{\mathcal{I}}$. This is called the reduced structure on $Z$
Your answer should be pretty clear once you recall that a scheme $Z$ is integral if and only if it's reduced and irreducible. So, to a closed irreducible subset $Z\subseteq X$ of comdimension $1$, there is a unique closed integral subscheme having replica cartier love bracelet $Z$ as its underlying space.
Correspondence Between Cartier Divisor and Weil Divisor (Hartshorne Proposition 6.11, Chapter 2)How to make sense of the inverse image of a Weil divisor?Why does restriction of Weil divisors preserve principal ness?Why do we have two definitions of Cartier divisor?Effective Weil divisors on a Surface and local equationsSupport of an replica Cartier love bracelet Platinum effective Cartier vs Weil divisorEquivalence of two definitions of proper morphism for separated varietiesDefinition of replica cartier love bracelet18k gold weil divisors.
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