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(1) Let $X$ be a Noetherian scheme of dimension 1 over a field $k$ and $supp(D)$ denote the support of a $Cartier$ divisor $D$ on $X$. Let $S\subseteq supp(D)$ be consists of closed points of $supp(D)$.Then why does $S$ has finite cardinality ?

(2) If $D$ is an effective $Cartier$ divisor , then why $S=supp(D)$ ?

I have tried to solve these problems but could not succeed.

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  • $\begingroup$ Also suggest me some reference where can i get these stuff. $\endgroup$ Apr 19, 2016 at 8:20

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The support of a Cartier-Divisor has codimension $1$. Hence, in your case it is $0$-dimensional. It is also noetherian, hence there are finitely many irreducible components. Thus it suffices to show that each of those components is finite. Actually each component is a point, since we have the following:

An irreducible $0$-dimensional space, which admits a closed point, is a point.

This is an easy exercise in topology.

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  • $\begingroup$ It means that every point of $supp(D)$ is closed point. $\endgroup$ Apr 19, 2016 at 17:30
  • $\begingroup$ How is the support of a Cartier divisor has codimension 1? @MooS $\endgroup$ Apr 19, 2016 at 17:32

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