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Edited: Let $X$ be an integral scheme and $R$ a Weil divisor of $X$. How can we view $R$ as a closed subscheme of $X$? What is the corresponding sheaf? My question is motivated by Hartshorne p. 301, where in the proof of proposition 2.3 he mentions that we can view the ramification divisor as a closed subscheme of the curve $X$.

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What are the hypothesis on $X$ ? To define a Weil divisor (cycle of codimension $1$), you need at least to assume $X$ is noetherian. – user18119 Nov 18 '12 at 23:35
@QiL: Yes, you can assume $X$ is Noetherian. I edited. – Manos Nov 19 '12 at 20:02
As explained Theodore, when $R$ is any positive Weil divisor, there is usually no canonical structure of closed subscheme, except when all multiplicities are one, then it is just endowed with the structure of reduced closed subscheme. If $R$ is associated to an effective Cartier divisor $D$ (automatic when $X$ is regular), then the sheaf $O_X(-D)$ is a sheaf of ideals and $R$ can be endowed with the structure of closed subscheme defined by $O_X(-D)$. – user18119 Nov 19 '12 at 20:50
up vote 4 down vote accepted

I guess this depends on what you mean by "divisor". Let me assume you mean Weil divisor. A Weil divisor is a finite sum $\sum_i n_i D_i$, where $D_i$ is an integral subscheme and $n_i$ is an integer. The support of this divisor is thus a closed subscheme. Now, there is just one important thing: multiplicities.

If you'd like, you can just ignore the multiplicities $n_i$. This way you obtain a reduced closed subscheme. If you don't do this, your subscheme is non-reduced once $n_i >1 $ or $n_i <-1$ for some $i$.

If you mean Cartier divisor, then for "nice" schemes these correspond to Weil divisors. And thus, the support of the corresponding Weil divisor gives a closed subscheme.

Also, the ramification divisor of a curve is a subtle notion. If you have $f:X\to Y$ a finite flat surjective morphism of regular integral schemes, then the ramification divisor is the Weil divisor supported on the ramification locus of $f$. It has multiplicities though (if $f$ is not etale) whereas the ramification locus is usually considered with its reduced closed subscheme structure.

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How do you define the support of a Weil divisor? I also edited my question to ask what is the corresponding sheaf of the Weil divisor viewed as a closed subscheme. – Manos Nov 18 '12 at 21:44
If $D=\sum n_i [D_i]$, then the support of $D$ is the union of all $D_i$. (This union is finite, thus it is a closed subscheme.) – Theodore Nov 18 '12 at 21:45
Thank you. What is the corresponding sheaf $O_D$? – Manos Nov 18 '12 at 21:51
Can i take as sheaf associated to $D$ any sheaf with same support as $D$? – Manos Nov 18 '12 at 22:29
Let $D$ be a Cartier divisor. Then the sheaf $\mathcal{O}_X(D)$ is defined using the local functions defining $D$. The sheaf $\mathcal{O}_D$ is the quotient of $\mathcal{O}_X$ by $\mathcal{O}_X(-D)$. – Theodore Nov 20 '12 at 22:04

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