# Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules

Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on $X$. Suppose $C$ is of the form $\mathrm{Proj}(S/I_C)$ for some ideal $I_C$. Is there any relation between the Castelnuovo-Mumford regularity of the coherent sheaf $\mathcal{O}_X(-C)$ and that of the module $I_C/I_X$?

EDIT Assume $X$ is irreducible but $C$ is not. $C$ might not even be reduced.

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Aren't they the same? You can compute the regularity of $I_C/I_X$ via local cohomology, the graded pieces are coherent cohomology of $O_X(-C)$, right? – Karl Schwede Sep 17 '13 at 16:53
@Schwede: That was the question. I could not find it mentioned in any literature. – Chen Sep 17 '13 at 16:54
@Schwede: It would be very helpful if you could cite some reference please. – Chen Sep 17 '13 at 16:56
Mel Hochster has some notes on local cohomology which do this in detail, but these are not freely available online. Various papers of Karen Smith also explain this in a lot of detail. Have you looked at any of her survey papers from the 90s? Finally, the relation between graded local cohomology and coherent cohomology is pretty easy to see if you consider Cech-cohomology. – Karl Schwede Sep 17 '13 at 16:58
Dear @Chen, you might also want to have a look at section 20.4 in M. Brodmann, R.Y. Sharp, Local cohomology (Cambridge studies in advanced mathematics 136). – Fred Rohrer Sep 18 '13 at 4:28