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Let R the polynomial ring in n variables with complex coefficients and I an ideal of R. Is it true that if R/I is CM also R/J is CM (where J is the radical of I)? Is there a relations between a resolution of R/J and one of R/I? What if I suppose that proj.dim(R/I)=2?

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Why projective dimension 2? Hartshorne's example $k[s^4,s^3t,st^3,t^4]$ works in char $p>0$. –  Hailong Dao Nov 27 '10 at 17:12
because I'm looking at a case in which the projective dimension is 2. Sorry, I'm not confident with Hartshorne's example....but I work in char p=0. –  Michele Torielli Nov 27 '10 at 17:19
Michele: I doubt it will be true in projective dimension 2. If you have a specific situation, you should post the details separately. This is not easy stuff to answer without knowing full details. –  Hailong Dao Nov 27 '10 at 18:07
sorry, you are right. In my case I is the jacobian ideal of a polynomial f such that R/I is CM of codim 2 and hence proj.dim 2 –  Michele Torielli Nov 27 '10 at 18:16
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2 Answers

up vote 9 down vote accepted

It is not true, but the example is not easy to find $I = (x_2^2-x_4x_5,x_1x_3-x_3x_4, x_3x_4-x_1x_5)$!

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ok, is it true adding some restriction on I? –  Michele Torielli Nov 26 '10 at 15:20
The only thing that jumps to my mind is hypersurfaces. In general, there's probably not much you can say. For example, many many varieties are locally set theoretic complete intersections (ie, the radicals are cut out by a regular sequence locally) but don't satisfy any nice properties themselves. Maybe Long will have some more suggestions? –  Karl Schwede Nov 26 '10 at 15:41
what if R is local?like power series? –  Michele Torielli Nov 26 '10 at 16:07
Michele: the same example works in the power series rings $k[[x_1,\cdots,x_5]]$. In addition to what Karl says, if $I$ is monomial, then what you say is true. –  Hailong Dao Nov 26 '10 at 16:24
With respect to what I said, it should be "(ie, they are cut out by regular sequences up to radical)." –  Karl Schwede Nov 26 '10 at 18:22
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Yes. From Eisenbud's Commutative Algebra: a ring $S$ is Cohen-Macaulay iff all the maximal ideal $m$ of $S$ satisfies codim($m$) = depth($m$). Now, the maximal ideals of $R/J$ are the same as $R/I$ and their depths and codimensions are the same as well.

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so it seems that also if R/J is CM then R/I is CM, but this is false if I'm not wrong. –  Michele Torielli Nov 26 '10 at 14:33
You are right. I guess you also need to observe to things: 1) a maximal $R/I$-sequence is also a maximal $R/J$ sequence, so that depth($R/J$) $\geq$ depth($R/I$), and 2) depth of any maximal ideal is less than or equal to its codimension. –  auniket Nov 26 '10 at 14:58
@auniket: I don't think that a $R/I$-sequence always becomes $R/J$-sequence. –  Hailong Dao Nov 26 '10 at 16:26
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