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Let $(R,m)$ be a Cohen-Macaulay local ring, I and J are ideals of height $r.$ Then we say $I$ is directly linked to $J$, i.e. $I \sim J$ if there exists an ideal K generated by a regular sequence $x_1,\ldots,x_r$ such that $K\subset I\cap J,$ $I=K:J$ and $J=K:I.$

We say $I$ is linked to $J$ if there exist ideals $I_1,\ldots,I_s$ of height $r$ and generated by regular sequences such that $I\sim I_1, I_1\sim I_2,\ldots, I_s\sim J.$

Question: Is the operation "$I$ is linked to $J$" an equivalence relation? Particularly I do not understand how it is reflexive?

enter link description here [Linkage and the Koszul Homology of Ideals - C. Huneke]

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  • $\begingroup$ A functorial definition of linkage for stable modules over Noetherian rings (i.e. in the module category modulo projectives) is presented in the following paper: [arxiv.org/pdf/1505.01112v3.pdf] ... a module $M$ over a Noetherian ring is linked if there is an isomorphism between $M$ and the 'syzygy transpose syzygy transpose' of $M$, for lack of a better name. Not sure if that's too high-flown to be of any help $\endgroup$
    – Samantha Y
    Sep 15, 2016 at 16:07
  • $\begingroup$ The relation is clearly symmetric, so if $I\sim J$, then $J\sim I$ and thus putting them together, one gets $I\sim I$. $\endgroup$
    – Mohan
    Sep 15, 2016 at 16:24
  • $\begingroup$ @Mohan Your comment presupposes that a given $I$ is directly linked to at east one $J$. $\endgroup$ Sep 15, 2016 at 18:13
  • $\begingroup$ @AndreasBlass Yes, I am supposing $R/I$ is Cohen-Macaulay, which is where linkage really plays a role. If this is the case, then such an $I$ is always directly linked to a $J$ with $R/J$ Cohen-Macaulay. $\endgroup$
    – Mohan
    Sep 15, 2016 at 18:38
  • $\begingroup$ @Mohan It will be extremely helpful for me if you explain your last comment that $R/I$ is Cohen-Macaulay. $\endgroup$
    – Cusp
    Sep 16, 2016 at 3:01

1 Answer 1

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If $I$ is an ideal with $R/I$ CM, pick a regular sequence $x_1,\ldots, x_r\in I$, where $r$ is the height of $I$. If we call $K$ the ideal generated by the $x_i$s, we can define $J=(K:I)$. One checks that $I=K:J$ and then we are done once we also check that $R/J$ is also CM.

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  • $\begingroup$ Could you please suggest me some reference for the proof. I know that the above statement is true when the ring is Gorenstein where it is used that injective hull of k=R/m is isomorphic to R when R is 0-dimensional Gorenstein ring. $\endgroup$
    – Cusp
    Sep 17, 2016 at 5:38
  • $\begingroup$ I should have said Gorenstein. I do not know whether this is true without the hypothesis. $\endgroup$
    – Mohan
    Sep 17, 2016 at 13:24

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