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Let $(A,m)$ be a commutative noetherian local ring, $E$ the injective hull of $A/m$, and $M$ a finitely generated $A$-module.

What is the connection between the support of $M$ and the support of the Matlis dual $M^* = Hom_A(M,E)$? are they the same? do they have the same Krull dimension?

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  • $\begingroup$ The Matlis dual is an artinian module over the completion $(\hat{A},\hat{m})$ of $(A,m)$. The support of the Matlis dual is thus obviously reduced to $\{\hat{m}\}$ (or $\emptyset$ if $M=0$). $\endgroup$
    – YCor
    Commented Nov 2, 2018 at 11:46
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    $\begingroup$ However, an article by Melkersson & Schenzel,and a later one by Andrew Richardson, deal with dual notions of co-support that are more suited to artiinian modules than support is. $\endgroup$
    – Neil Epstein
    Commented Nov 3, 2018 at 23:45

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