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Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{m}$ for every maximal ideal $\mathfrak{m}\subseteq R$.

Consider the following alternate condition for an $R$-module $M$: each $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{m}$ and finitely many copies of $\mathfrak{m}$.

Does this type of module have a global characterization or a name?

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    $\begingroup$ Such a module could be quite bad if $R$ is nonreduced, e.g., a local Artinian ring. If the scheme is a nodal curve, such modules come up as limits of locally free modules in the Altman-Kleiman compactification of the moduli space of vector bundles. If $R$ is $S2$, then the reflexive hull of such a module is locally free, and the cokernel is a direct sume of (reduced) skyscraper sheaves / modules. $\endgroup$
    – Jason Starr
    Commented Feb 18, 2018 at 13:58
  • $\begingroup$ @Jason Thanks for this very interesting comment! In the example I have in mind, the ring is, in fact, reduced. Can something more be said under this assumption? $\endgroup$
    – Ben Knudsen
    Commented Mar 2, 2018 at 18:02

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