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Let $R$ be a commutative noetherian local ring (with 1) and let $M$ be a finitely generated $R$-module. Consider a minimal free presentation of $M$ as follows: $R^{\beta_2}\rightarrow R^{\beta_1}\rightarrow M\rightarrow0$. Is there a nice characterization of modules with $\beta_1>\beta_2$? What are these modules?

I had naively thought perhaps they are modules with annihilator equal to zero, but that's not true as you can see in a previous question here.

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  • $\begingroup$ Perhaps this question is too general to have a definite answer. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Dec 18, 2011 at 0:43
  • $\begingroup$ Well, what kind of characterization you are hoping for? $\endgroup$
    – Hailong Dao
    Commented Dec 18, 2011 at 3:14
  • $\begingroup$ Anything that is special about these modules could be interesting. What are some conditions (if any) that are equivalent to having $\beta_1>\beta_2$? $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Dec 18, 2011 at 4:19
  • $\begingroup$ Do you know what kind of rings is $R$? $\endgroup$
    – Hailong Dao
    Commented Dec 18, 2011 at 5:58
  • $\begingroup$ $R$ is arbitrary, but if it is necessary to have extra assumptions for some equivalent conditions, you can add those assumptions. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Dec 18, 2011 at 15:11

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