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Let R be a ring. For example, take $R=k[x_1,\ldots,x_n]$ or, if possible, $R = \Bbb{Z}[x_1,\ldots,x_n]$.

Consider a sequence of free R-modules $$R^a \stackrel{f}\to R^b \stackrel{g}\to R^c$$ where $f$ and $g$ are explicitly given by appropriately sized matrices.

How does one check (in practice, presumably(?) on a computer) whether or not this sequence is exact?

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    $\begingroup$ Macaulay2 has "kernel" and "image" commands. Don't they do it? math.uiuc.edu/Macaulay2/doc/Macaulay2-1.8.1/share/doc/Macaulay2/… $\endgroup$
    – Lazzaro Campeotti
    Jul 10, 2015 at 7:31
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    $\begingroup$ To be more explicit about the mathematics involved, computer-algorithms for calculating images and kernels (and a host of other things) over commutative rings all use Groebner Bases techniques. However, for an easier necessary condition, if $gf=0$, you can tensor with the field of fractions of $R$ and do linear algebra. This cannot prove exactness, but it will quickly eliminate things that aren't exact. $\endgroup$
    – Aaron
    Jul 10, 2015 at 14:22
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    $\begingroup$ If you want to check by hand, you can use Buchsbaum-Eisenbud criterion. You can find it in their paper: What makes a complex exact? I think published in 1973. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Jul 11, 2015 at 15:29

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