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In an article, I encounter an isomorphism relation as follows:

Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism":

$K[x,S]$ isom. to $\Sigma^{-1} Hom_S (K[x,S],S)$, where \Sigma is the translation functor.

I guess this mean we should regard S as a chain complex concentrated at 0, and look at the homomophism between chain complexes, but I cannot figure out the relation.

Explicitly, how to identify the homomophism of chain complexes with K[x,S] as categories?

Thanks a lot.

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  • $\begingroup$ Just write down what the complex $\hom_S(K[x,S],S)$ is, using the fact that the dual of a free module of rank $1$ is of free of rank $1$. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 24, 2011 at 2:45
  • $\begingroup$ @Mariano Suárez-Alvarez: I am just a starter of comm. and homological algebra. Do I understand $Hom_S (K[x,S],S)$ correctly? Thank you for your patience. $\endgroup$
    – AlgRev
    Commented Oct 24, 2011 at 2:52

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