$A$ a local ring and $a\_{1}$, ..., $a\_{n}$ elements in its maximal ideal, $M$ a finitely generated $A$-module. In this case apparently the homologies from the Koszul complex are finitely generated as $A$-module. Is there a simple explanation for this? Or is it some deep result? Thanks!



Homology of koszul complex is finitely generated?
If your local rings are Noetherian it's obvious. The Koszul complex
consists of finitely generated free modules and the homology
modules are subquotients of it so also finitely generated.


For non-Noetherian rings it's not true, even when $n=1$. In
this case the $H\_1$ is the annihilator of $a\_1$ which may not
be finitely generated. As an example consider the quotient
of $k[x\_1,x\_2,\ldots]$ by the ideal generated by all the
$x\_i x\_j$. The annihilator of $x\_1$ is the
non-finitely generated maximal ideal.


