= Koszul resolution
{c}
For a <regular sequence> $f_1,\ldots,f_r$ in a commutative <ring> $R$, its Koszul complex is the finite free <projective resolution> of $R/(f_1,\ldots,f_r)$ with terms $\bigwedge^jR^r$ and <boundary map> contracting by $(f_1,\ldots,f_r)$. Exactness follows by induction: append $f_r$ using the <mapping cone> for multiplication by $f_r$, which is injective on the preceding quotient. For a <polynomial ring> viewed as a <bimodule>, use the <regular sequence> $X_{i,\ell}-X_{i,r}$ in the enveloping <algebra>.
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