= Koszul complex
{c}
{title2=$K_\bullet(f_1,\ldots,f_r;R)$}
{wiki}
The Koszul complex on $f_1,\ldots,f_r$ has term $K_i=\bigwedge^iR^r$ and differential $\partial(e_{j_1}\wedge\cdots\wedge e_{j_i})=\sum_a(-1)^{a-1}f_{j_a}e_{j_1}\wedge\cdots\wedge\widehat{e_{j_a}}\wedge\cdots\wedge e_{j_i}$. Its degree-zero <homology> is $R/(f_1,\ldots,f_r)$. For a <regular sequence> it is exact in positive degrees, hence a <Koszul resolution> of that quotient. The variables of a <polynomial ring> form a <regular sequence>, so this resolves the coefficient <field> in length equal to the number of variables.
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