The Koszul complex on has term and differential . Its degree-zero homology is . 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.
For a possibly noncommutative ring and central , the formal exterior basis gives free bimodules with differential . Centrality makes this a bimodule differential. Pairwise cancellation proves .
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The Koszul complex is a construction in algebraic topology and commutative algebra that arises in the study of modules over a ring and their syzygies. It is particularly useful for understanding the structure of modules over a polynomial ring and in computing homological properties.