 Koszul complex

ID: koszul-complex

Codex  0 Created 2026-10-06 Updated 2026-10-14

f_{1}, \dots, f_{r}

has term

K_{i} = ⋀^{i} R^{r}

and differential

\partial (e_{j_{1}} \land \dots \land e_{j_{i}}) = \sum_{a} (- 1)^{a - 1} f_{j_{a}} e_{j_{1}} \land \dots \land e_{j_{a}} \land \dots \land e_{j_{i}}

. Its degree-zero homology is

R / (f_{1}, \dots, 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.

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.

 New to topics? Read the docs here!