Algebraic Geometry
《Algebraic Geometry》 · 第 245 页 · Chapter III，Section 7，Proposition 7.10A 前
Let $f_1,\ldots,f_r$ be elements of a ring $A$. Let $K_1$ be the free $A$-module on generators $e_1,\ldots,e_r$, let $K_p=\bigwedge^pK_1$, and define $d:K_p\to K_{p-1}$ by $d(e_{i_1}\wedge\cdots\wedge e_{i_p})=\sum_{j=1}^p(-1)^{j-1}f_{i_j}e_{i_1}\wedge\cdots\wedge\widehat e_{i_j}\wedge\cdots\wedge e_{i_p}$. The resulting complex is called the Koszul complex associated to $f_1,\ldots,f_r$.