The Stacks project

Proof. We urge the reader to prove this for themselves. Denote $K(R, f_1^ n, \ldots , f_ r^ n)$ the Koszul complex of Definition 15.28.2 viewed as a cochain complex sitting in degrees $0, \ldots , r$. Thus we have

with the term $\wedge ^ r(R^{\oplus r})$ sitting in degree $0$. Let $e^ n_1, \ldots , e^ n_ r$ be the standard basis of $R^{\oplus r}$. Then the elements $e^ n_{j_1} \wedge \ldots \wedge e^ n_{j_{r - p}}$ for $1 \leq j_1 < \ldots < j_{r - p} \leq r$ form a basis for the term in degree $p$ of the Koszul complex. Further, observe that

by our construction of the Koszul complex in Section 15.28. The transition maps of our system

are given by the rule

where the indices $1 \leq i_0 < \ldots < i_{p - 1} \leq r$ are such that $\{ 1, \ldots r\} = \{ i_0, \ldots , i_{p - 1}\} \amalg \{ j_1, \ldots , j_{r - p}\}$. We omit the short computation that shows this is compatible with differentials. Observe that the transition maps are always $1$ in degree $0$ and equal to $f_1 \ldots f_ r$ in degree $r$.

Denote $K^ p(R, f_1^ n, \ldots , f_ r^ n)$ the term of degree $p$ in the Koszul complex. Observe that for any $f \in R$ we have

Hence we see that in degree $p$ we obtain

Here the element $e^ n_{j_1} \wedge \ldots \wedge e^ n_{j_{r - p}}$ of the Koszul complex above maps in the colimit to the element $(f_{i_0} \ldots f_{i_{p - 1}})^{-n}$ in the summand $R_{f_{i_0} \ldots f_{i_{p - 1}}}$ where the indices are chosen such that $\{ 1, \ldots r\} = \{ i_0, \ldots , i_{p - 1}\} \amalg \{ j_1, \ldots , j_{r - p}\}$. Thus the differential on this complex is given by

Thus if we consider the map of complexes given in degree $p$ by the map

determined by the rule

then we get an isomorphism of complexes from $\mathop{\mathrm{colim}}\nolimits K(R, f_1^ n, \ldots , f_ r^ n)$ to the extended alternating Čech complex defined in this section. We omit the verification that the signs work out. $\square$