Lemma 15.29.6. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$. The extended alternating Čech complex

$R \to \bigoplus \nolimits _{i_0} R_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r}$

is a colimit of the Koszul complexes $K(R, f_1^ n, \ldots , f_ r^ n)$; see proof for a precise statement.

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

$K(R, f_1^ n, \ldots , f_ r^ n) : 0 \to \wedge ^ r(R^{\oplus r}) \to \wedge ^{r - 1}(R^{\oplus r}) \to \ldots \to R^{\oplus r} \to R \to 0$

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

$d(e^ n_{j_1} \wedge \ldots \wedge e^ n_{j_{r - p}}) = \sum (-1)^{a + 1} f_{j_ a}^ n e^ n_{j_1} \wedge \ldots \wedge \hat e^ n_{j_ a} \wedge \ldots \wedge e^ n_{j_{r - p}}$

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

$K(R, f_1^ n, \ldots , f_ r^ n) \to K(R, f_1^{n + 1}, \ldots , f_ r^{n + 1})$

are given by the rule

$e^ n_{j_1} \wedge \ldots \wedge e^ n_{j_{r - p}} \longmapsto f_{i_0} \ldots f_{i_{p - 1}} e^{n + 1}_{j_1} \wedge \ldots \wedge e^{n + 1}_{j_{r - p}}$

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

$R_ f = \mathop{\mathrm{colim}}\nolimits (R \xrightarrow {f} R \xrightarrow {f} R \to \ldots )$

Hence we see that in degree $p$ we obtain

$\mathop{\mathrm{colim}}\nolimits K^ p(R, f_1^ n, \ldots f_ r^ n) = \bigoplus \nolimits _{1 \leq i_0 < \ldots < i_{p - 1} \leq r} R_{f_{i_0} \ldots f_{i_{p - 1}}}$

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 $\{ 1, \ldots r\} = \{ i_0, \ldots , i_{p - 1}\} \amalg \{ j_1, \ldots , j_{r - p - 2}\}$. Thus the differential on this complex is given by

$d(1\text{ in }R_{f_{i_0} \ldots f_{i_{p - 1}}}) = \sum \nolimits _{i \not\in \{ i_0, \ldots , i_{p - 1}\} } (-1)^{i - t}\text{ in } R_{f_{i_0} \ldots f_{i_ t} f_ i f_{i_{t + 1}} \ldots f_{i_{p - 1}}}$

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

$\bigoplus \nolimits _{1 \leq i_0 < \ldots < i_{p - 1} \leq r} R_{f_{i_0} \ldots f_{i_{p - 1}}} \longrightarrow \bigoplus \nolimits _{1 \leq i_0 < \ldots < i_{p - 1} \leq r} R_{f_{i_0} \ldots f_{i_{p - 1}}}$

determined by the rule

$1\text{ in }R_{f_{i_0} \ldots f_{i_{p - 1}}} \longmapsto (-1)^{i_0 + \ldots + i_{p - 1} + p}\text{ in }R_{f_{i_0} \ldots f_{i_{p - 1}}}$

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. $\square$

Comment #6206 by Owen on

Toward the end of the proof it is written the indices are chosen so that $\{1,\ldots,r\}=\{i_0,\ldots,i_{p-1}\}\coprod\{j_1,\ldots,j_{r-p-2}\}$. I think $\{1,\ldots,r\}=\{i_0,\ldots,i_{p-1}\}\coprod\{j_1,\ldots,j_{r-p}\}$ is meant.

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