Lemma 35.4.16 (08WZ)—The Stacks project

Lemma 35.4.16. For $(M,\theta ) \in DD_{S/R}$, the diagram

35.4.16.1

\begin{equation} \label{descent-equation-equalizer-M} \xymatrix@C=8pc{ M \ar[r]^{\theta \circ (1_ M \otimes \delta _0^1)} & M \otimes _{S, \delta _1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta _2^2) \circ (1_ M \otimes \delta ^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta ^2_1} & M \otimes _{S, \delta _{12}^1} S_3 } \end{equation}

is a split equalizer.

Proof. Define the ring homomorphisms $\sigma ^0_0: S_2 \to S_1$ and $\sigma _0^1, \sigma _1^1: S_3 \to S_2$ by the formulas

\begin{align*} \sigma ^0_0 (a_0 \otimes a_1) & = a_0a_1 \\ \sigma ^1_0 (a_0 \otimes a_1 \otimes a_2) & = a_0a_1 \otimes a_2 \\ \sigma ^1_1 (a_0 \otimes a_1 \otimes a_2) & = a_0 \otimes a_1a_2. \end{align*}

We then take the auxiliary morphisms to be $1_ M \otimes \sigma _0^0: M \otimes _{S, \delta _1^1} S_2 \to M$ and $1_ M \otimes \sigma _0^1: M \otimes _{S,\delta _{12}^1} S_3 \to M \otimes _{S, \delta _1^1} S_2$. Of the compatibilities required in (35.4.2.1), the first follows from tensoring the cocycle condition (35.4.14.1) with $\sigma _1^1$ and the others are immediate. $\square$

The Stacks project

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