# The Stacks Project

## Tag 08WZ

Lemma 34.4.16. For $(M,\theta) \in DD_{S/R}$, the diagram $$\tag{34.4.16.1} \xymatrix@C=8pc{ M \ar[r]^{\theta \circ (1_M \otimes \delta_0^1)} & M \otimes_{S, \delta_1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta^2_1} & M \otimes_{S, \delta_{12}^1} S_3 }$$ 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 (34.4.2.1), the first follows from tensoring the cocycle condition (34.4.14.1) with $\sigma_1^1$ and the others are immediate. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1106–1120 (see updates for more information).

\begin{lemma}
\label{lemma-equalizer-M}
For $(M,\theta) \in DD_{S/R}$, the diagram

\label{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
}

is a split equalizer.
\end{lemma}

\begin{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 (\ref{equation-split-equalizer-conditions}),
the first follows from tensoring the cocycle condition
(\ref{equation-cocycle-condition}) with $\sigma_1^1$
and the others are immediate.
\end{proof}

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