# The Stacks Project

## Tag 08X1

Lemma 34.4.17. For $(M, \theta) \in DD_{S/R}$, the diagram $$\tag{34.4.17.1} \xymatrix@C=8pc{ C(M \otimes_{S, \delta_{12}^1} S_3) \ar@<1ex>[r]^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))} \ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta^2_1)} & C(M \otimes_{S, \delta_1^1} S_2 ) \ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(M). }$$ obtained by applying $C$ to (34.4.16.1) is a split coequalizer.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1140–1155 (see updates for more information).

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

\label{equation-coequalizer-CM}
\xymatrix@C=8pc{
C(M \otimes_{S, \delta_{12}^1} S_3)
\ar@<1ex>[r]^{C((\theta \otimes \delta_2^2) \circ (1_M \otimes \delta^2_0))}
\ar@<-1ex>[r]_{C(1_{M \otimes S_2} \otimes \delta^2_1)} &
C(M \otimes_{S, \delta_1^1} S_2 )
\ar[r]^{C(\theta \circ (1_M \otimes \delta_0^1))} & C(M).
}

obtained by applying $C$ to (\ref{equation-equalizer-M}) is a split
coequalizer.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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