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Tag 08WZ

Chapter 34: Descent > Section 34.4: Descent for universally injective morphisms

Lemma 34.4.16. For $(M,\theta) \in DD_{S/R}$, the diagram \begin{equation} \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 } \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 (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
    \begin{equation}
    \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
    }
    \end{equation}
    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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