## Tag `08WS`

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

Remark 34.4.11. We will use frequently the standard adjunction between $\mathop{\rm Hom}\nolimits$ and tensor product, in the form of the natural isomorphism of contravariant functors \begin{equation} \tag{34.4.11.1} C(\bullet_1 \otimes_R \bullet_2) \cong \mathop{\rm Hom}\nolimits_R(\bullet_1, C(\bullet_2)): \text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R \end{equation} taking $f: M_1 \otimes_R M_2 \to \mathbf{Q}/\mathbf{Z}$ to the map $m_1 \mapsto (m_2 \mapsto f(m_1 \otimes m_2))$. See Algebra, Lemma 10.13.5. A corollary of this observation is that if $$ \xymatrix@C=9pc{ C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P) } $$ is a split coequalizer diagram in $\text{Mod}_R$, then so is $$ \xymatrix@C=9pc{ C(M \otimes_R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes_R Q) \ar[r] & C(P \otimes_R Q) } $$ for any $Q \in \text{Mod}_R$.

The code snippet corresponding to this tag is a part of the file `descent.tex` and is located in lines 968–994 (see updates for more information).

```
\begin{remark}
\label{remark-adjunction}
We will use frequently the standard adjunction between $\Hom$ and tensor
product, in the form of the natural isomorphism of contravariant functors
\begin{equation}
\label{equation-adjunction}
C(\bullet_1 \otimes_R \bullet_2) \cong \Hom_R(\bullet_1, C(\bullet_2)):
\text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R
\end{equation}
taking $f: M_1 \otimes_R M_2 \to \mathbf{Q}/\mathbf{Z}$ to the map $m_1 \mapsto
(m_2 \mapsto f(m_1 \otimes m_2))$. See
Algebra, Lemma \ref{algebra-lemma-hom-from-tensor-product-variant}.
A corollary of this observation is that if
$$
\xymatrix@C=9pc{
C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P)
}
$$
is a split coequalizer diagram in $\text{Mod}_R$, then so is
$$
\xymatrix@C=9pc{
C(M \otimes_R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes_R Q) \ar[r] & C(P
\otimes_R Q)
}
$$
for any $Q \in \text{Mod}_R$.
\end{remark}
```

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