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Tag 08WS

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 $$\tag{34.4.11.1} C(\bullet_1 \otimes_R \bullet_2) \cong \mathop{\rm Hom}\nolimits_R(\bullet_1, C(\bullet_2)): \text{Mod}_R \times \text{Mod}_R \to \text{Mod}_R$$ 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 $Q \in \text{Mod}_R$.

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\begin{remark}
We will use frequently the standard adjunction between $\Hom$ and tensor
product, in the form of the natural isomorphism of contravariant functors

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

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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