The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.75 Functorialities for Tor

In this section we briefly discuss the functoriality of $\text{Tor}$ with respect to change of ring, etc. Here is a list of items to work out.

  1. Given a ring map $R \to R'$, an $R$-module $M$ and an $R'$-module $N'$ the $R$-modules $\text{Tor}_ i^ R(M, N')$ have a natural $R'$-module structure.

  2. Given a ring map $R \to R'$ and $R$-modules $M$, $N$ there is a natural $R$-module map $\text{Tor}_ i^ R(M, N) \to \text{Tor}_ i^{R'}(M \otimes _ R R', N \otimes _ R R')$.

  3. Given a ring map $R \to R'$ an $R$-module $M$ and an $R'$-module $N'$ there exists a natural $R'$-module map $\text{Tor}_ i^ R(M, N') \to \text{Tor}_ i^{R'}(M \otimes _ R R', N')$.

Lemma 10.75.1. Given a flat ring map $R \to R'$ and $R$-modules $M$, $N$ the natural $R$-module map $\text{Tor}_ i^ R(M, N)\otimes _ R R' \to \text{Tor}_ i^{R'}(M \otimes _ R R', N \otimes _ R R')$ is an isomorphism for all $i$.

Proof. Omitted. This is true because a free resolution $F_\bullet $ of $M$ over $R$ stays exact when tensoring with $R'$ over $R$ and hence $(F_\bullet \otimes _ R N)\otimes _ R R'$ computes the Tor groups over $R'$. $\square$

The following lemma does not seem to fit anywhere else.

Lemma 10.75.2. Let $R$ be a ring. Let $M = \mathop{\mathrm{colim}}\nolimits M_ i$ be a filtered colimit of $R$-modules. Let $N$ be an $R$-module. Then $\text{Tor}_ n^ R(M, N) = \mathop{\mathrm{colim}}\nolimits \text{Tor}_ n^ R(M_ i, N)$ for all $n$.

Proof. Choose a free resolution $F_\bullet $ of $N$. Then $F_\bullet \otimes _ R M = \mathop{\mathrm{colim}}\nolimits F_\bullet \otimes _ R M_ i$ as complexes by Lemma 10.11.9. Thus the result by Lemma 10.8.8. $\square$


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