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\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*}

Lemma 10.74.5. Let $R$ be a ring. For any $i \geq 0$ the functors $\text{Mod}_ R \times \text{Mod}_ R \to \text{Mod}_ R$, $(M, N) \mapsto \text{Tor}_ i^ R(M, N)$ and $(M, N) \mapsto \text{Tor}_ i^ R(N, M)$ are canonically isomorphic.

Proof. Let $F_\bullet $ be a free resolution of the module $M$ and let $G_\bullet $ be a free resolution of the module $N$. Consider the double complex $(A_{i, j}, d, \delta )$ defined as follows:

  1. set $A_{i, j} = F_ i \otimes _ R G_ j$,

  2. set $d_{i, j} : F_ i \otimes _ R G_ j \to F_{i-1} \otimes G_ j$ equal to $d_{F, i} \otimes \text{id}$, and

  3. set $\delta _{i, j} : F_ i \otimes _ R G_ j \to F_ i \otimes G_{j-1}$ equal to $\text{id} \otimes d_{G, j}$.

This double complex is usually simply denoted $F_\bullet \otimes _ R G_\bullet $.

Since each $G_ j$ is free, and hence flat we see that each row of the double complex is exact except in homological degree $0$. Since each $F_ i$ is free and hence flat we see that each column of the double complex is exact except in homological degree $0$. Hence the double complex satisfies the conditions of Lemma 10.74.3.

To see what the lemma says we compute $R(A)_\bullet $ and $U(A)_\bullet $. Namely,

\begin{eqnarray*} R(A)_ i & = & \mathop{\mathrm{Coker}}(A_{1, i} \to A_{0, i}) \\ & = & \mathop{\mathrm{Coker}}(F_1 \otimes _ R G_ i \to F_0 \otimes _ R G_ i) \\ & = & \mathop{\mathrm{Coker}}(F_1 \to F_0) \otimes _ R G_ i \\ & = & M \otimes _ R G_ i \end{eqnarray*}

In fact these isomorphisms are compatible with the differentials $\delta $ and we see that $R(A)_\bullet = M \otimes _ R G_\bullet $ as homological complexes. In exactly the same way we see that $U(A)_\bullet = F_\bullet \otimes _ R N$. We get

\begin{eqnarray*} \text{Tor}_ i^ R(M, N) & = & H_ i(F_\bullet \otimes _ R N) \\ & = & H_ i(U(A)_\bullet ) \\ & = & H_ i(R(A)_\bullet ) \\ & = & H_ i(M \otimes _ R G_\bullet ) \\ & = & H_ i(G_\bullet \otimes _ R M) \\ & = & \text{Tor}_ i^ R(N, M) \end{eqnarray*}

Here the third equality is Lemma 10.74.3, and the fifth equality uses the isomorphism $V \otimes W = W \otimes V$ of the tensor product.

Functoriality. Suppose that we have $R$-modules $M_\nu $, $N_\nu $, $\nu = 1, 2$. Let $\varphi : M_1 \to M_2$ and $\psi : N_1 \to N_2$ be morphisms of $R$-modules. Suppose that we have free resolutions $F_{\nu , \bullet }$ for $M_\nu $ and free resolutions $G_{\nu , \bullet }$ for $N_\nu $. By Lemma 10.70.4 we may choose maps of complexes $\alpha : F_{1, \bullet } \to F_{2, \bullet }$ and $\beta : G_{1, \bullet } \to G_{2, \bullet }$ compatible with $\varphi $ and $\psi $. We claim that the pair $(\alpha , \beta )$ induces a morphism of double complexes

\[ \alpha \otimes \beta : F_{1, \bullet } \otimes _ R G_{1, \bullet } \longrightarrow F_{2, \bullet } \otimes _ R G_{2, \bullet } \]

This is really a very straightforward check using the rule that $F_{1, i} \otimes _ R G_{1, j} \to F_{2, i} \otimes _ R G_{2, j}$ is given by $\alpha _ i \otimes \beta _ j$ where $\alpha _ i$, resp. $\beta _ j$ is the degree $i$, resp. $j$ component of $\alpha $, resp. $\beta $. The reader also readily verifies that the induced maps $R(F_{1, \bullet } \otimes _ R G_{1, \bullet })_\bullet \to R(F_{2, \bullet } \otimes _ R G_{2, \bullet })_\bullet $ agrees with the map $M_1 \otimes _ R G_{1, \bullet } \to M_2 \otimes _ R G_{2, \bullet }$ induced by $\varphi \otimes \beta $. Similarly for the map induced on the $U(-)_\bullet $ complexes. Thus the statement on functoriality follows from the statement on functoriality in Lemma 10.74.3. $\square$


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