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*}

15.56 Computing Tor

Let $R$ be a ring. We denote $D(R)$ the derived category of the abelian category $\text{Mod}_ R$ of $R$-modules. Note that $\text{Mod}_ R$ has enough projectives as every free $R$-module is projective. Thus we can define the left derived functors of any additive functor from $\text{Mod}_ R$ to any abelian category.

This applies in particular to the functor $ - \otimes _ R M : \text{Mod}_ R \to \text{Mod}_ R$ whose left derived functors are the Tor functors $\text{Tor}_ i^ R(-, M)$, see Algebra, Section 10.74. There is also a total left derived functor

15.56.0.1
\begin{equation} \label{more-algebra-equation-derived-tensor-module} -\otimes _ R^{\mathbf{L}} M : D^{-}(R) \longrightarrow D^{-}(R) \end{equation}

which is denoted $-\otimes _ R^{\mathbf{L}} M$. Its satellites are the Tor modules, i.e., we have

\[ H^{-p}(N \otimes _ R^{\mathbf{L}} M) = \text{Tor}_ p^ R(N, M). \]

A special situation occurs when we consider the tensor product with an $R$-algebra $A$. In this case we think of $- \otimes _ R A$ as a functor from $\text{Mod}_ R$ to $\text{Mod}_ A$. Hence the total right derived functor

15.56.0.2
\begin{equation} \label{more-algebra-equation-derived-tensor-algebra} -\otimes _ R^{\mathbf{L}} A : D^{-}(R) \longrightarrow D^{-}(A) \end{equation}

which is denoted $-\otimes _ R^{\mathbf{L}} A$. Its satellites are the tor groups, i.e., we have

\[ H^{-p}(N \otimes _ R^{\mathbf{L}} A) = \text{Tor}_ p^ R(N, A). \]

In particular these Tor groups naturally have the structure of $A$-modules.


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