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

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

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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