The Stacks project

15.57 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 10.75. There is also a total left derived functor

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

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

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

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

We will generalize the material in this section to unbounded complexes in the next few sections.