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
\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
\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.

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

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 064F. Beware of the difference between the letter 'O' and the digit '0'.