## 22.12 Tensor product

Let $R$ be a ring. Let $A$ be an $R$-algebra (see Section 22.2). Given a right $A$-module $M$ and a left $A$-module $N$ there is a tensor product

$M \otimes _ A N$

This tensor product is a module over $R$. As an $R$-module $M \otimes _ A N$ is generated by symbols $x \otimes y$ with $x \in M$ and $y \in N$ subject to the relations

$\begin{matrix} (x_1 + x_2) \otimes y - x_1 \otimes y - x_2 \otimes y, \\ x \otimes (y_1 + y_2) - x \otimes y_1 - x \otimes y_2, \\ xa \otimes y - x \otimes ay \end{matrix}$

for $a \in A$, $x, x_1, x_2 \in M$ and $y, y_1, y_2 \in N$. We list some properties of the tensor product

In each variable the tensor product is right exact, in fact commutes with direct sums and arbitrary colimits.

The tensor product $M \otimes _ A N$ is the receptacle of the universal $A$-bilinear map $M \times N \to M \otimes _ A N$, $(x, y) \mapsto x \otimes y$. In a formula

$\text{Bilinear}_ A(M \times N, Q) = \mathop{\mathrm{Hom}}\nolimits _ R(M \otimes _ A N, Q)$

for any $R$-module $Q$.

If $A$ is a $\mathbf{Z}$-graded algebra and $M$, $N$ are graded $A$-modules then $M \otimes _ A N$ is a graded $R$-module. Then $n$th graded piece $(M \otimes _ A N)^ n$ of $M \otimes _ A N$ is equal to

$\mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{r + t + s = n} M^ r \otimes _ R A^ t \otimes _ R N^ s \to \bigoplus \nolimits _{p + q = n} M^ p \otimes _ R N^ q \right)$

where the map sends $x \otimes a \otimes y$ to $x \otimes ay - xa \otimes y$ for $x \in M^ r$, $y \in N^ s$, and $a \in A^ t$ with $r + s + t = n$. In this case the map $M \times N \to M \otimes _ A N$ is $A$-bilinear and compatible with gradings and universal in the sense that

$\text{GradedBilinear}_ A(M \times N, Q) = \mathop{\mathrm{Hom}}\nolimits _{\text{graded }R\text{-modules}}(M \otimes _ A N, Q)$

for any graded $R$-module $Q$ with an obvious notion of graded bilinar map.

If $(A, \text{d})$ is a differential graded algebra and $M$ and $N$ are left and right differential graded $A$-modules, then $M \otimes _ A N$ is a differential graded $R$-module with differential

$\text{d}(x \otimes y) = \text{d}(x) \otimes y + (-1)^{\deg (x)}x \otimes \text{d}(y)$

for $x \in M$ and $y \in N$ homogeneous. In this case the map $M \times N \to M \otimes _ A N$ is $A$-bilinear, compatible with gradings, and compatible with differentials and universal in the sense that

$\text{DifferentialGradedBilinear}_ A(M \times N, Q) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}(R)}(M \otimes _ A N, Q)$

for any differential graded $R$-module $Q$ with an obvious notion of differential graded bilinar map.

## Comments (3)

Comment #4215 by Frank on

It seems better to clarify the second property of graded tensor product: the $p,q$ in $x\in M^p,y\in N^q$ are different from those in the direct sum $\bigoplus_{p+q=n}$. In other words, here $p+q$ is only assumed to be no greater than $n$. It seems better to choose different names for them.

Comment #4397 by on

Yes, you are completely right, this is too confusing. Fixed here.

Comment #5871 by Josh on

Should we not speak of $A$-balanced maps, rather than $A$-bilinear maps, in a context where $Q$ is not an $A$-module but merely an $R$-module?

## 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 09LL. Beware of the difference between the letter 'O' and the digit '0'.