## 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*

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

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

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

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

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

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

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

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