## 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.

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?

Comment #6083 by on

Can somebody else please confirm that "$A$-balanced" is the more correct terminology to use here? Thanks!

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