Proposition 3

Let $C,\ C_1$ be two complexes of right $A$-modules, $C',\ C'_1$ two complexes of left $A$-modules, and $u : C \to C_1,\ v : C \to C_1,\ u' : C' \to C'_1,\ v' : C' \to C'_1$ be morphisms of complexes.

a) If $u$ and $u'$ are homotopic to $v$ and $v'$ respectively, then the two morphisms $u \otimes u'$ and $v \otimes v'$ from $C \otimes_A C'$ to $C_1 \otimes_A C'_1$ are homotopic.

b) If $u$ and $u'$ are homotopy equivalences, $u \otimes u'$ is a homotopy equivalence.

c) If $C$ or $C'$ is homotopic to zero, $C \otimes_A C'$ is homotopic to zero.

Let us denote by the same letter $d$ the differentials of the complexes $C,\ C_1,\ C',\ C'_1$ and by $D$ the differentials of the complexes $C \otimes_A C'$ and $C_1 \otimes_A C'_1$.

If $u$ (resp. $u'$) is homotopic to $v$ (resp. $v'$), there exists a graded homomorphism of degree 1 $s:C\to C_1$ (resp. $s':C'\to C'_1$) such that

Let $S:C\otimes_A C'\to C_1\otimes_A C'_1$ be the unique graded homomorphism of degree 1 such that, for $x\in C_p,\ y\in C'_q,\ p,q\in\mathbf Z$, we have

Then, with the previous notations:

This gives $DS+SD=u\otimes u'-v\otimes v'$, whence a).

Let us prove b). If $u$ and $u'$ are homotopy equivalences, there exist homomorphisms of complexes $\alpha:C_1\to C$ and $\alpha':C'_1\to C'$ such that $u\circ\alpha,\ \alpha\circ u',\ \alpha'\circ u'$ are homotopic respectively to $\mathrm{Id}_{C_1},\ \mathrm{Id}_C,\ \mathrm{Id}_{C'_1},\ \mathrm{Id}_{C'}$. Then $(u\otimes u')\circ(\alpha\otimes\alpha')$, which is equal to $(u\circ\alpha)\otimes(u'\circ\alpha')$, is homotopic by a) to $\mathrm{Id}_{C_1}\otimes\mathrm{Id}_{C'_1}=\mathrm{Id}_{C_1\otimes C'_1}$, while $(\alpha\otimes\alpha')\circ(u\otimes u')$ is homotopic to $\mathrm{Id}_{C\otimes C'}$, whence b). Finally, c) follows from b) applied to the case where $C_1$ or $C'_1$ is zero.