Proposition 12

The mapping $\overline{\varepsilon}_M : B(A, M) \to M$ is a homotopism of complexes of $k$-modules. In particular, the complex $B(A, M)$ is split over $k$, and $(B(A, M), \overline{\varepsilon}_M)$ is a left resolution of the $A$-module $M$.

For $n \geqslant 0$, let us define a $k$-linear mapping $s_n : B_n \to B_{n+1}$ by the formula:

This is a homomorphism of right $A$-modules, which satisfies the identities:

and consequently one has

(16)

Moreover, one has

(17)

Let $\eta : A \to A \otimes_k A$ denote the mapping defined by $\eta(a) = 1 \otimes a$, and $\overline{\eta} : A \to B(A)$ the morphism of complexes which coincides with $\eta$ in degree 0. It is clear that $\overline{\varepsilon}_A \circ \overline{\eta} = 1_A$; formulas (16) and (17) show that $d \circ s + s \circ d = 1_{B(A)} - \overline{\eta} \circ \overline{\varepsilon}_A$. Putting $\overline{\eta}_M = \overline{\eta} \otimes 1_M$, $d_M = d \otimes 1_M$ and $s_M = s \otimes 1_M$, we deduce that $\overline{\varepsilon}_M \circ \overline{\eta}_M = 1_M$ and $d_M \circ s_M + s_M \circ d_M = 1_{B(A,M)} - \overline{\eta}_M \circ \overline{\varepsilon}_M$. In other words, (X, p. 33, Definition 5), $\overline{\varepsilon}_M$ is a homotopism of complexes of $k$-modules. The other assertions of the proposition follow from this at once.