Exercise 1, § 9

Let $A$ be a ring, $L$ an $A$-module, $K$ the complex $S(L) \otimes_A \Lambda(L)$ (X, p. 151, example 1).

a) Now prove that there exists an $A$-linear endomorphism $s$ of $K$, graded of degree zero, such that

for all integers $p, q \geq 0$ and all $x \in S^p L \otimes_A \Lambda^q L$.

b) Suppose that $A$ is a $Q$-algebra. Now prove that $K$ defines a resolution of the $A$-module $A$.