Exercise 2, § 9

Let $A$ be a ring, $u : L \to M$ an homomorphism of $A$-modules.

(a) We denote by $\varepsilon$ the commutation factor on $\mathbf{Z}^2$ defined by $\varepsilon(\alpha, \beta) = \alpha_2 \beta_2$ for $\alpha, \beta \in \mathbf{Z}^2, \alpha = (\alpha_1, \alpha_2), \beta = (\beta_1, \beta_2)$. Now prove that there exists a unique (A, $\varepsilon$)-derivation (III, p. 118) $d$ of the bigraded algebra $S(M) \otimes_A \Lambda(L)$, of degree $(1, -1)$, such that one has $d(1 \otimes x) = u(x) \otimes 1$ for $x \in L$. Now prove that $d \circ d = 0$, so that $S(M) \otimes_A \Lambda(L)$, endowed with the graduation deduced from that of $\Lambda(L)$ and the differential $d$, defines a complex $K(u)$.

(b) If $M = L$ and $u = 1_{d_L}$, now prove that $K(u)$ is equal to the complex defined in No. 3, p. 151.

(c) Let $B = S_A(M)$, and let $\overline{u} : B \otimes_A L \to B$ be the $B$-homomorphism deduced from $u$. Now prove that $K(u)$ identifies canonically with the complex $K^B(\overline{u})$.

(d) Let $u' : L' \to M'$ be a second $A$-homomorphism. Now prove that the complexes $K(u \oplus u')$ and $K(u) \otimes_A K(u')$ are isomorphic.

(e) Let $f : L \to L'$ and $g : M \to M'$ be two $A$-homomorphisms such that $g \circ u = u' \circ f$. Now prove that the homomorphism $S(g) \otimes \Lambda(f)$ is a morphism of complexes from $K(u)$ into $K(u')$. In particular, if $v : M \to P$ is an $A$-homomorphism such that $v \circ u = 0$, one obtains a morphism of complexes $h : K(u) \to S_A(P)$.

(f) Suppose that the sequence $0 \to L \xrightarrow{u} M \xrightarrow{v} P \to 0$ is an exact sequence of flat $A$-modules. Now prove that $h$ is an isomorphism (reduce to the case where $L, M, P$ are free of finite type, and use $d$). Deduce for every $n \geq 1$ an exact sequence