Exercise 3, § 3

Let $B$ be a ring, $a$ a two-sided ideal of $B$, $b$ a left ideal of $B$ containing $a$. We denote by A the ring $B/a$, by $p : B \to A$ the homomorphism passing to the quotient and by b’ the ideal $p(b) \subset A$.

a) We consider the sequence of canonical injections:

Now prove that the sequence obtained by tensor product with $B/a$

defines a left resolution of the A-module $A/b'$.

b) Suppose that the ideals a and b are free left A-modules. Now prove that the preceding resolution is a free resolution of the A-module $A/b'$.

c) Let G be a group, k a commutative ring, $A = k^{(G)}$ the algebra of G over k, $\varepsilon : A \to k$ the homomorphism of k-algebras such that $\varepsilon(e_g) = 1$ for every $g \in G$. We endow k with the structure of A-module (left) associated with the homomorphism $\varepsilon$. Let $(g_i)_{i \in I}$ be a generating family of G, F the free group constructed on I, $B = k^{(F)}$, $\pi : F \to G$ the homomorphism such that $\pi(i) = g_i$ for $i \in I$, $p : B \to A$ the homomorphism of k-algebras deduced from $\pi$. Now prove that the ideals $a = \mathrm{Ker}\,(p)$ and $b = \mathrm{Ker}\,(\varepsilon \circ p)$ are free left B-modules; deduce a free resolution of the $k^{(G)}$-module k.

(From l, p. 147, Exercise 20, the group $\mathrm{Ker}\,(\pi)$ admits a basic family $(r_\alpha)_{\alpha \in J}$; now prove that the elements $(e_{r_\alpha} - 1)$, for $\alpha \in J$ (resp. $(e_i - 1)$, for $i \in I$) form a basis of the left B-module a (resp. b).)

d) With the notations of c), put $R = \mathrm{Ker}\,(\pi)$; let $\varphi : R \to R/(R, R)$ be the mapping passing to the quotient. We endow the k-module $k \otimes_Z R/(R, R)$ with the structure of A-module such that

for every $r \in R, f \in F$. Now prove that there exists an exact sequence of A-modules: