Exercise 7, § 5

Let I be an increasing filtered ordered set admitting a countable cofinal subset and A a ring (not necessarily commutative). We shall show that every inductive limit, over I, of projective A-modules (on the left) has projective dimension $\leq 1$ (A, X, p. 134). a) Let $((Q_i)_{i \in I}, (\varphi_{i,j})_{i < j})$ be an inductive system of projective A-modules and let $Q = \lim_{\longrightarrow} Q_i$. To show that Q has projective dimension $\leq 1$, reduce to the case where $I = \mathbf{N}$. We then have an exact sequence

where for every $n \in \mathbf{N}$ and every $x \in Q_n$, we have

b) Let $((P_i)_{i \in I}, (\varphi_{i,j})_{i < j})$ be an inductive system of projective A-modules such that for every pair $(i, j)$ with $i < j$, $\varphi_{i,j}$ is injective and the module $\mathrm{Coker}(\varphi_{i,j})$ is projective. Show that the module $P = \lim_{\longrightarrow} P_i$ is projective. (Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of A-modules. Then

is an exact sequence of projective systems of commutative groups and the projective system $i \mapsto \mathrm{Hom}_A(P_i, M')$ has the Mittag-Leffler property for the discrete topology (TG, II, p. 18). We shall deduce from this that the sequence of projective limits is exact.) c) Let $(Q_i)_{i \in I}$ be an inductive system of projective A-modules. For every $i \in I$, let $R_i = \bigoplus_{j \leq i} Q_j$ and for $i < i'$ let $\psi_{i,i'} : R_i \to R_{i'}$ be the canonical injection. Define an exact sequence of inductive systems $0 \to (P_i) \to (R_i) \to (Q_i) \to 0$ such that $(P_i)$ satisfies the hypotheses of b). Deduce another proof of a).