Proposition 7

The following conditions are equivalent : (i) $\mathrm{dh}(A) \leq 1$, (ii) every submodule of a projective module is projective, (ii') every ideal of $A$ is projective, (iii) every quotient of an injective a-module is injective, (iv) for every projective complex $C$, there exists a homomorphism $\varphi : C \to H(C)$ such that $H(\varphi) = 1_{H(C)}$, (v) for every injective complex $C$, there exists a homomorphism $\psi : H(C) \to C$ such that $H(\psi) = 1_{H(C)}$.

(i) ⇔ (ii) ⇔ (iii) : this follows from Prop. 4 of X, p. 138. (ii) ⇒ (iv) : let $C$ be a projective complex. If (ii) is satisfied, the submodule $B(C)$ of $C$ is projective, hence (iv) by X, p. 35, Remark b). (iii) ⇒ (v) : let $C$ be an injective complex. If (iii) is satisfied, the quotient $B_n(C)$ of $C_{n+1}$ is injective for all $n$, hence (v) by X, p. 35, Remark a). (iv) ⇒ (ii) : let $P$ be a projective a-module, $M$ a submodule of $P$, and $i : M \to P$ the canonical injection. Let $p : L \to M$ be a surjective homomorphism from a free module $L$ onto $M$. Consider the projective complex $C$ such that $C_1 = L$, $C_0 = P$, $C_i = 0$ for $i \ne 0, 1$, and $d_1 = i \circ p$. If (iv) is satisfied, let $\varphi : C \to H(C)$ be a homomorphism such that $H(\varphi) = 1_{H(C)}$. Since $H_1(C) = \mathrm{Ker}\ p$, $\varphi_1$ is a projector of $L$ onto $\mathrm{Ker}\ p$, so the exact sequence

is split and $M$ is isomorphic to a direct factor of $L$, hence projective.

$(v)\Rightarrow(iii)$ : let $I$ be an injective module, $M$ a quotient of $I$, and $\pi : I \to M$ the canonical projection. Let $i : M \to J$ be an injective homomorphism of $M$ into an injective module $J$. Consider the injective complex $C$ such that $C^0 = I$, $C^1 = J$, $C^i = 0$ for $i \ne 0, 1$, and $d^0 = i \circ \pi$. If (v) is satisfied, let $\psi : H(C) \to C$ be a homomorphism such that $H(\psi) = 1_{H(C)}$. Since

$\psi^1$ is a section of the canonical projection $J \to \operatorname{Coker}i$, and $M$ is a direct factor in $J$, hence injective.

$(ii)\Rightarrow(ii')$ : this is trivial.

$(ii')\Rightarrow(ii)$ : this follows from VII, § 3, Corollary 1 to Theorem 1.