global dimension is detected on cyclic modules

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

Left and right global dimension of a ring: For a ring $R$, define $\mathrm{l.gl.dim}R=\sup \{{\mathrm{pd}}_{R}M:M\text{ is a left }R\text{-module}\},$ and $\mathrm{r.gl.dim}R=\sup \{{\mathrm{pd}}_{R^{\mathrm{o}\mathrm{p}}}M:M\text{ is a right }R\text{-module}\}\mathrm{.}$ They are separately defined extended natural numbers; their equality is not part of the notation.

Baer's criterion for injective modules: Assume the Axiom of Choice. A left $R$-module $I$ is injective if and only if every homomorphism $f:J\to I$ from a left ideal $J\le R$ extends to a homomorphism $R\to I$. The forward implication is choice-free. The converse uses AC through Zorn's lemma.

Injective dimension at most n iff higher Ext vanishes: Assume enough injectives. For an object $N$ and $n\ge 0$, $\mathrm{id}(N)\le n$ if and only if ${\mathrm{Ext}}^k(M,N)=0$ for every object $M$ and every $k>n$.

Projective dimension at most n iff higher Ext vanishes: Assume the Axiom of Dependent Choice. In an abelian category with enough projectives and enough injectives, fix supplied projective and injective resolution data on all objects. Let $M$ be an object and $n\ge 0$. The following are equivalent: 1. $\mathrm{pd}(M)\le n$; 2. ${\mathrm{Ext}}^k(M,N)=0$ for every object $N$ and every $k>n$; 3. ${\mathrm{Ext}}^{n+1}(M,N)=0$ for every object $N$.

Ext dimension shifting in the second variable: Assume the Axiom of Dependent Choice. Let $\mathcal{A}$ be abelian with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all its objects. If $0\to N\to I\to \mathrm{\Sigma }N\to 0$ is an injective copresentation, then for $q\ge 1$ there are natural isomorphisms ${\mathrm{Ext}}^{q+1}(M,N)\cong {\mathrm{Ext}}^q(M,\mathrm{\Sigma }N)$; its low-degree part is $0\to \mathrm{Hom}(M,N)\to \mathrm{Hom}(M,I)\to \mathrm{Hom}(M,\mathrm{\Sigma }N)\to {\mathrm{Ext}}^1(M,N)\to 0$.

Module categories have enough injectives: Assume the Axiom of Choice. For every unital ring $R$ and every left $R$-module $M$, there is an injective left $R$-module $I$ and a monomorphism $M\to I$. Thus left $R$-modules have enough injectives. For commutative $R$, one explicit functorial target is $J(M)={\left(R^{(M^{\vee })}\right)}^{\vee }\cong {\prod }_{\varphi \in M^{\vee }}{R}^{\vee },$ where $X^{\vee }={\mathrm{Hom}}_{\mathbb{Z}}(X,\mathbb{Q}\mathrm{/}\mathbb{Z})$; the embedding is $M\to M^{\vee \vee }\to J(M)$. Here $X^{\vee }$ is a left $R$-module by $(r\varphi )(x)=\varphi (rx)$.