The integers have weak and global dimension one

Under The Axiom of Choice, Higher Tor over the integers vanishes supplies a projective resolution of length at most one for every abelian group. Its terms are flat by Projective left and right modules are flat over an arbitrary ring, so the same sequence is a flat resolution. Thus every projective and flat dimension is at most one, by Projective dimension of an object and The flat dimension of a module.

Put $T=\mathbb{Z}\mathrm{/}n\mathbb{Z}$. The tensor presentation in The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums gives inverse maps $\mathbb{Z}{\otimes }_{\mathbb{Z}}T\to T$, $a\otimes t\mapsto at$, and $t\mapsto 1\otimes t$: their composites fix generators. Tensoring the injection $\mathbb{Z}\stackrel{\times n}{\to }\mathbb{Z}$ with $T$ therefore gives multiplication by $n$ on $T$, which is zero on the nonzero class of $1$. This is not injective, so $T$ is not flat. Projective modules are flat by Projective left and right modules are flat over an arbitrary ring, hence $T$ is not projective either. No choice is used in this calculation.

A length-zero flat or projective resolution would identify $T$ with a flat or projective module respectively. Thus step 1.2 excludes dimension zero; step 1.1 gives ${\mathrm{fd}}_{\mathbb{Z}}T={\mathrm{pd}}_{\mathbb{Z}}T=1$ under AC. Taking the suprema in Left and right global dimension of a ring and Left and right weak global dimension yields both dimensions exactly one. Since left and right modules over the commutative ring $\mathbb{Z}$ are the same abelian groups with the same integer action, the argument proves all four stated dimensions.