Vanishing Tor one does not require a projective factor

The module $\mathbb{Q}$ is nonprojective by the explicitly choice-free counterinstance in Flat modules need not have projective dimension zero. The module $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ is nonprojective by the choice-free nonflatness clause of The integers have weak and global dimension one, with $n=2$. Neither use imports the AC dimension assertion.

The exact sequence $0\to \mathbb{Z}\stackrel{\times 2}{\to }\mathbb{Z}\to \mathbb{Z}\mathrm{/}2\mathbb{Z}\to 0$ is a projective resolution $P_{\bullet }$ of $M$: its terms are rank-one free and hence projective without AC by the finite-basis clause of Free modules are projective, with the exact choice boundary.

By 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, $\mathbb{Q}\otimes \mathbb{Z}\cong \mathbb{Q}$ via $q\otimes a\mapsto aq$, with inverse $q\mapsto q\otimes 1$. Thus $\mathbb{Q}\otimes P_{\bullet }$ is $0\to \mathbb{Q}\stackrel{\times 2}{\to }\mathbb{Q}\to 0$ in degrees one and zero. Its degree-one homology is the kernel of multiplication by $2$, which is zero. This proves the choice-free specified left-resolution calculation of Tor from a projective resolution of the left module.

Under The Axiom of Choice, Higher Tor over the integers vanishes supplies a projective resolution of $\mathbb{Q}$, while step 1.2 supplies one of $M$. To obtain DC locally, given an entire relation $E$ on a nonempty set $X$ and $x_0\in X$, AC chooses $s(x)\in \{y:xEy\}$ for every $x\in X$; The recursion theorem then gives $x_{n+1}=s(x_n)$ from the prescribed $x_0$. This proves the exact DC hypothesis of The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain. Hence The balanced Tor bifunctor identifies the calculated zero group with ${\mathrm{Tor}}_1^{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}\mathrm{/}2\mathbb{Z})$. Together with step 1.1 this refutes the proposed implication.