Tor one of a cyclic abelian group detects n-torsion

Let $Q_1=Q_0=\mathbb{Z}$, with $d_1(k)=nk$, zero terms above degree one, and augmentation the quotient map to $\mathbb{Z}\mathrm{/}n$. Multiplication by $n$ is injective and its image is the augmentation kernel. The module $\mathbb{Z}$ is projective without choice: to lift a map $\mathbb{Z}\to B$ through $E\twoheadrightarrow B$, choose one preimage of the image of $1$ and extend by integer multiplication. The zero module is projective by its unique map. Thus $Q$ is a projective resolution, also when $n=1$.

AC supplies DC as follows. For any nonempty set $X$ with an entire relation and specified start $a$, apply AC to the family of its nonempty successor sets to obtain a successor function $s:X\to X$. Iterate $s$ from $a$ using [L3]. The resulting sequence is precisely the chain required by The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain. This is the use of AC; it is not an appeal to a later choice theorem.

The maps $\mathbb{Z}{\otimes }_{\mathbb{Z}}M\to M$, $k\otimes m\mapsto km$, and $M\to \mathbb{Z}{\otimes }_{\mathbb{Z}}M$, $m\mapsto 1\otimes m$, are inverse: the first descends by balance and the composites are identities because $k\otimes m=1\otimes km$. Under these maps $d_1\otimes 1$ is multiplication by $n$. Hence $Q\otimes M$ is $0\to M\stackrel{n}{\to }M\to 0$ in degrees one and zero.

There are no degree-two boundaries, so $H_1(Q\otimes M)=\ker (n:M\to M)=\{m:nm=0\}$. For a homomorphism $v:M\to M^{\mathrm{\prime }}$, the map $1_Q\otimes v$ is $v$ in both degrees and induces its restriction on these kernels. This proves the calculation and its naturality without choice; for $n=1$ the kernel is zero.

Apply [L2] using the DC supplied in step 1.2. The right-resolution calculation in step 3.1 agrees canonically with that from any other supplied right resolution and, when a left resolution of $M$ is supplied, with the left-resolution calculation. Naturality and coherence of these identifications transport the explicit natural kernel map. Thus the displayed balanced-Tor formula holds under the stated conventions without assuming any unprovided resolution-existence theorem.