Localization is flat and has vanishing positive Tor

In the fraction definition Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions, $(a,2^k)$ and $(b,2^l)$ are equivalent precisely when some $2^j$ annihilates $2^{l}a-2^{k}b$. Since no nonzero integer is annihilated by $2^j$, this is equivalent to $2^{l}a=2^{k}b$. Therefore sending the class of $(a,2^k)$ to the rational $a\mathrm{/}{2}^k$ is a well-defined injective homomorphism with image $\mathbb{Z}[1\mathrm{/}2]$. This proves the asserted identification and makes $L$ torsion-free.

By Torsion-free abelian groups are flat, $L$ is flat. Thus tensoring with $L$ preserves short exact sequences, as in Left and right flat modules over an arbitrary ring. This proof of flatness uses no general localization-exactness or unproved fraction-tensor isomorphism.

Let $Q_{\bullet }\to N$ be a supplied projective resolution, and fix $i>0$. Write $Z_i=\ker d_i$ and $B_{i-1}=\mathrm{im}{d}_i$. Tensor the short exact sequence $0\to Z_i\to Q_i\to B_{i-1}\to 0$ with $L$. Tensoring the inclusion $B_{i-1}\to Q_{i-1}$ is also injective by flatness. Since $d_i$ factors through these two maps, $\ker (d_i\otimes 1)$ is the image of $Z_i\otimes L$ in $Q_i\otimes L$. Exactness of the resolution gives a surjection $Q_{i+1}\to Z_i$; tensoring preserves this surjection, so the same image is $\mathrm{im}(d_{i+1}\otimes 1)$. Thus $H_i(Q_{\bullet }\otimes L)=0$ in every positive degree. This proves the specified-resolution clause of Tor from a projective resolution of the right module, without dimension shifting or choice.

Under The Axiom of Choice, Higher Tor over the integers vanishes supplies projective resolutions of $N$ and $L$. 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. The interface The balanced Tor bifunctor therefore identifies the homology computed in step 3.1 with the usual balanced Tor. Its vanishing holds for all $i>0$, not just for $i=1$.