Tensor products and Tor functors

For a possibly noncommutative ring, take a right module $M$ and a left module $N$. Resolve $M$ by projective right modules, tensor the chain complex with $N$, and take homology. The output is an abelian group unless additional compatible actions are supplied. For a commutative ring, both factors can be treated as modules over the same ring and Tor naturally carries that action. Writing a tensor product of two arbitrary left modules over a noncommutative ring is not justified by notation alone.

The zeroth homology is the ordinary tensor product, while the positive homology groups are the higher Tor groups. Comparison maps and their homotopies make the construction independent of the resolution. One can instead resolve the left factor by projective left modules: the balanced Tor theorem identifies the results. This does not mean tensor product has become exact in both arguments. It means either suitably chosen resolution recovers the same derived failure of exactness.

A short exact sequence in either compatible module argument produces a long exact Tor sequence. Its low-degree end contains a map from a first Tor group into the tensor product of the submodule, followed by the map that may have lost injectivity. Exactness identifies the image of that connecting map with the kernel of the failed injection. The connecting map need not itself be injective, so Tor should not automatically be identified with that kernel unless the preceding term is known to vanish.

Over a commutative ring, a module is flat precisely when its first Tor with every module vanishes; equivalently all its positive Tor groups vanish against every module. Vanishing against one test module proves much less. For instance, tensoring with the ring itself creates no positive Tor for any module, including nonflat ones. Universal criteria contain a universal quantifier, and an example involving only a single quotient must be interpreted as a computation rather than a complete certificate of flatness.

For an integer quotient, the length-one free resolution reduces every Tor calculation to a kernel and a cokernel of one multiplication map. It proves that higher groups vanish without requiring a separate calculation in each degree. Over the dual-number ring, the periodic resolution behaves differently: tensoring with the residue field kills every differential because the nilpotent element acts as zero. Homology then survives in every nonnegative degree, despite the original augmented resolution being exact.

That contrast is an early diagnostic of singular behavior. It also illustrates why counting generators of the original module does not predict resolution length. The construction of a resolution and the calculation after tensoring are distinct stages: exactness before tensoring is a prerequisite, while failure afterward is the intended output. A finite drawing of the first few periodic terms suggests a pattern, but the proof that the pattern continues comes from the uniform kernel-image calculation in every degree.