Definition

Let be R-modules and a projective resolution. The ith Tor group is the group

where is the ith homology group.

Note this is independent of the choice of projective resolution up to isomorphism.

As a functor

Let be an -module. Then the above defines a functor

For every short exact sequence of -modules

there is a (long) exact sequence of -modules

If we take an arbitrary short exact sequence then we know that the bottom row

If is flat then since the exact sequence under the functor is exact. Thus, measures the obstructions to the tensor product functor being exact.

Let

be a short exact sequence and be a projective resolution of .

Using the tensor product functor, we get a complex of complexes

In each degree , we have

which is exact since is projective. Thus, by the long exact sequence of homology theorem which is by definition the exact sequence desired.

If is a projective R-module then for . (We can see this via the projective resolution )