For a right -module , -bimodule , and -right module , the two triple tensor products

Let and , . To make and into -bimodules, we define the following actions:

However,

One of these is -dimensional and the other is just . The actions don’t “match up”.

Let and recall that , where and are vector spaces and is linear. We defined a functor

There exists a -dimensional -module such that .

If and are -algebras, where and is a subalgebra of both and such that and , then we can form . This is an algebra with

Let be the quaternions, the -dimensional -algebra with basis , and multiplication defined by the relations

We have

If is a finite Galois extension, then

The functor

is right exact but not exact in general.

This is implied by the theoremtheorem below.

Let and be a left and right adjoint pair. Then is right exact. In general, we can take adjoint functors between abelian categories.

Suppose

be exact. We want to show that

is exact. But now, by the propositionproposition, this is equivalent to showing that

is exact for all modules , i.e., by adjunction, that

is exact. But this is true by left-exactness of .

The sequence

is exact if and only if the sequence

is exact for all modules .

Take . The goal is to show that . Now consider the composition

Since is injective and is mapped to , then . So , and is surjective.

First, take . Then the exact sequence takes on the form

Under this composition, is mapped to

Exactness at means that .

Now take . Consider the composition

Tor is the left derived functor of .

A right -module is flat if is exact.

Take and .

A chain complex is a sequence

such that for all . A cochain complex is a sequence

such that for all .

The homology of a chain complex are the groups

The cohomology of a cochain complex are the groups

A morphism of chain complexes

is a diagram

such that all squares commute.

Let be a morphism of chain complexes. Then induces a map on homology

for all .