8]), we need the following

commutative diagram similar to the diagram in Section 3.

If M is an A-G-module where G acts rationally on A and M is a rational G-module, finitely generated as an A-module, then an A-free (projective) resolution of M can be lifted to an A-G-free resolution, that is a

commutative diagramThese three isometries form the

commutative diagram in Figure 3.

It is clear from the

commutative diagram (30) that the equivalence P sends relative projectives into relative projective; similarly, it is clear from the commutative rectangle (31) that relative injectives are sent by an equivalence into relative injectives.

Given an abelian category A assume that we have a

commutative diagram in A with exact lines:

we have the following

commutative diagram of (compact) topological groups with exact rows

greater than or equal to]8](K), we have a

commutative diagramMathematical Expression Omitted]) follows from the

commutative diagramsummation of^K, of sufficiently high power, there is a

commutative diagramSuppose we are given any homotopy

commutative diagram in T:

Then we obtain a

commutative diagram of exact sequences

11] Consider a

commutative diagram of unbroken arrows: