Homology Groups with Other Coefficients

The complete algebraic apparatus described in Subsection 3.2.3 can be set up with an arbitrary commutative ring 7 . with unit taking the role played by Z. In this scenario, G (Z TV) are free 7 -modules generated by the simplices of A. The boundary operator is still defined by (3.4) and (3.5) and then extended by TZ-linearity. The 7 -modules Zn A TZ) and Bn A-,TV) are also defined in the same way, for all n 0. 

Again there is some abuse of terminology here, since Hn A]TZ) actually has the inherited stmctiu e of an 72-module. 

In general, passing between the homology groups with different coefficients requires some work. Just to give an example, let A be the real projective plane 

The description of the exact mechanism of going between various homology groups requires some homological algebra, which lies beyond the intended scope of this book. 

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