TITLE: Why do we study homology over different rings (that are not even fields)?
QUESTION [7 upvotes]: I am a beginner in this regard, but I believe to have developed a general (geometrically oriented) idea for how to think about homology groups of topological spaces. However, one feature of homology, that I do not yet know what to make of, is that one can develop it over many different fields or rings. That is, the $C_i$ in the chain complex are free $R$-modules for some more or less nicely behaved ring (or even field) $R$.

Questions:

People seem to like studying homology over $R=\Bbb Z$ or other rings that are not fields. But why? Modules over fields (that is, vector spaces) are so much better behaved.
If $R$ is a field, how much does the choice of field matter. Is there something important about the field besides its characteristic? For example, when does it make a difference whether I work over $\Bbb Q,\Bbb R$ or $\Bbb C$?
What is the general idea behind choosing different $R$? When should I choose which? In the case of fields, how do I know what characteristic to work with?

REPLY [3 votes]: Homology over $\Bbb Z$ is the most natural and geometric. Chains and homology classes represent geometric loops or higher dimensional holes. Homology over $\Bbb Z_2$ is the next most natural, due to technical reasons, many formulas simplify and make it possible to define cohomology operations like the Steenrod squares. Usually homology over $\Bbb Q$ is studied to eliminate torsion. So basically I think you have it backward, understanding the groups over $\Bbb Z$ is the ultimate goal, being the closest to geometric reality. Over fields is a simplification.