Homology is a functor into Groups or R-Modules

Question

From what I understood, taking the homology was a functor from the category of chain complexes to abelian groups. I am reading through Davis' lectures on algebraic topology and it states that homology is a functor from chain complexes to graded $R$-modules.

Is this a more general kind of homology, or is this because we can view any abelian group as a $\mathbb Z$ module?

Answer 1

Here's a brief outline of how all this goes.

When you first learn homology you are really learning homology with $\mathbb Z$-coefficients.

At a more advanced level, given an abelian group $A$, you can learn homology with coefficients in $A$. Commonly studied examples are $A = \mathbb Z / 2 \mathbb Z$, $\mathbb Q$, $\mathbb R$, or $\mathbb C$. The idea is that when defining the chain groups $C_n$, each simplex is assigned a coefficient from the abelian group $A$ instead of a coefficient from $\mathbb Z$, thus making each chain group into a free abelian group.

Now "abelian groups" and "$\mathbb Z$-modules" are more-or-less the same kind of mathematical objects. So when you learn homology with coefficients in an abelian group $A$, you are really learning homology with coefficients in $\mathbb Z$-modules. Each chain group with coefficients in $A$ is still an abelian group, i.e. it is a $\mathbb Z$-module (in fact, a free $\mathbb Z$-module, if $A$ itself is free). Each boundary map is an abelian group homomorphism, i.e. it is a $\mathbb Z$-module homomorphism. The kernels (cycle groups), boundary groups (image groups), and homology groups (quotients of cycle groups modulo image groups) are all abelian groups, i.e. all $\mathbb Z$-modules.

And then, given a commutative ring $R$ with unit, you can study homology with coefficients in an $R$-module $M$: when defining the chain groups $C_n$, each simplex is assigned a coefficient from $M$. The chain groups themselves are then $R$-modules (in fact, free $R$-module, if $M$ itself is free), the boundary maps are $R$-module homomorphisms, and the chain groups, boundary groups, and homology groups are all $R$-modules.