TITLE: Computing homology groups over different fields
QUESTION [6 upvotes]: First of all, my background in Homology and Algebraic Topology is very limited and doesn't include much except for the basic concepts; I'm trying to understand some Homology techniques as applications relevant to a course in combinatorics I'm taking.
Anyway, in class we've seen the example of computing the homology groups of $\mathbb{P}^2$ - The projective plane over $\mathbb{R}$. It is known that $\beta_0(\mathbb{P}^2 ; \mathbb{F})=1$ since the projective plane is connected. Now, it turns out that $H_2(\mathbb{P}^2 ; \mathbb{Q})=0$ while $H_2(\mathbb{P}^2 ; \mathbb{F}_2) \cong \mathbb{F}_2$, but the lecturer hasn't really explained how it is computed.
Any explanation on the computing method would be greatly appreciated, as will be any intuition on this whole concept, or references to further reading (I know Hatcher's book on Algebraic Topology, but didn't find it very helpful).

REPLY [3 votes]: I think that we can get along just fine without spectral sequences or the universal coefficient theorem :) (even though both of those are probably more useful in general- it's nice to actually do a real computation at least once in your life!).
For the case of $\mathbb{R}P^2$ we happen to have a nice description of this as a $\Delta$-complex (I'm assuming you saw this in Hatcher). So you can compute the homology groups just as Hatcher described for a $\Delta$-complex in general, except that when he uses $\mathbb{Z}$ (i.e. looking at the "free abelian group generated by") just replace it with $\mathbb{Z}/2$ or $\mathbb{Q}$ and look at the vector space. 
More specifically: Given a $\Delta$-complex $X$, let $\Delta_n(X, F)$ be the vector space with basis the open $n$-simplices of $X$. Now look at Hatcher's Example 2.4 and see what happens with these new coefficients!