I heard this definition: Given a map f : M → N of modules, and projective resolutions P• → M and Q• → N, a map of projective resolutions covering f is a map g: P• → Q• of chain complexes such that $H_0(g)$ ≈ f.
But I don't understand what $H_0(g)$ is? Or maybe, can you give an alternative explanation about "cover" here? Thank you!
A map of chain complexes $g:P_\cdot\to Q_\cdot$ induces maps on homology $H_n(P_\cdot)\to H_n(Q_\cdot)$. These are denoted $H_n(g)$. Explicitly, if $\partial_n^P:P_n\to P_{n-1}$ and $\partial_n^Q:Q_n\to Q_{n-1}$ are the differentials, then you can check that $g_n(\ker\partial_n^P)\subseteq\ker\partial_n^Q$ and $g_n(\mathrm{im\,}\partial_{n+1}^P)\subseteq\mathrm{im\,}\partial_{n+1}^Q$, so we get a map $$ H_n(g):H_n(P)=\ker\partial_n^P/\mathrm{im\,}\partial_{n+1}^P\to \ker\partial_n^Q/\mathrm{im\,}\partial_{n+1}^Q=H_n(Q). $$ In particular we have a map $H_0(g):H_0(P)\to H_0(Q)$. But $H_0(P)\cong M$ and $H_0(Q)\cong N$, so we can think of $H_0(g)$ as a map $M\to N$.