When can I have "short" projective resolution?

Question

When we use universal coefficient theorem, we only need to compute Tor$(A,B)$. But suddenly, later on, like in more advanced homological algebra. We start to construct Tor$_n$, Ext$_n$... Can't we just use a short free resolution to substitute compute only one Tor or Ext as before?

Answer 1

The classical universal coefficient theorem is a theorem about $\mathbb{Z}$-modules (or abelian groups, if you prefer). The ring $\mathbb{Z}$ has the very special property that every submodule of a free $\mathbb{Z}$-module is free. As a result, any $\mathbb{Z}$-module $M$ has a "short" free resolution: we can pick an epimorphism $f:F\to M$ from a free module $F$, and then $\ker f=G$ is another free module giving a free resolution $$0\to G\to F\to M\to 0.$$ As a result, $\operatorname{Tor}_n^\mathbb{Z}$ and $\operatorname{Ext}^n_\mathbb{Z}$ are trivial for $n>1$ (and for $n=1$, they are sometimes abbreviated as just $\operatorname{Tor}$ and $\operatorname{Ext}$).

However, none of this works in general over an arbitrary ring $R$. A submodule of a free $R$-module need not be free, and so an $R$-module need not have a short free resolution. So, $\operatorname{Tor}_n^R$ and $\operatorname{Ext}^n_R$ may be nontrivial for $n>1$.