Homological methods in algebraic geometry

Solution 1:

I imagine one could write books on the subject, but probably the first moment one encounters homological algebra in algebraic geometry is the following.

If $f: X\to Y$ is a proper map of varieties, then given a coherent sheaf $\mathcal{F}$ on $Y$ one can form the pull-back $f^{*} \mathcal{F}$. Now, the pull-back functor $f^{*}: \mathcal{C}oh(Y) \rightarrow \mathcal{C}oh(X)$ is often exact (under a technical assumption of flatness) and so one has some control over it even without homological algebra.

However, given a coherent sheaf $\mathcal{G}$ on $X$, one can also push it forward to obtain $f_{*} \mathcal{G}$. Notably, the functor $f_{*}$ fails to be exact (it is only left exact) and so its right derived functors are of crucial importance.

Already very interesting case is given by taking $Y$ to be a point. Then $f_{*} = \Gamma$, the global sections functor and its derived functors are called sheaf cohomology, denoted by $H^{i}$. These vector spaces are important in their own right and turn out to be finite-dimensional if $X$ is projective. Thus, their dimension is a powerful numerical invariant of the coherent sheaf $\mathcal{G}$ and, by extension, of $X$.

However, the usual recipe to compute $H^{i}$ (resolve your sheaf by injective sheaves) is almost useless in practice, as injective sheaves are very complicated beasts. There turns out to be another, very direct way to compute $H^{i}$ using so called $\check{C}$ech cohomology. One needs a substantial amount of (admittedly, not very deep, but that depends on who you are talking to...) homological algebra to prove that these very computable $\check{C}$ech cohomology groups agree with the derived functors of $\Gamma$, which have excellent formal properties. This is all nicely explained in the standard textbook by R. Hartshorne.