# Intro, Motivations (Monday, January 10)

:::{.remark}
Topic: cohomology of sheaves and derived categories.
The plan:

- Sheaves (see ELC notes)
- Derived functors and coherent sheaves (see ELC notes)
- Derived categories (Gelfand-Manin, Tohoku)

References:

- Valery's notes (see ELC)
- Gelfand-Manin, *Methods of Homological Algebra*.

:::

:::{.remark}
Compare (genus $g$) Riemann surfaces in the classical topology to (genus $g$, projective) algebraic curves over $\CC$ in the Zariski topology.
Recall that
\[
H^*(\Sigma_g; \ZZ) =
\begin{cases}
\ZZ & *=0, 2 
\\
\ZZ^{2g} & *=1
\\
0 & \text{else}.
\end{cases}
\]
Note that this is a linear invariant in the sense that the constituents are free abelian groups, and we can extract a numerical invariant.
For surfaces up to homeomorphism, this distinguishes them completely.

For algebraic curves, note that the topology is very different: the only closed sets are finite.
In this topology, 
\[
H^*(X; \ZZ)=
\begin{cases}
\ZZ & *=0 
\\
0 & \text{else},
\end{cases}
\]
which doesn't see the genus at all.
In fact all such curves are homeomorphic in this topology, witnessed by picking any bijection and noting that it sends closed sets to closed sets.
The linear replacement: $H^*(X; \OO_X)$ for $\OO_X$ the structure sheaf, which yields
\[
H^*(X; \OO_X)
= 
\begin{cases}
\CC & *=0 
\\
\CC^g & *=1
\\
0 & \text{else}.
\end{cases}
\]
These surfaces can be parameterized by the moduli space $\mg$, which is dimension $3g-3$ for $g \geq 2$.
:::

:::{.remark}
The POV in classical topology is to fix the coefficients: $\ZZ, \RR, \CC, \ZZ/n$, or $R$ a general ring.
A minor variation is to consider a local system $\mcl$, which are locally constant but may have nontrivial monodromy around loops.
For example, one might have $\RR$ locally, but traversing a loop induces an automorphism $f\in \Aut(\RR) = \RR\units$.
In this setting, we have a functor $F(\wait) = H(\wait; R)$.
For sheaf cohomology, instead fix $X$ and take $G(\wait) = H(X; \wait)$.
In general, one can take sheaves of abelian groups, $\OO_X\dash$modules, quasicoherent sheaves, or coherent sheaves:
\[
\Sh(X, \Ab\Grp) \injects \mods{\OO_X}\injects \QCoh(X) \injects \Coh(X)
.\]
:::

:::{.remark}
We'll be looking at three kinds of topologies:

- The order topology: start with a poset and define the open sets to be the *decreasing/lower sets*, i.e. subsets $U_{x_0}$ that contain every element below a point $x_0$.
  In other words, if $x\in U$ and $y\leq x$, then $y\in U$.

- The Zariski topology: let $R$ be a DVR, so $\spec R = \ts{ \gens{0}, \mfm}$.
  E.g. for $R \da \CC\powerseries{t}$, $\mfm = \gens{t}$, and the open sets are $\ts{\gens 0}, \spec R$, corresponding to the poset $\pt\to\pt$.

- The classical topology, usually paracompact and Hausdorff.

One can define sheaves in all three cases, which have different properties.
For posets, e.g. one can take $C^0(\wait, R)$ for $R = \RR, \CC, \ZZpadic$.
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:::{.remark}
Some computational tools:

- Vanishing theorems
- Riemann-Roch

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