# Computing Cohomology (Monday, March 28)

:::{.remark}
Upcoming topics related to $\cocomplex{H}(X; \mcf)$:

- General vanishing theorems
- Čech cohomology
- Riemann-Roch

:::

## Vanishing Theorems

:::{.theorem title="Grothendieck"}
If $X$ is a Noetherian space, then $\tau_{\geq n+1}\cocomplex{H}(X; \mcf) = 0$ for $n\da \dim X$.
:::

:::{.remark}
\envlist

- Note that the theorem statement uses the Zariski topology, and so doesn't contradict that ${H}^{2d}_{\sing}(X; \ZZ) \neq 0$ for (say) $X$ a compact complex manifold.
  - The theorem uses algebraic dimension $d \da \dim_\CC X$, which is generally twice the real dimension.
- Recall that $X$ is Noetherian iff $X$ satisfies the DCC on closed sets.
- Algebraic varieties with the Zariski topology are Noetherian, since dimension strictly decreases on proper closed subsets.
- Affine schemes over Noetherian rings are Noetherian, since closed subsets corresponds to radical ideals, which satisfy the ACC.
- $\dim X$ is defined as $\sup \ts{d \st Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_d }$.
- Noetherian spaces can have infinite dimension (see examples by Nagata)
- Schemes are nonsingular if the completions of local rings are formal power series.
- Smallest class of nice rings in AG: referred to as "Japanese rings" in the literature, finitely generated rings over DVRs, plus localizations, completions, direct sums, etc.

:::

:::{.definition title="Quasicoherent sheaves"}
A sheaf $\mcf\in \Sh(X, \mods{\OO_X})$ is **quasicoherent** if for all $U = \spec R \subseteq X$, the restrictions $\ro{\mcf}{U} \cong \tilde M$ for $M\in \rmod$.
Recall that $\OO_X(D(f)) = R\invert{f}$, and we define $\tilde M(D(f)) \da M\invert{f}$, so e.g. $\tilde R = \OO_X$.
:::

:::{.theorem title="Serre"}
A sheaf $\mcf \in \Sh(X, \mods{\OO_X})$ is quasicoherent iff 
\[
\OO_X \sumpower{J} \to \OO_X\sumpower{I} \to \mcf \to 0
.\]
:::

:::{.remark}
Analogy:

- Quasicoherent: arbitrary modules $M$
- Coherent: finitely presented modules $M$.

:::

:::{.example title="Coherent sheaves"}
Examples of coherent sheaves

- For $X \subseteq \PP^N$ projective (or quasiprojective, i.e. open in a projective), the **twisting sheaves** $\OO_X(d)$ whose local sections are $p(\vector x)/q(\vector x)$ for $p,q$ homogeneous where $\deg p - \deg q = d$.
- For any $Z \subseteq X$ as above, the **ideal sheaf** $\mci_Z \subseteq \OO_Z$ and their twists $\mci_Z(d) \da \mci_Z \tensor_{\OO_X} \OO(d)$.
- Tangent sheaves $\T_X$ and cotangent sheaves $\T\dual_X$, and their tensor powers, e.g. $\Omega^n_X$.

:::

:::{.theorem title="Serre Vanishing 1"}
\[
\mcf \in \QCoh(X), \, X\in \Aff\Sch\slice k \implies
\tau_{\geq 1} \cocomplex{H}(X; \mcf) = 0
.\]
:::

:::{.theorem title="Serre Vanishing 2"}
\[
\mcf \in \Coh(X),\, X\in \Proj\Sch\slice k \implies 
\tau_{\geq 1} \cocomplex{H}(X; \mcf(n) ) = 0 \text{ for some } n\gg 0 
.\]
:::

:::{.remark}
Affine schemes correspond to general rings, and projective schemes correspond to graded rings.
In the second statement, coherence is used as a kind of finiteness.
:::

## Čech Cohomology

:::{.definition title="The Cech complex an differential"}
For open covers, write $\mcu\covers X$ iff $X = \union_i U_i$.
Define $U_{i_0, i_1,\cdots, i_p} \da U_{i_1} \intersect U_{i_1} \intersect \cdots \intersect U_{i_p}$.
Define a complex
\[
0 \to \Cc^0(\mcu; \mcf) = \bigoplus_{i_0\in I} \globsec{\mcf; U_{i_0}} \mapsvia{\del_1} \bigoplus _{i_1 < i_2} \globsec{\mcf; U_{i_0, i_1}} \mapsvia{\del_2}  \cdots
.\]
where we specify where elements land componentwise:
\[
\ro{\del_i}{i_1 < \cdots < i_{p+1}}: \bigoplus _{i_0 < \cdots < i_p} \mcf(U_{i_0, \cdots , i_p}) \\
f &\mapsto \sum_{0\leq k \leq p+1} (-1)^k \ro{f}{i_0 < \cdots \hat{k} < i_{p+1} }\mid_{U_{i_0, \cdots, i_{p+1}}}
.\]

:::

:::{.remark}
Why $\del^2 = 0$: if $k < \ell$, forget $\ell$ first and then $k$ to get a sign $(-1)^\ell (-1)^k$, or forget $k$ first then $\ell$ to get $(-1)^k (-1)^{\ell - 1}$ due to the shift.
So these contributions cancel.
:::

:::{.theorem title="?"}
Suppose that for all inclusions $j_{i_0, \cdots, i_p}: U_{i_0, \cdots, i_p} \to X$, the pushforwards of $\mcf$ 
\[
\ro{ (j_{i_0, \cdots, i_p})_* \mcf }{U_{i_0, \cdots, i_p}}
\]
have vanishing cohomology in degrees $p\geq 1$.
Then
\[
\cocomplex{H}(X; \mcf) \iso \cocomplex{\Hc}(\mcu; \mcf)
.\]

This is true for all affine schemes if $\mcf\in \QCoh(X)$, e.g. for algebraic varieties or separated schemes.
:::