# Wednesday, April 27

## Cohomological Functors

:::{.remark}
Recall that for $X \mapsvia{f} Y$, $\cone(f) \approx X[1] \oplus Y$ and $\Cyl(f) \approx X \oplus X[1] \oplus Y$ with differential
\[
d_{\Cyl(f)} \da
\begin{bmatrix}
d_X & -1 & 
\\
 & d_X[1] & 
\\
 & f[1] & d_Y
\end{bmatrix}
\actson \tv{x_i, x_{i+1}, y_i} \in \Cyl(f)^i
.\]

> Note: I use $\approx$ above because these formulas hold levelwise, but the SESs they fit into may not be split exact, so $\cone(f), \Cyl(f)$ may not such direct sums.

There are related exact triples, here the first and second rows:

\begin{tikzcd}
	&& Y && {\Cone(f)} && {X[1]} \\
	\\
	X && {\Cyl(f)} && {\Cone(f)} \\
	\\
	X && Y
	\arrow[from=5-1, to=5-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=1-3, to=1-5]
	\arrow[Rightarrow, no head, from=1-5, to=3-5]
	\arrow["\alpha", from=1-3, to=3-3]
	\arrow["\beta", from=3-3, to=5-3]
	\arrow[Rightarrow, no head, from=3-1, to=5-1]
	\arrow[from=1-5, to=1-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMiwwLCJZIl0sWzIsNCwiWSJdLFs0LDAsIlxcQ29uZShmKSJdLFs2LDAsIlhbMV0iXSxbNCwyLCJcXENvbmUoZikiXSxbMiwyLCJcXEN5bChmKSJdLFswLDIsIlgiXSxbMCw0LCJYIl0sWzcsMV0sWzYsNV0sWzUsNF0sWzAsMl0sWzIsNCwiIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCw1LCJcXGFscGhhIl0sWzUsMSwiXFxiZXRhIl0sWzYsNywiIiwxLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMiwzXV0=)

Here $\beta\alpha = \id_Y$ and $\alpha\beta \homotopic \id_{\Cyl(f)}$.
:::

:::{.definition title="Cohomological functors"}
A functor $H\in [\cat C, \cat A]$ with $\cat C\in \triang\Cat, \cat A\in \Ab\Cat$ (where $\cat A$ is not necessarily related to $\cat C$) is a **cohomological functor** iff every distinguished triangle $A\to B\to C\in \cat{C}$ is sent to an exact sequence $HA\to HB\to HC\in \cat{A}$.
:::

:::{.corollary title="?"}
If $H$ is cohomological, there is an associated LES 
\[
\cdots \to HA \to HB\to HC \to H(A[1]) \to H(B[1]) \to \cdots
.\]
:::

:::{.lemma title="?"}
The functor $H: \derivedcat{A}\to A$ where $X\mapsto H^0(X)$ is cohomological, noting that $H^i(X)$ can be written as $H^0(X[i])$.
:::

:::{.definition title="Ext for triangulated categories"}
\[
\Ext^i(X, Y) \da \Hom_{\cat{C}}(X, Y[1])
.\]
:::

:::{.lemma title="?"}
\[
\Ext_{\cat A}^i(X, Y) \cong \Ext_{\derivedcat{\cat A}}(\iota X, \iota Y)
\]
where $\iota: \cat{A} \to \Ch\cat{A}$ is given by $\iota(A) = \cdots \to 0\to A\to 0 \to\cdots$ supported in degree zero.
:::

:::{.theorem title="?"}
For all $\cat{C}\in\triang\Cat$, for all $X,Y\in \cat{C}$ the (co)representable hom functors are cohomological:
\[
h_Y &\da \Hom_{\cat C}(\wait, Y) \qquad\text{covariant} \\
(\wait)^X &\da \Hom_{\cat C}(X, \wait) \qquad\text{contravariant}
.\]


:::

:::{.proof title="?"}
The proof uses the octahedral axiom TR3.
To show that applying homs yields a complex, show that the maps on homs square to zero using the following:

\begin{tikzcd}
	X && X && 0 && {X[1]} \\
	\\
	A && B && C && {A[1]} \\
	\\
	{[X, A]} && {[X, B]} && {[X, C]} \\
	&& {}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow["u", from=3-1, to=3-3]
	\arrow["v", from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow["f"', from=1-1, to=3-1]
	\arrow["fu"', from=1-3, to=3-3]
	\arrow["{\exists 0}"', from=1-5, to=3-5]
	\arrow["{(\wait)^X(u)}", from=5-1, to=5-3]
	\arrow["{(\wait)^X(v)}", from=5-3, to=5-5]
	\arrow["{\therefore 0}"', curve={height=24pt}, dashed, from=5-1, to=5-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTIsWzAsMCwiWCJdLFsyLDAsIlgiXSxbNCwwLCIwIl0sWzAsMiwiQSJdLFsyLDIsIkIiXSxbNCwyLCJDIl0sWzYsMiwiQVsxXSJdLFs2LDAsIlhbMV0iXSxbMCw0LCJbWCwgQV0iXSxbMiw0LCJbWCwgQl0iXSxbNCw0LCJbWCwgQ10iXSxbMiw1XSxbMCwxXSxbMSwyXSxbMiw3XSxbMyw0LCJ1Il0sWzQsNSwidiJdLFs1LDZdLFswLDMsImYiLDJdLFsxLDQsImZ1IiwyXSxbMiw1LCJcXGV4aXN0cyAwIiwyXSxbOCw5LCIoXFx3YWl0KV5YKHUpIl0sWzksMTAsIihcXHdhaXQpXlgodikiXSxbOCwxMCwiXFx0aGVyZWZvcmUgMCIsMix7ImN1cnZlIjo0LCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)


:::


## Exceptional Collections

:::{.definition title="Exceptional collections"}
For $\cat{C}\in \triang\Cat$, an **exceptional collection/sequence** is a chain of morphisms
\[
\mce_1\to \mce_2 \to\cdots\to \mce_n \in \cat{C}
\]

such that

1. Self-Exts are supported only in degree zero, i.e. $\Hom(\mce_i, \mce_i[k]) = 0$ for $k\neq 0$.
2. There are no homs in the opposite direction, i.e. $\Hom(\mce_j, \mce_i[m]) = 0$ for $j > i$ and for any $m$.


:::

:::{.example title="?"}
From a paper of Valery's: let $X$ be a smooth projective surface with $H^1(\OO_X) = H^2(\OO_X) = 0$, which cohomologically look like rational surfaces. 
Examples: $X$ rational with $\abs{n K_X} = \emptyset$ (so "negative" canonical class), or $X$ of general type with $q=p_g=0$ and $\abs{n K_X}$ big for $n \gg 0$.
In these cases, there are line bundles $\mce$ with $\Ext^i(\mce, \mce) = H^1(\OO, \OO) = H^i(\OO_X)$ and one can use that $\Ext(\mce_i, \mce_j) = H^i(\mce_i \tensor \mce_j\inv)$.
:::

:::{.theorem title="Beilinson, Bondal, Kapranov"}
If $\cat{C}' \leq \cat{C}$ is the full subcategory generated by $\mce_1,\cdots, \mce_n$, then $\cat{C}' \homotopic \Db (Q)$ for $Q$ a quiver.
In particular, if $\ts{\mce_i}$ is a full exceptional collection, $\cat{C} = \cat{C}'$.
:::