# Friday, April 22

:::{.remark}
Recall that for $S\subseteq \Mor(\cat C)$, there is a localized category $\cat{C}\localize{S}$ whose morphisms are chains $s_0\inv \circ f_0 \circ s_1\inv \circ \cdots$ modulo an equivalence, and if $S$ is **localizing** then 

- Morphisms are single roofs (i.e. we can collect the product fraction involving $s_i, f_i$ into a single fraction).
  - Note that roofs can be multiplied, and roofs are equivalent when they admit a common roof:

\begin{tikzcd}
	&& \bullet && \times && \bullet \\
	& \bullet && B && B && \bullet \\
	\\
	&&& \bullet \\
	{=} && \bullet && \bullet \\
	& \bullet && B && \bullet
	\arrow["{s_1}"', from=1-3, to=2-2]
	\arrow["{f_1}", from=1-3, to=2-4]
	\arrow["{s_2}", from=1-7, to=2-6]
	\arrow["{f_2}"', from=1-7, to=2-8]
	\arrow["{s_1}"', from=5-3, to=6-2]
	\arrow["{f_1}", from=5-3, to=6-4]
	\arrow["{s_2}"', from=5-5, to=6-4]
	\arrow["{f_2}", from=5-5, to=6-6]
	\arrow["\exists"{description}, dashed, from=4-4, to=5-3]
	\arrow["\exists"{description}, dashed, from=4-4, to=5-5]
	\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=18pt}, from=4-4, to=6-2]
	\arrow[color={rgb,255:red,214;green,92;blue,92}, curve={height=-18pt}, from=4-4, to=6-6]
\end{tikzcd}

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

  - Morphisms are equivalent when they admit a common roof:

\begin{tikzcd}
	& \bullet \\
	\\
	\bullet && \bullet \\
	\\
	A && B
	\arrow[from=3-1, to=5-1]
	\arrow[from=3-1, to=5-3]
	\arrow[from=3-3, to=5-1]
	\arrow[from=3-3, to=5-3]
	\arrow[dashed, from=1-2, to=3-1]
	\arrow[dashed, from=1-2, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCw0LCJBIl0sWzAsMiwiXFxidWxsZXQiXSxbMiwyLCJcXGJ1bGxldCJdLFsyLDQsIkIiXSxbMSwwLCJcXGJ1bGxldCJdLFsxLDBdLFsxLDNdLFsyLDBdLFsyLDNdLFs0LDEsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs0LDIsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

- If $\cat{C}\in \Add\Cat$ then $\cat{C}\localize{S} \in \Add\Cat$, where the calculus of fractions behaves as in ring localization: ${f_1\over s} + {f_2\over s} = {f_1+f_2\over s}$.
- If $I \leq \cat{C}$ is a full subcategorya nd $S$ is *compatible*, i.e. $S \intersect \Mor(I)$ is localizing, then $I\localize{S} \leq \cat{C}\localize{S}$ is a full subcategory.

:::

:::{.remark}
$\Ch\cat{C}$ with $S$ quasi-isomorphisms yields $\DD\cat{A}\da \cat{C}\localize{S}$.
:::

:::{.theorem title="?"}
The collection $S$ of quasi-isomorphisms is localizing.
:::

:::{.corollary title="?"}
$\DD\cat{A}$ is additive and morphisms are roofs in $\Ch\cat{A}$.
:::

:::{.theorem title="?"}
$I$ defined as $\ho\Ch\cat{C}^{\mathrm{inj}}$, the homotopy category of complexes of injective objects, is compatible with $S$.
:::

:::{.theorem title="?"}
$I\localize{S} \leq \cat{A}\localize{S} = \DD\cat{A}$, with an equivalence if $\cat A$ has enough injectives.
:::

:::{.warnings}
These last two theorems do *not* hold just for $I = \Ch\cat{C}^{\mathrm{inj}}$.
:::

:::{.remark}
An application: for $F\in \Ab\Cat(\cat A, \cat B)$ additive (with no left/right exactness conditions), there is a derived functor $\DD F\in \cat{\DD^+ \cat A, \DD^+ \cat B}$ if $\cat A$ has enough injectives.
Note that $\DD\cat{A}$ is never abelian but admits a triangulated structure.
:::

:::{.example title="?"}
For $X\in \smooth\Proj\Var\slice k$, the usual notation is $\DD(X) \da \DD^b\Coh(X)$.
Global sections $\Gamma\in \Cat(\Coh X\to \Ab\Grp)$ induce a derived functor $\RR\Gamma\in \Cat(\DD X \to \DD^b\Ab\Grp)$.
Note that $\Coh X \embeds \DD(X)$ by $\mcf \mapsto \mcf[0]$.
:::

:::{.remark}
For $X\Proj\Var\slice k$, recall $\K_0 X \da \K_0 \Coh X$ where $[b] = [a] + [c]$ for $0\to a\to b\to c$, and $\K^0 X \da \K^0 \Sh^{\loc\free}(X)$.
If $X$ is smooth, these are isomorphic, but generally they are not if $X$ is singular.
In general, $\DD X \da \DD^+ \Coh X$ replaces $\K_0(X)$, and $\DD^+ \Sh^{\loc\free}(X)$ replaces $\K^0 X$. 
:::

:::{.theorem title="?"}
$\DD\cat{A}\in \triang\Cat$.
:::

:::{.remark}
Although these do not have SESs, there are distinguished triangles for which any morphism $X\to Y$ can be completed to $X\to Y\to Z\to \shift{X}{1}$.
This can be accomplished using mapping cylinders/cones:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Spring/SheafCohomology/sections/figures/2022-04-22_11-13.xoj -->

![](figures/2022-04-22_11-15-10.png)

:::

:::{.remark}
See tilting of complexes, exceptional sequences.
:::