# Wednesday, April 20

## Derived Categories

:::{.remark}
Recall how to construct derived functors.
It is advantageous to embed $\cat{C} \embeds \Ch\cat{C}$ and resolve by nicer objects.
A complex contains strictly more information than homology: e.g. $0\to \ZZ \mapsvia{\cdot 2} \ZZ \to 0$ and $0 \to \ZZ \injects \ZZ \oplus {\ZZ\over 2\ZZ}\to 0$ have isomorphic homology but aren't isomorphic as complexes.
:::

:::{.definition title="Quasi-isomorphism"}
A morphism $f\in \Ch\cat{C}(A, B)$ is a **quasi-isomorphism** iff the induced map $f^*\in \Ch\cat{C}(\cocomplex H A, \cocomplex HB)$ is an isomorphism.
:::

:::{.definition title="The derived category"}
There is a category $\DD \cat{C}$ and a functor $\Ch\cat{C}\to \DD \cat{C}$ with the following universal property: if $\Ch\cat{C} \to \cat B$ is any functor sending quasi-isomorphisms to isomorphisms, there is a unique functor $\DD\cat{C} \to B$ factoring it.
We call $\DD\cat{C}$ the **derived category** of $\cat C$.
:::

:::{.remark}
The basic morphisms in $\DD\cat{C}$ are given by usual chain maps $f:A\to B$, and if $f$ is a quasi-isomorphism we formally add inverses $X_f: B\to A$.
A general morphism is a sequence of morphisms $\bullet \to \bullet \to \cdots \to \bullet$ where we quotient by 

- $\bullet \mapsvia{f} \bullet \mapsvia{g} \bullet \sim \bullet \mapsvia{g f} \bullet$ 
- $A \mapsvia{f} B \mapsvia{X_f} A \sim A \mapsvia{\id} A$
- $B \mapsvia{X_f} A \mapsvia{f} B \sim B \mapsvia{\id} B$.

One would like a calculus of fractions, so define:
:::

:::{.definition title="Localizing morphisms"}
Given $\cat{C}\in \Cat$, and subset $S \subseteq \Mor(\cat{ C})$ of morphisms is **localizing** iff 


- $\id_A \in S$ for all objects $A$
- $S$ is closed under compositions
- For every roof with $f$ arbitrary and $s\in S$, there exist arrows:

\begin{tikzcd}
	&& \bullet \\
	\\
	& \bullet && \bullet \\
	\\
	\bullet && \bullet && \bullet
	\arrow[from=3-2, to=5-1]
	\arrow["f"', from=3-2, to=5-3]
	\arrow["{s\in S}", from=3-4, to=5-3]
	\arrow[from=3-4, to=5-5]
	\arrow["\exists"{description}, dashed, from=1-3, to=3-2]
	\arrow["\exists"{description}, dashed, from=1-3, to=3-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMSwyLCJcXGJ1bGxldCJdLFswLDQsIlxcYnVsbGV0Il0sWzIsNCwiXFxidWxsZXQiXSxbMywyLCJcXGJ1bGxldCJdLFs0LDQsIlxcYnVsbGV0Il0sWzIsMCwiXFxidWxsZXQiXSxbMCwxXSxbMCwyLCJmIiwyXSxbMywyLCJzXFxpbiBTIl0sWzMsNF0sWzUsMCwiXFxleGlzdHMiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNSwzLCJcXGV4aXN0cyIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)


As a corollary, arrows in $\cat{C}\invert{S}$ are roofs modulo equivalence.
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:::{.remark}
The set $S$ of quasi-isomorphisms in $\Ch\cat{A}$ is localizing.

Note that we can take 

- $\Ch\cat{C}:$ all complexes,
- $\Ch^+\cat{C}:$ complexes bounded from below,
- $\Ch^-\cat{C}:$ complexes from above,
- $\Ch^b\cat{C}:$ complexes from above and below.

These yield derived categories $\DD\cat{C}, \DD^+\cat{C}, \DD^-\cat{C}, \DD^b\cat{C}$.
Note: frequently $\DD\cat{C}$ actually means $\DD^+\cat{C}$ in the literature.
When $\DD^b\cat{C}$ is used: if $\mcf \in \Coh(X)$ and $X$ is projective, which corresponds to a graded module which (by Hilbert) has a finite resolution.

One can similarly define homotopy categories $\ho\Ch\cat{C}, \ho\Ch^+\cat{C}, \ho\Ch^-\cat{C}, \ho\Ch^b\cat{C}$ with $\Ob( \ho\Ch\cat{C} ) \da \Ob(\Ch\cat{C})$ and $\Mor(\ho\Ch\cat{C}) \da \Mor(\ho\Cat{C})/\sim$ where $\sim$ denotes chain homotopy equivalence.
:::

:::{.theorem title="?"}
$\DD^+\cat{A} \cong \ho\Ch^+\cat{\cat{ I_A} }$ where $\cat{I_A}$ is the homotopy category of complexes of injective objects in $\Ch\cat{A}$.
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:::{.remark}
Generally there is a functor $\ho\Ch\cat{A} \embeds \DD\cat{A}$ since chain homotopy equivalences induce isomorphisms on homology (where we apply the universal property of $\DD\cat{A}$)
There is also a functor $\DD\cat{A}\to \ho\Ch\cat{A}$ where $A\mapsto \Tot(\cobicomplex{I})$ is a quasi-isomorphism.
:::