# Thursday, February 03

## Projective Resolutions and Chain Maps

:::{.remark}
Also check that $\homotopic$ is an equivalence relation, i.e. it is symmetric, transitive, and reflexive.
For transitivity: given
\[
\alpha_i - \beta_i &= d_{i+1}' s_i +s_{i-1} d_i \\
\beta_i - \gamma_i &= d_{i+1}' t_{i} + t_{i-1} d_i
,\]
one can write
\[
\alpha_i - \gamma_i &= d_{i+1}'(s_i + t_i) + (s_{i-1} + t_{i-1} ) d_i
.\]

:::

:::{.theorem title="?"}
Let $\alpha, \beta \in \Ch\cat C(A, B)$ with induced maps $\hat\alpha, \hat\beta \in \Ch\cat C(H^* A, H^* B)$ on homology.
If $\alpha \homotopic \beta$, then $\hat \alpha = \hat \beta$.
:::

:::{.proof title="?"}
A computation:
\[
\hat{\alpha}_{1}(&\left.z_{1}+B_{i}\right)=\alpha_{1}\left(z_{i}\right)+B_{i}^{\prime} \\
&=\beta_{i}\left(z_{i}\right)+\delta_{i+1}^{\prime} s_{1}\left(z_{i}\right)+s_{i-1}^{\prime \prime} \delta_{i}\left(z_{i}\right) + B_i'\\
&=\beta_{i}\left(z_{i}\right)+B_{i}^{\prime} \\
&=\hat{\beta}_{i}\left(z_{i}+B_{i}\right)
\]
:::

:::{.remark}
Roadmap:

- Homological algebra
- Commutative rings
- Support theory
- Tensor triangular geometry

:::

:::{.definition title="?"}
Let $M\in \rmod$.
A **projective complex** for $M$ is a chain complex $(C_i, d_i)_{i\in \ZZ}$, indexed homologically:
\[
\cdots \to C_2 \mapsvia{d_2} C_1 \mapsvia{d_1} C_0 \mapsvia{d_0\da \eps} 0
.\]

In particular, $d^2 = 0$, but this complex need not be exact.
A **projective resolution** of $M$ is an *exact* projective complex in the following sense:

- $H_{k\geq 1}(\complex{C}) = 0$
- $H_0(\complex{C}) = C_0/d(C_1) = C_0/\ker \eps \cong M$.

:::

:::{.example title="?"}
Some projective resolutions:

- For $M\in \rmod$, projective resolutions exist since we can find covers by free modules:

\begin{tikzcd}
	\cdots & {F_2} & {F_1} & {F_0} & M & 0 \\
	&& {\ker d_1} & {\ker \eps}
	\arrow[from=1-4, to=2-4]
	\arrow[from=2-4, to=1-5]
	\arrow[from=1-3, to=2-3]
	\arrow[from=2-3, to=1-4]
	\arrow[from=1-2, to=1-3]
	\arrow["{d_1}", from=1-3, to=1-4]
	\arrow["\eps", two heads, from=1-4, to=1-5]
	\arrow[from=1-5, to=1-6]
	\arrow[from=1-1, to=1-2]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJcXGNkb3RzIl0sWzEsMCwiRl8yIl0sWzIsMCwiRl8xIl0sWzMsMCwiRl8wIl0sWzQsMCwiTSJdLFs1LDAsIjAiXSxbMywxLCJcXGtlciBcXGVwcyJdLFsyLDEsIlxca2VyIGRfMSJdLFszLDZdLFs2LDRdLFsyLDddLFs3LDNdLFsxLDJdLFsyLDMsImRfMSJdLFszLDQsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzQsNV0sWzAsMV1d)

- For $M\in \mods{Z}$, every module has a 2-stage resolution:

\begin{tikzcd}
	0 & {\ker \eps \cong \ZZ\sumpower{m}} & {\ZZ\sumpower{n}} & M & 0
	\arrow[from=1-4, to=1-5]
	\arrow["\eps", two heads, from=1-3, to=1-4]
	\arrow[from=1-2, to=1-3]
	\arrow[from=1-1, to=1-2]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbNCwwLCIwIl0sWzMsMCwiTSJdLFsyLDAsIlxcWlpcXHN1bXBvd2Vye259Il0sWzEsMCwiXFxrZXIgXFxlcHMgXFxjb25nIFxcWlpcXHN1bXBvd2Vye219Il0sWzAsMCwiMCJdLFsxLDBdLFsyLDEsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMl0sWzQsM11d)

:::

:::{.theorem title="?"}
For $\mu \in \cat C(M, M')$ and $C \da (\complex{C}, d) \surjects M, C' \da (\complex{C}', d')\surjects M'$, there is an induced chain map $\alpha \in \Ch\cat{C}(C, C')$.
Moreover, any other chain map $\beta$ is chain homotopic to $\alpha$.

> Note that $C$ can in fact be any projective complex over $M$, not necessarily a resolution.

:::

:::{.proof title="?"}
Using that $C_0$ is projective, there is a lift of the following form:

\begin{tikzcd}
	{C_0} && M \\
	\\
	{C_0'} && {M'}
	\arrow["\mu", from=1-3, to=3-3]
	\arrow["\eps"', two heads, from=3-1, to=3-3]
	\arrow["\eps", two heads, from=1-1, to=1-3]
	\arrow["{\exists \alpha_0}"', dashed, from=1-1, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJDXzAnIl0sWzIsMiwiTSciXSxbMiwwLCJNIl0sWzAsMCwiQ18wIl0sWzIsMSwiXFxtdSJdLFswLDEsIlxcZXBzIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFxlcHMiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywwLCJcXGV4aXN0cyBcXGFscGhhXzAiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

Now inductively, we want to construct the following lift:

\begin{tikzcd}
	{C_n} && {C_{n-1}} && {C_{n-2}} \\
	\\
	{C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\
	& {\im d_n' = \ker d_{n-1}'}
	\arrow["{d_n}", from=1-1, to=1-3]
	\arrow["{d_{n-1}}", from=1-3, to=1-5]
	\arrow["{\alpha_{n-2}}", from=1-5, to=3-5]
	\arrow["{d_{n-1}'}"', from=3-3, to=3-5]
	\arrow["{d_n'}"', from=3-1, to=3-3]
	\arrow["{\alpha_{n-1}}", from=1-3, to=3-3]
	\arrow["\exists"', dashed, from=1-1, to=3-1]
	\arrow[two heads, from=3-1, to=4-2]
	\arrow[hook, from=4-2, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMCwwLCJDX24iXSxbMiwwLCJDX3tuLTF9Il0sWzQsMCwiQ197bi0yfSJdLFs0LDIsIkNfe24tMn0nIl0sWzIsMiwiQ197bi0xfSciXSxbMCwyLCJDX3tufSciXSxbMSwzLCJcXGltIGRfbicgPSBcXGtlciBkX3tuLTF9JyJdLFswLDEsImRfbiJdLFsxLDIsImRfe24tMX0iXSxbMiwzLCJcXGFscGhhX3tuLTJ9Il0sWzQsMywiZF97bi0xfSciLDJdLFs1LDQsImRfbiciLDJdLFsxLDQsIlxcYWxwaGFfe24tMX0iXSxbMCw1LCJcXGV4aXN0cyIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs1LDYsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFs2LDQsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d)

STS $\im \alpha_{n-1} d_n \subseteq \ker d_{n-1}'$, which follows from
\[
d_{n-1}' \alpha_{n-1} d_n(x) = \alpha_{n-1} d_{n-1} d_n(x)
.\]

So there is a map $C_n \to \im d_n'$, and using projectivity produces the desired lift by the same argument as in the case case:

\begin{tikzcd}
	{C_n} && {C_{n-1}} && {C_{n-2}} \\
	\\
	{C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\
	& {\im d_n' = \ker d_{n-1}'}
	\arrow["{d_n}", from=1-1, to=1-3]
	\arrow["{d_{n-1}}", from=1-3, to=1-5]
	\arrow["{\alpha_{n-2}}", from=1-5, to=3-5]
	\arrow["{d_{n-1}'}"', from=3-3, to=3-5]
	\arrow["{d_n'}"'{pos=0.4}, from=3-1, to=3-3]
	\arrow["{\alpha_{n-1}}", from=1-3, to=3-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\exists \text{ by projectivity}}"', dashed, from=1-1, to=3-1]
	\arrow[two heads, from=3-1, to=4-2]
	\arrow[hook, from=4-2, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "\exists"{description}, curve={height=-18pt}, dashed, from=1-1, to=4-2]
	\arrow[shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
\end{tikzcd}

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

To see that any two such maps are chain homotopic, set $\gamma \da \alpha - \beta$, then
\[
\eps'( \gamma_0) = \eps'( \alpha_i - \beta_i) = \mu\eps - \mu \eps =0
,\]
and 
\[
d_n'(\gamma_n) 
&- d_n'( \alpha_n - \beta_n) \\
&= d_n' \alpha_n - d_n' \beta_n \\
&= \alpha_{n-1} d_n - \beta_{n-1} d_n \\
&= \gamma_{n-1} d_n
,\]
so $\gamma$ yields a well-defined chain map.

We'll now construct the chain homotopy inductively.
There is a lift $s_0$ of the following form:

\begin{tikzcd}
	&& {C_0} \\
	\\
	{C_1'} && {C'_0} & {M'} & 0 \\
	& {\im d_1'}
	\arrow["{d_1'}"', two heads, from=3-1, to=4-2]
	\arrow[hook, from=4-2, to=3-3]
	\arrow["{\gamma_0}"', from=1-3, to=3-3]
	\arrow["{\exists s_0}"', dashed, from=1-3, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow["{\eps'}", two heads, from=3-3, to=3-4]
	\arrow[from=3-4, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMywyLCJNJyJdLFs0LDIsIjAiXSxbMiwyLCJDJ18wIl0sWzEsMywiXFxpbSBkXzEnIl0sWzIsMCwiQ18wIl0sWzAsMiwiQ18xJyJdLFs1LDMsImRfMSciLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywyLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFs0LDIsIlxcZ2FtbWFfMCIsMl0sWzQsNSwiXFxleGlzdHMgc18wIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzUsMl0sWzIsMCwiXFxlcHMnIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMV1d)

This follows because $\im d_1' = \ker \eps'$ and $\eps' \gamma_0 = 0$ by the previous calculation.

Assuming all $s_{i\leq n-1}$ are constructed, set $\gamma_i = d_{i+1}' s_i + s_{i-1} d_i$.
Setting $\gamma_n - s_{n-1}d_n: C_n \to C_n'$, then
\[
d_n'( \gamma_n - s_{n-1} d_n)
&= d_n' \gamma_n - d_n' s_{n-1} d_n \\
&= \gamma_{n-1} d_n - d_n' s_{n-1} d_n \\
&= (\gamma_{n-1} - d_n' s_{n-1})d_n \\
&= s_{n-2} d_{n-1} d_n \\
&= 0
,\]
using $d^2 = 0$.
Now there is a lift $s_n$ of the following form:

\begin{tikzcd}
	&& {C_n} \\
	\\
	{C_{n+1}'} && {C_n'} && {C_{n-1}} \\
	& {\im d_{n+1} = \ker d_n'}
	\arrow["{\gamma_n - s_{n-1} d_N}", from=1-3, to=3-3]
	\arrow["{d_n'}", from=3-3, to=3-5]
	\arrow[from=3-1, to=3-3]
	\arrow["{d_{n+1}}"', from=3-1, to=4-2]
	\arrow[from=4-2, to=3-3]
	\arrow["{s_n}"', dashed, from=1-3, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJDX24iXSxbMiwyLCJDX24nIl0sWzQsMiwiQ197bi0xfSJdLFswLDIsIkNfe24rMX0nIl0sWzEsMywiXFxpbSBkX3tuKzF9ID0gXFxrZXIgZF9uJyJdLFswLDEsIlxcZ2FtbWFfbiAtIHNfe24tMX0gZF9OIl0sWzEsMiwiZF9uJyJdLFszLDFdLFszLDQsImRfe24rMX0iLDJdLFs0LDFdLFswLDMsInNfbiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Thus follows from the fact that $\im \gamma_n - s_{n-1} d_n \subseteq \ker d_n'$ and projectivity of $C_n$.
:::

:::{.remark}
Dually one can construct injective resolutions $0 \to M \injectsvia{\eta} \complex{D}$
:::

## Derived Functors

:::{.remark}
Setup: $F: \rmod \to \zmod$ is an additive covariant functor, e.g. $(\wait) \tensor_R N$ or $M\tensor_R(\wait)$, and $\complex{C}\surjectsvia{\eps} M$ a complex over $M$.
We define the left-derived functors as $(L_n F)(M) \da H_n(F(\complex{C}))$.
:::