# Friday, March 18

:::{.remark}
Given effacements:

\begin{tikzcd}
	0 && {A_1} && {M_1} \\
	\\
	0 && {A_2} && {M_2}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow["g", from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzIsMCwiQV8xIl0sWzQsMCwiTV8xIl0sWzQsMiwiTV8yIl0sWzIsMiwiQV8yIl0sWzAsMiwiMCJdLFswLDFdLFsxLDJdLFs1LDRdLFs0LDNdLFsxLDQsImciXV0=)


There exists an effacement extending $g$.
Use

\begin{tikzcd}
	0 && {A_1} && {M_1\oplus M_2} \\
	\\
	0 && {A_2} & {} & {M_2}
	\arrow[from=1-1, to=1-3]
	\arrow["{(i_1, gi_2)}", from=1-3, to=1-5]
	\arrow[from=3-1, to=3-3]
	\arrow["{i_2}", from=3-3, to=3-5]
	\arrow["g", from=1-3, to=3-3]
	\arrow["{(0, \id)}", from=1-5, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMCwwLCIwIl0sWzIsMCwiQV8xIl0sWzQsMCwiTV8xXFxvcGx1cyBNXzIiXSxbNCwyLCJNXzIiXSxbMiwyLCJBXzIiXSxbMCwyLCIwIl0sWzMsMl0sWzAsMV0sWzEsMiwiKGlfMSwgZ2lfMikiXSxbNSw0XSxbNCwzLCJpXzIiXSxbMSw0LCJnIl0sWzIsMywiKDAsIFxcaWQpIl1d)

There is a factorization:

\begin{tikzcd}
	&& {S^i A_2} \\
	\\
	{S^i A_1} &&&& {S^iM_2}
	\arrow["0", dashed, from=1-3, to=3-5]
	\arrow[dashed, from=3-1, to=1-3]
	\arrow[from=3-1, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJTXmkgQV8xIl0sWzQsMiwiU15pTV8yIl0sWzIsMCwiU15pIEFfMiJdLFsyLDEsIjAiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwxXV0=)

> ?? Concludes theorem from last time.z

:::

:::{.remark}
Recall that $\Hom(C, \wait)$ is left exact covariant and $\Hom(\wait, C)$ is left exact contravariant.
For left exact functors, 

- Right derived functors are computed with injective resolutions.
- $\cat C$ needs enough injectives

For right exact functors, 

- Left derived functors are computed with projective resolutions.
- $\cat C$ needs enough projectives

:::

:::{.remark}
Projective sheaves are locally free.
:::

:::{.exercise title="?"}
Show:

- Injectives are closed under $\prod$,
- Projectives are closed under $\bigoplus$.

:::

:::{.proof title="?"}
\begin{tikzcd}
	0 && A && B \\
	\\
	&&& {I_i} \\
	\\
	&&& {\prod I_i}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, color={rgb,255:red,214;green,92;blue,92}, from=1-3, to=3-4]
	\arrow[from=3-4, to=5-4]
	\arrow[""{name=1, anchor=center, inner sep=0}, color={rgb,255:red,214;green,92;blue,92}, curve={height=30pt}, from=1-3, to=5-4]
	\arrow[""{name=2, anchor=center, inner sep=0}, "\exists"', color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-5, to=3-4]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{\therefore \exists}", color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, dashed, from=1-5, to=5-4]
	\arrow[color={rgb,255:red,214;green,92;blue,92}, shorten <=7pt, shorten >=7pt, Rightarrow, 2tail reversed, from=1, to=0]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, shorten <=7pt, shorten >=7pt, Rightarrow, dashed, 2tail reversed, from=2, to=3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiQSJdLFs0LDAsIkIiXSxbMywyLCJJX2kiXSxbMyw0LCJcXHByb2QgSV9pIl0sWzAsMV0sWzEsMl0sWzEsMywiIiwwLHsiY29sb3VyIjpbMCw2MCw2MF19XSxbMyw0XSxbMSw0LCIiLDAseyJjdXJ2ZSI6NSwiY29sb3VyIjpbMCw2MCw2MF19XSxbMiwzLCJcXGV4aXN0cyIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX0sWzI0MCw2MCw2MCwxXV0sWzIsNCwiXFx0aGVyZWZvcmUgXFxleGlzdHMiLDAseyJjdXJ2ZSI6LTUsImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX0sWzI0MCw2MCw2MCwxXV0sWzksNywiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwiY29sb3VyIjpbMCw2MCw2MF0sInN0eWxlIjp7InRhaWwiOnsibmFtZSI6ImFycm93aGVhZCJ9fX1dLFsxMCwxMSwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn0sImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

:::

:::{.exercise title="?"}
Show that in $\rmod$, $M$ is projective $\iff M$ is a direct summand of a free module iff $M$ is locally free.
:::

:::{.solution}
Some hints:

\begin{tikzcd}
	&& P \\
	\\
	F && P && 0
	\arrow[two heads, from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[Rightarrow, no head, from=1-3, to=3-3]
	\arrow[dashed, from=1-3, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJGIl0sWzIsMiwiUCJdLFs0LDIsIjAiXSxbMiwwLCJQIl0sWzAsMSwiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMl0sWzMsMSwiIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMywwLCIiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

:::


:::{.exercise title="?"}
Show

- $\RR\Hom_{\zmod}(C_n, M) = M[n] \oplus \Sigma^1(M/nM)$ using $0\to \ZZ \mapsvia{\times n} \ZZ \to \ZZ/n\ZZ \to 0$.
  - Conclude that divisible module has vanishing $\Ext^1(C_n, \wait)$.

- If $R$ is a PID, then $M\in \rmod$ is injective $\iff M$ is divisible.

- For all rings $R$, $R$ is injective iff

\begin{tikzcd}
	0 && N && R \\
	\\
	&&&& I
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-3, to=3-5]
	\arrow[dashed, from=1-5, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCIwIl0sWzIsMCwiTiJdLFs0LDAsIlIiXSxbNCwyLCJJIl0sWzAsMV0sWzEsMl0sWzEsM10sWzIsMywiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

:::