# Thursday, February 03


:::{.remark}
Exercises:

- Find a way to remember which $\hom$ is covariant vs contravariant.
- Show $P$ is projective $\iff \Hom(P, \wait)$ is exact, $I$ is injective $\iff \Hom(\wait, I)$ is exact (sends injections to surjections).
- Find a non-free projective module.
- Show that $P$ is projective iff $P$ is a direct summand of a free module.
  Use that $0\to K\to A\sumpower{I} \to P\to 0$ and lift $P \mapsvia{\id_P} P$ to get $A\sumpower{I} = K \oplus P$.
  For the other direction:

\begin{tikzcd}
	&&&&& N \\
	{A\sumpower{m}} && {P\oplus K} && P & M \\
	&&&&& 0
	\arrow[two heads, from=1-6, to=2-6]
	\arrow[from=2-6, to=3-6]
	\arrow["{\exists \text{ by freeness}}", curve={height=-18pt}, dotted, from=2-1, to=1-6]
	\arrow["\cong"', Rightarrow, no head, from=2-1, to=2-3]
	\arrow["{\pr_1}"', from=2-3, to=2-5]
	\arrow[from=2-5, to=2-6]
	\arrow["{\exists \iota_1}"', curve={height=12pt}, dotted, from=2-5, to=2-3]
	\arrow["{\therefore \exists}", dashed, from=2-5, to=1-6]
\end{tikzcd}

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

- Show that in $\zmod$, $I$ is injective iff divisible.
  - For one direction, show $ni' = i$ using the following diagram:

\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,92;blue,214}{1} & \ZZ && \ZZ & \textcolor{rgb,255:red,92;green,92;blue,214}{1} \\
	\\
	\textcolor{rgb,255:red,92;green,92;blue,214}{i} & I & \textcolor{rgb,255:red,92;green,92;blue,214}{i'}
	\arrow["{\times n}", from=1-2, to=1-4]
	\arrow[from=1-2, to=3-2]
	\arrow["\exists", dashed, from=1-4, to=3-2]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=1-1, to=3-1]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=1-5, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMSwwLCJcXFpaIl0sWzEsMiwiSSJdLFswLDIsImkiLFsyNDAsNjAsNjAsMV1dLFswLDAsIjEiLFsyNDAsNjAsNjAsMV1dLFszLDAsIlxcWloiXSxbMiwyLCJpJyIsWzI0MCw2MCw2MCwxXV0sWzQsMCwiMSIsWzI0MCw2MCw2MCwxXV0sWzAsNCwiXFx0aW1lcyBuIl0sWzAsMV0sWzQsMSwiXFxleGlzdHMiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMywyLCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzYsNSwiIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==)

  - For the other direction, produce a map:

\begin{tikzcd}
	0 && X && Y \\
	\\
	&& I
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-3, to=3-3]
	\arrow["{\exists ?}"{description}, dashed, from=1-5, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCIwIl0sWzIsMCwiWCJdLFs0LDAsIlkiXSxbMiwyLCJJIl0sWzAsMV0sWzEsMl0sWzEsM10sWzIsMywiXFxleGlzdHMgPyIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

  - Use Zorn's lemma on pairs $(Y, f)$  where $(Y, f) \leq (Y', g) \iff Y \subseteq Y'$ and $\ro{g}{Y} = f$.
    Show every chain has an upper bound by setting $Y_\infty = \Union Y_i$ and $f_\infty = \union f_i$, thus producing $Y_{\mathrm{max}}, f_{\mathrm{max}}$
    Take a SES $0 \to \gens{y} \to \gens{y, Y_{\mathrm{max}}} \to \gens{y, Y_{\mathrm{max}}}/\gens{y}\to 0$ and map the last term to $I$.

- Recall that projective resolutions are complexes $\complex{P} = \cdots P_1\to P_0\to 0$ with $H_{i>0}\complex{P} = 0$ and $H_0\complex{P} = M$, equivalently an exact complex $\cdots\to P_1\to P_0\to M\to 0$.
- Find a projective resolution of $C_2\in\zmod$ and $k[t]/\gens{t^2}\in \mods{k}$. 
  Why must the latter necessarily be infinite length?
- Compute $\Tor_*^\ZZ(C_2, C_2) \in \zmod$ and $\Tor_*^\ZZ(\ZZ\sumpower 2, C_2)$.
- Show $\Tor_*^{R}(k, k) = \bigoplus_{i\geq 0} k$ for $R = k[t]/\gens{t^2}$?
  Use the resolution $P_i = k[t]/\gens{t^2}$ with maps $(\wait)\times t$, using that $t\actson k$ by zero.
- Show that if $f\homotopic g$, then $f-g$ induces the zero map in homology.
  Use $d_{i+1} s_i + s_{i-1} d_i: H_i(C)\to H_i(D)$, pick $\bar x\in C_i$ with $d_i \bar x = 0$ and check $(d_{i+1} s_i + s_{i-1} d_{i})\bar x\in \im d_{i+1}$.

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