# Thursday, January 20

:::{.remark}
Last time: $\Hom(N, \wait)$ and $\Hom(\wait, N)$ are left exact.
We can explicitly describe
\[
\Hom_A(A/I, M) = \ts{m\in M \st im = 0 \forall i\in I}
,\]
which is the $I\dash$torsion in $M$.
Using that $0\to I\to A \to A/I \to 0$ is short exact, we'll get a long exact sequence
\[
0 \to \Hom(I, M) \to \Hom(A, M) \mapsvia{f} \ts{I\dash\text{torsion in } M} \to \Ext^1_A(I, M) \to \cdots
,\]
where the $\Ext$ term measures failure of surjectivity of $f$.
:::

:::{.exercise title="?"}
Show that $\Hom(N, \wait)$ and $\Hom(\wait, N)$ are left exact.
:::

:::{.remark}
Recall the snake lemma:

\begin{tikzcd}
	0 & {\ker a} & {\ker b} & {\ker c} \\
	0 & {A_1} & {B_1} & {C_1} & 0 \\
	\\
	0 & {A_2} & {B_2} & {C_2} & 0 \\
	& {\coker a} & {\coker b} & {\coker c} & 0
	\arrow["a", from=2-2, to=4-2]
	\arrow["b", from=2-3, to=4-3]
	\arrow["c", from=2-4, to=4-4]
	\arrow[from=2-2, to=2-3]
	\arrow[from=2-3, to=2-4]
	\arrow[from=4-2, to=4-3]
	\arrow[from=4-3, to=4-4]
	\arrow[from=2-1, to=2-2]
	\arrow[from=2-4, to=2-5]
	\arrow[from=4-1, to=4-2]
	\arrow[from=4-4, to=4-5]
	\arrow[from=1-1, to=1-2]
	\arrow[from=1-2, to=1-3]
	\arrow[from=1-3, to=1-4]
	\arrow[from=5-2, to=5-3]
	\arrow[from=5-3, to=5-4]
	\arrow[from=5-4, to=5-5]
	\arrow[curve={height=12pt}, dashed, from=1-4, to=5-2]
	\arrow[from=1-2, to=2-2]
	\arrow[from=1-3, to=2-3]
	\arrow[from=1-4, to=2-4]
	\arrow[from=4-2, to=5-2]
	\arrow[from=4-3, to=5-3]
	\arrow[from=4-4, to=5-4]
\end{tikzcd}

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

The recipe for the connecting morphism:

- Start with $x\in \ker(C_1\to C_2)$, choose a preimage in $B_1$
- Push to $B_2$
- Use exactness to pull to $A_2$
- Project along $A_2\to \coker(A_1\to A_2)$
:::

:::{.definition title="Finite presentation"}
An object $M\in\mods{R}$ is of **finite presentation** iff there is an exact sequence
\[
R^m \to R^n\to M \to 0
,\]
i.e. there are finitely many generators and finitely many relations.
:::

:::{.remark}
A nice application of the snake lemma: for finitely presented modules $M, N$, one can extend a morphism $M\to N$ to

\begin{tikzcd}
	0 && {R^{m_1}} && {R^{n_1}} && M \\
	\\
	0 && {R^{m_2}} && {R^{n_2}} && N
	\arrow["f", from=1-7, to=3-7]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=1-5, to=3-5]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMiwwLCJSXnttXzF9Il0sWzQsMCwiUl57bl8xfSJdLFs2LDAsIk0iXSxbMiwyLCJSXnttXzJ9Il0sWzQsMiwiUl57bl8yfSJdLFs2LDIsIk4iXSxbMCwwLCIwIl0sWzAsMiwiMCJdLFsyLDUsImYiXSxbMCwxXSxbMSwyXSxbMyw0XSxbNCw1XSxbMSw0XSxbMCwzXSxbNiwwXSxbNywzXV0=)

Since $R\in \mods{R}$ is free, the extended maps can be represented by matrices, which is a significant simplification.
In fact, $f$ can be recovered uniquely by knowing the map on generators.
:::

:::{.exercise title="?"}
Prove the snake lemma, and show exactness at all 6 places.
:::

:::{.remark}
Recall the universal property of $M\tensor_R N$ in $\rmod$ in terms of bilinear morphisms.
:::

:::{.exercise title="?"}
Prove uniqueness of any object satisfying a universal property using the Yoneda lemma.
:::

:::{.exercise title="?"}
Prove using universal properties:

- $R\tensor_R R \cong R$, using universal properties. Why is this unique?
- $R\tensor_R M \cong M$.
- $(M_1 \oplus M_2) \tensor_R N \cong (M_1 \tensor_R N) \oplus (M_2\tensor_R N)$.

:::