# Wednesday, March 03

## Baer Sum and Higher Exts

Last time: Baer sum.

:::{.remark}

\begin{tikzcd}
	{\xi':} && 0 && B && {X'} && A && 0 \\
	\\
	{\text{Ref}:} && 0 && M && P && A && 0
	\arrow[from=1-3, to=1-5]
	\arrow["{\iota'}", from=1-5, to=1-7]
	\arrow["{\sigma'}"', from=3-7, to=1-7]
	\arrow["\pi", from=3-7, to=3-9]
	\arrow["{\pi'}", from=1-7, to=1-9]
	\arrow[from=1-9, to=1-11]
	\arrow[from=3-9, to=3-11]
	\arrow["{\beta'}", from=3-5, to=1-5]
	\arrow["j", from=3-5, to=3-7]
	\arrow[from=3-3, to=3-5]
	\arrow[equals, from=3-9, to=1-9]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTIsWzQsMCwiQiJdLFs2LDAsIlgnIl0sWzgsMCwiQSJdLFsyLDAsIjAiXSxbNCwyLCJNIl0sWzYsMiwiUCJdLFs4LDIsIkEiXSxbMTAsMCwiMCJdLFsxMCwyLCIwIl0sWzIsMiwiMCJdLFswLDAsIlxceGknOiJdLFswLDIsIlxcdGV4dHtSZWZ9OiJdLFszLDBdLFswLDEsIlxcaW90YSciXSxbNSwxLCJcXHNpZ21hJyIsMl0sWzUsNiwiXFxwaSJdLFsxLDIsIlxccGknIl0sWzIsN10sWzYsOF0sWzQsMCwiXFxiZXRhJyJdLFs0LDUsImoiXSxbOSw0XSxbNiwyLCIiLDEseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d)

\begin{tikzcd}
	&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\one_B} && \textcolor{rgb,255:red,92;green,92;blue,214}{\Theta(\xi')} \\
	\cdots && {\Hom(X', B)} && {\Hom(B, B)} && {\Ext^1_R(A, B)} \\
	\\
	\cdots && {\Hom(P, B)} && {\Hom(M, B)} && {\Ext^1_R(A, B)} \\
	&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\beta'} && \textcolor{rgb,255:red,92;green,92;blue,214}{\bd(\beta') = \Theta(\xi')}
	\arrow["{\sigma'_*}", from=2-3, to=4-3]
	\arrow["{\beta'_*}", from=2-5, to=4-5]
	\arrow[from=4-1, to=4-3]
	\arrow[from=2-1, to=2-3]
	\arrow[from=2-3, to=2-5]
	\arrow[from=2-5, to=2-7]
	\arrow[from=4-3, to=4-5]
	\arrow[from=4-5, to=4-7]
	\arrow[equals, from=2-7, to=4-7]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=5-5, to=5-7]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=1-5, to=1-7]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, maps to, from=1-5, to=5-5]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, maps to, from=1-7, to=5-7]
\end{tikzcd}

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

We want to define $\xi' \oplus \xi''$, 
An important takeaway is that $\Theta$ can alternatively be defined as a map induced by the original boundary map coming from the SES, i.e. $\bd(\beta') = \Theta(\xi')$.
This fits into the diagram as follows:

\begin{tikzcd}
	{\xi':} && 0 && B && {X'} && A && 0 \\
	\\
	{\text{Ref}:} && 0 && M && P && A && 0 \\
	\\
	{\xi'':} && 0 && B && {X''} && A && 0
	\arrow[from=1-3, to=1-5]
	\arrow["{\iota'}", from=1-5, to=1-7]
	\arrow["{\sigma'}", from=3-7, to=1-7]
	\arrow["\pi", from=3-7, to=3-9]
	\arrow["{\pi'}", from=1-7, to=1-9]
	\arrow[from=1-9, to=1-11]
	\arrow[from=3-9, to=3-11]
	\arrow["{\beta'}", from=3-5, to=1-5]
	\arrow["j", from=3-5, to=3-7]
	\arrow[from=3-3, to=3-5]
	\arrow[equals, from=3-9, to=1-9]
	\arrow[from=5-3, to=5-5]
	\arrow["{\iota''}", from=5-5, to=5-7]
	\arrow["{\pi''}", from=5-7, to=5-9]
	\arrow[from=5-9, to=5-11]
	\arrow["{\beta''}"', from=3-5, to=5-5]
	\arrow["{\sigma''}"', from=3-7, to=5-7]
	\arrow[equals, from=3-9, to=5-9]
\end{tikzcd}

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


We define
\[
\tilde X \da \ts{ (x', x'') \in X' \cross X'' \st \pi'(x') = \pi''(x'') } \surjects Y
,\]
and note that we had a skew diagonal $\tilde\diagonal \subseteq \tilde X$.
This yields a YES
\[
\phi: 0 \to B \to Y \to Y/B \cong A \to 0
.\]

:::

:::{.corollary title="?"}
The set of equivalence classes of extensions of $A$ by $B$ is an abelian group under the Baer sum, where 
\[
[\xi] \oplus [\xi'] \da [\varphi]
,\]
where the identity element $0$ is the class of split extensions.
The map $\Theta$ is an isomorphism of abelian groups.
:::

:::{.remark}
One should check that this is well-defined since we're using equivalence classes.
There is a fast way to do both at once, i.e. showing $\Theta$ is well-defined and also a group morphism.
:::

:::{.proof title="?"}
We'll show that 
\[
\Theta(\varphi) = \Theta(\xi) + \Theta(\xi'') \in \Ext^1_R(A, B)
,\]
which will make it a group isomorphism since $\Theta$ was already a set bijection.
Considering commutativity in the 3-row diagram, we can get a well-defined map
\[
\sigma\da \sigma' \oplus \sigma'': P \to \tilde{X}
.\]
So let \( \bar{ \sigma}: P\to Y \) be the induced map.
The restriction of \( \bar{ \sigma} \) to $M$ is induced by the map
\[
\beta' + \beta'': M\to (B \cross 0) + (0 \cross B) \subseteq \tilde X
.\]
These both map to $B$ in $Y$ under the SES $0\to B\to Y\to Y/B\to 0$.
This gives a commutative diagram


\begin{tikzcd}
	0 && M && P && A && 0 \\
	\\
	0 && B && Y && A && 0
	\arrow[equals, from=1-7, to=3-7]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow[from=3-7, to=3-9]
	\arrow[from=1-7, to=1-9]
	\arrow["{\beta'+\beta''}"', from=1-3, to=3-3]
	\arrow["{\bar{\sigma}}"', from=1-5, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIk0iXSxbNCwwLCJQIl0sWzYsMCwiQSJdLFswLDIsIjAiXSxbMiwyLCJCIl0sWzQsMiwiWSJdLFs2LDIsIkEiXSxbOCwyLCIwIl0sWzgsMCwiMCJdLFszLDcsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMCwxXSxbMSwyXSxbMiwzXSxbNCw1XSxbNSw2XSxbNiw3XSxbNyw4XSxbMyw5XSxbMSw1LCJcXGJldGEnK1xcYmV0YScnIiwyXSxbMiw2LCJcXGJhcntcXHNpZ21hfSIsMl1d)

We then have $\Theta(\varphi) = \bd( \beta' + \beta'') = \bd(\beta') + \bd(\beta'')$ using that $\bd \in \Mor(\rmod)$.
But this is equal to $\Theta(\xi') + \Theta(\xi'')$, which is what we wanted to show.
:::

:::{.remark}
What about the 0 element for split SESs?
Recall that additive functors preserve split exact sequences, since these are just in terms of sums of maps composing to the identity.
Then applying the hom functor to the original SES produces another SES, which in particular has no Ext correction term.
:::

:::{.remark}
Similarly, $\Ext^n(A, B)$ is identified with equivalence classes of longer sequences with $n+2$ terms, and an equivalence is a sequence of maps that result in commuting squares:

\begin{tikzcd}
	{\xi:} && 0 & B & {X_n} & \cdots & {X_1} & A & 0 \\
	\\
	{\xi':} && 0 & B & {X_n'} & \cdots & {X_1'} & A & 0
	\arrow[from=1-3, to=1-4]
	\arrow[from=1-4, to=1-5]
	\arrow[from=1-5, to=1-6]
	\arrow[from=1-6, to=1-7]
	\arrow[from=1-7, to=1-8]
	\arrow[from=1-8, to=1-9]
	\arrow[equals, from=1-4, to=3-4]
	\arrow[equals, from=1-8, to=3-8]
	\arrow[dashed, from=1-5, to=3-5]
	\arrow[dashed, from=1-7, to=3-7]
	\arrow[dashed, from=1-6, to=3-6]
	\arrow[from=3-3, to=3-4]
	\arrow[from=3-4, to=3-5]
	\arrow[from=3-5, to=3-6]
	\arrow[from=3-6, to=3-7]
	\arrow[from=3-7, to=3-8]
	\arrow[from=3-8, to=3-9]
\end{tikzcd}

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

Note that if $\complex{P} \to A\to 0$ is a projective resolution, then the comparison theorem yields maps and a commutative diagram

\begin{tikzcd}
	{\phi:} && 0 & M & {P_{n-1}} & \cdots & {P_0} & A & 0 \\
	\\
	{\xi':} && 0 & B & {X_n'} & \cdots & {X_1'} & A & 0
	\arrow[from=1-3, to=1-4]
	\arrow[from=1-4, to=1-5]
	\arrow[from=1-5, to=1-6]
	\arrow[from=1-6, to=1-7]
	\arrow[from=1-7, to=1-8]
	\arrow[from=1-8, to=1-9]
	\arrow["{\exists \beta}", dashed, from=1-4, to=3-4]
	\arrow["{\one_A}", equals, from=1-8, to=3-8]
	\arrow["\exists", dashed, from=1-5, to=3-5]
	\arrow["\exists", dashed, from=1-7, to=3-7]
	\arrow[dashed, from=1-6, to=3-6]
	\arrow[from=3-3, to=3-4]
	\arrow[from=3-4, to=3-5]
	\arrow[from=3-5, to=3-6]
	\arrow[from=3-6, to=3-7]
	\arrow[from=3-7, to=3-8]
	\arrow[from=3-8, to=3-9]
\end{tikzcd}


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

Then the dimension shifting theorem (Exc. 2.4.3) and its proof yields an exact sequence
\[
\Hom(P_{n-1}, B) \to \Hom(M, B) \mapsvia{\bd} \Ext^n(A, B) \to 0
,\]
and the asserted bijection is then given by $\Theta(\xi) \da \bd(\beta)$.
:::

## 3.6: Kunneth and Universal Coefficient Theorems

:::{.observation}
If $R$ is a field $F$ then $\Tor_n^F(A, B) = 0$ for all $n>0$, i.e. every module over a field is a complex space, hence free, hence projective, hence flat, and so $A\tensor_F \wait$ is exact.
:::

:::{.question}
If $\complex P \in \Ch(\modr)$ is a complex of of right \(R\dash\)modules and $M \in \rmod$ is a left \(R\dash\)module, how is the homology of $\complex P$ and that of $\complex P \tensor_R M$ related?
:::

:::{.lemma title="?"}
Given a 5-term exact sequence
\[
A_1 \mapsvia{\alpha} A_2 \mapsvia{f} B \mapsvia{g} C_1 \mapsvia{\gamma} C_2
,\]
there is a corresponding SES

\begin{tikzcd}
	0 & A & B & C & 0 \\
	& {\substack{ A_2/\ker f = A_2/\im\alpha \\ = \coker \alpha} } && {\im g = \ker f}
	\arrow[from=1-1, to=1-2]
	\arrow["{\bar f}", from=1-2, to=1-3]
	\arrow["g", from=1-3, to=1-4]
	\arrow[from=1-4, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMCwwLCIwIl0sWzEsMCwiQSJdLFsyLDAsIkIiXSxbMywwLCJDIl0sWzQsMCwiMCJdLFszLDEsIlxcaW0gZyA9IFxca2VyIGYiXSxbMSwxLCJBXzIvXFxrZXIgZiA9IEFfMi9cXGltXFxhbHBoYSA9IFxcY29rZXIgXFxhbHBoYSJdLFswLDFdLFsxLDIsIlxcYmFyIGYiXSxbMiwzLCJnIl0sWzMsNF1d)

In particular, we can always take $A = \coker \alpha$ and $C = \ker \gamma$ in any abelian category.
:::

:::{.theorem title="The Kunneth Formula"}
Let $\complex P\in \Ch(\modr)$ be a chain complex of flat right \(R\dash\)modules such that each boundary module $dP_n$ is again flat.
Then for every $M\in \rmod$ and all $N$, there is an exact sequence 

\begin{tikzcd}
	0 && {H_n(\complex P)\tensor_R M} && {H_n(\complex P \tensor_R M)} && {\Tor^1_R(H_{n-1}(\complex P), M)} && 0
	\arrow[from=1-7, to=1-9]
	\arrow[from=1-5, to=1-7]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiSF9uKFxcdmVjdG9yIFApXFx0ZW5zb3JfUiBNIl0sWzQsMCwiSF9uKFxcdmVjdG9yIFAgXFx0ZW5zb3JfUiBNKSJdLFs2LDAsIlxcVG9yXjFfUihIX3tuLTF9KFxcdmVjdG9yIFApLCBNKSJdLFs4LDAsIjAiXSxbMyw0XSxbMiwzXSxbMSwyXSxbMCwxXV0=)

:::

:::{.remark}
Note that the correction term vanishes if $R$ is a field.
:::

:::{.proof title="?"}
Let $Z_n \da Z_n(\complex{P})$, there there is a SES
\[
0 \to Z_n \to P_n \mapsvia{d} dP_n \to 0
.\]
Since $P_n, dP_n$ are flat by assumption, by Exc. 3.2.2, $Z_n$ is also flat.
Taking the LES from applying $\wait \tensor_R M$, noting that $M$ is arbitrary yields

\begin{tikzcd}
	&&&& 0 \\
	{Z_n\tensor_R M} && {P_n\tensor_R M} && {dP_n\tensor_R M} \\
	\\
	&& \cdots && {\Tor_1(dP_n, M)}
	\arrow[from=4-5, to=2-1, out=0, in=180]
	\arrow[from=2-1, to=2-3]
	\arrow[from=2-3, to=2-5]
	\arrow[from=4-3, to=4-5]
	\arrow[from=2-5, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbNCwxLCJkUF9uXFx0ZW5zb3JfUiBNIl0sWzIsMSwiUF9uXFx0ZW5zb3JfUiBNIl0sWzAsMSwiWl9uXFx0ZW5zb3JfUiBNIl0sWzQsMywiXFxUb3JfMShkUF9uLCBNKSJdLFsyLDMsIlxcY2RvdHMiXSxbNCwwLCIwIl0sWzMsMl0sWzIsMV0sWzEsMF0sWzQsM10sWzAsNV1d)

Here $\Tor_1(dP_n, M)=0$ since $dP_n$ is flat, noting that one could also apply $\Tor(dP_n, \wait)$ to get a similar LES.
So this lifts to a SES of complexes
\[
0 \to \complex{Z}\tensor M \to \complex{P}\tensor M \to \complex{dP}\tensor M \to 0
,\]
where we can consider $d\tensor \one$ in the middle.
We'll pick this up next time!
:::