# Tuesday, January 25

:::{.exercise title="?"}
Do A&M:

- 2.12, 24, 25
- 3.1, 4, 12

See website: <https://www.daniellitt.com/commutative-algebra>
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:::{.remark}
Last time: 

- Defined $M\tensor_R N$, need to show it exists.
- Showed $R\tensor_R M \cong M$
- Showed sums commute with tensor products

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:::{.exercise title="?"}
Show that if $A \mapsvia{f}  B \mapsvia{g}  C\to 0$ is exact and $A\tensor_R N, B\tensor_R N$ exist, then $C\tensor_R N$ also exists.

> Hint: Use the universal property to produce a map $\psi: A\tensor N\to B\tensor N$ and set $C\tensor N \da \coker \psi$.

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:::{.exercise title="?"}
Prove that $M\tensor_R N$ exists by constructing it.
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:::{.solution}
Some hints:
Construction 1: use $0\to \ker f\to R\sumpower{I} \mapsvia{f} M\to 0$, and find $R\sumpower{J}\to \ker f$ to assemble an exact sequence 
\[
R\sumpower{J}\to R\sumpower{I}\to M\to 0
.\]
Now apply $(\wait)\tensor_R N$ to exhibit $M\tensor_R N$ as a cokernel $N\sumpower{J}\to N\sumpower{I}\to M\tensor_R N\to 0$.
Separately, use the hands-on construction and prove it satisfies the universal property.

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:::{.exercise title="?"}
Prove $C_2 \tensor_\ZZ C_3 \cong 0$ using construction 1 above.
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:::{.solution}
Sketch:

Take $\ZZ \mapsvia{\times 2} \ZZ \to C_2\to 0$, apply $(\wait)\tensor C_3$, and check that $\coker(C_3 \mapsvia{\times 2} C_3) = 0$ since multiplication by 2 is invertible.
Alternatively, use bilinearity:
\[
B(x,y) = 3B(x, y) - 2B(x, y) = B(x, 3y) - B(2x, y) = B(x, 0) - B(0, y) = 0
.\]

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

- $(\wait)\tensor_R N$ is right exact
- Tensoring is associative, distributive, commutative
- There is a canonical isomorphism $R\tensor_R M\to M$ induced by $r\tensor m \mapsto r.m$.
- Morphisms $f: A\to B\in \CRing$ induce functors $f^\sharp: \mods{A} \to \mods{B}$.
  Also show that $M\tensor_A B$ has a $B\dash$module structure given by $b_1(m\tensor b_2) \da m\tensor (b_1 b_2)$.
- For $N\in\mods{B}$, there is an isomorphism $\Hom_{A}(M, N) \iso \Hom_{B}(M\tensor_A B, N)$.
  - Use $f\mapsto (m\tensor b \mapsto bf(m)) \in \mathrm{Bil}_B(M\times B, N)$, with inverse $Q\mapsto Q(\wait, 1)$.
- Show that $M\tensor_R R/I \cong M/IM$, using $R/I = \coker(I\injects R)$ and applying $(\wait)\tensor M$.
  Also show that $I\tensor_R M\iso IM$ canonically.
- Show that $k[t]\sumpower{2} \tensor_{k[t]} {k[t]\over\gens{t^2}} \cong \qty{k[t]\over \gens{t^2}}\sumpower{2}$.
- Show that $R/I \tensor_R R/J \cong {R/I \over I \cdot R/J} \cong {R\over I+J}$.
- Tensoring need not be left exact.
- $C_p \not\in \mods{\ZZ}^\flat$ .
- $R\in\mods{R}^\flat$.
- $\mods{R}^\flat$ is closed under $\tensor_R$ and $\oplus$.
- $\QQ\in\mods{\ZZ}^\flat$ but not in $\mods{\ZZ}^\free$

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:::{.definition title="$A\dash$algebras"}
$\Alg\slice A$ is the coslice category $\CRing\coslice{A}$: objects are rings $B$ equipped with ring morphisms $A\to B$, and morphisms are cones under $A$:

\begin{tikzcd}
	& A \\
	\\
	B && C
	\arrow[from=1-2, to=3-1]
	\arrow[from=1-2, to=3-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMSwwLCJBIl0sWzAsMiwiQiJdLFsyLDIsIkMiXSxbMCwxXSxbMCwyXSxbMSwyXV0=)

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:::{.example title="?"}
Examples of algebras:

- $k[t] \in\Alg\slice k$
- $R\in \CRing \implies R\in\Alg\slice \ZZ$
- Every $B\in \Alg\slice A$ is a quotient of some polynomial algebra $A[t_1,\cdots]$ on potentially infinitely many generators.

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:::{.definition title="Finiteness"}
An object $B\in\Alg\slice A$ is **finitely generated** if it is a quotient of some $A[t_1, \cdots, t_n]$.
Equivalently, there exist $x_1,\cdots, x_n\in B$ such that any subring containing the $x_i$ and the image of $A$ is all of $B$.

$B$ is **finite** if $B\in \mods{A}^\fg$.
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:::{.example title="?"}
Examples:

- $k[t] \in \kalg$ is finitely generated but not finite.
- $k[t]/\gens{t^2}\in \kalg$ is finitely generated and finite.
- $(\wait)\tensor_A (\wait) \in \Fun(\Alg\slice{A}\cartpower{2}, \Alg\slice A)$, defined by $(b_1 \tensor c_1)(b_2\tensor c_2) \da b_1 b_2 \tensor c_1 c_2$.
- $\Hom_{\Alg\slice A}(B\tensor_A C, S) = \ts{f: B\to S, g: C\to S \st \ro{f}{A} = \ro{g}{A}}$.
- $k[t_1] \tensor_k k[t_2] \cong k[t_1, t_2]$ is not isomorphic to $k[t_1]\times k[t_2]$ via $(f(t_1), g(t_2)) \mapsto f(t_1) \cdot g(t_2)$, since e.g. $h(t_1, t_2) \da t_1 + t_2$ is not in the image of this map.

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