# Thursday, January 06

> Topics: Localization and completion, Nakayama's lemma, Dedekind domains, Hilbert's basis theorem, Hilbert's Nullstellensatz, Krull dimension, depth and Cohen-Macaulay rings, regular local rings.

:::{.remark}
References:

- Atiyah-MacDonald, *Commutative Algebra*.
  Be sure to check the erratum!

- Chambert-Loir, [Mostly Commutative Algebra](https://webusers.imj-prg.fr/~antoine.chambert-loir/enseignement/2014-15/ga/commalg.pdf)
- Miles Reid, *Undergraduate Algebraic Geometry*
- Altman-Kleiman, [A Term of Commutative Algebra](https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf)
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:::{.example title="?"}
Some examples of module morphisms:

- $\Hom_\Ring(\ZZ, S) = \ts{\pt}$, since $1\to 1$ is necessary.

- $\Hom_\Ring(\ZZ[x], S) \cong S$.
  Why? Since $1\to 1$ is forced, $x\mapsto s$ can be sent to any $s\in S$.

- $\Hom_\Ring\qty{ { \ZZ[x, y]\over \gens{y^2-x^3-1} }, S} = \ts{(a, b)\in S\cartpower{2} \st a^2-b^3=1}$.
- $\Hom_\Ring(R/I, S) = \ts{f\in \Hom_\Ring(R, S) \st f(I) = 0}$

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:::{.exercise title="?"}
Show that $\Id(R/I)\mapstofrom \ts{J\in \Id(R) \st J \contains I}$ using
\[
\Pi: R &\to R/I \\
x &\mapsto [x] \\
\pi\inv(J) &\mapsfrom J
.\]
Show that $\pi\inv(J)$ is in fact an ideal, construct a proposed inverse $\Pi\inv$, and show $\Pi\circ \Pi\inv = \Pi\inv \circ \Pi = \id$.
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:::{.exercise title="?"}
Show that $R$ is a field iff $R$ is a simple ring iff any ring morphism $R\to S$ is injective.
For $3\implies 1$, directly shows that every nonzero element is a unit.
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:::{.exercise title="?"}
Chapter 1 of A&M:

- 1,8,10,13,15,16,19.

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