# Wednesday January 8

> Course text: [http://math.uga.edu/~pete/integral2015.pdf](http://math.uga.edu/~pete/integral2015.pdf)

Summary:
The study of commutative rings, ideals, and modules over them.

The chapters we'll cover: 

- 1 (Intro), 
- 2 (Modules, partial), 
- 3 (Ideals, CRT), 
- 7 (Localization), 
- 8 (Noetherian Rings), 
- 11 (Nullstellensatz), 
- 12 (Hilbert-Jacobson rings), 
- 13 (Spectrum), 
- 14 (Integral extensions), 
- 17 (Valuation rings), 
- 18 (Normalization),
- 19 (Picard groups),
- 20 (Dedekind domains),
- 22 (1-dim Noetherian domains)

In number theory, arises in the study of $\ZZ_k$, the ring of integers over a number field $k$, along with *localizations* and *orders* (both preserve the fraction field?).

In algebraic geometry, consider $R = k[t_1, \cdots, t_n]/I$ where $k$ is a field and $I$ is an ideal.

Some preliminary results:

1. In $\ZZ_k$, ideals factor uniquely into primes (i.e. it is a Dedekind domain).
2. $\ZZ_k$ has an integral basis (i.e. as a $\ZZ\dash$modules, $\ZZ_k \cong \ZZ^{[k: \QQ]}$).
3. The Nullstellensatz: there is a bijective correspondence 
\begin{align*}
\correspond{\text{Irreducible Zariski closed subsets of } \CC^n} \iff
\correspond{\text{Prime ideals in } \CC[t_1, \cdots, t_n]}
.\end{align*}

4. Noether normalization (a structure theorem for rings of the form $R$ above).

All of these results concern particularly "nice" rings, e.g. $\ZZ_k, \CC[t_1, \cdots, t_n]$.
These rings are

- Domains
- Noetherian
- Finitely generated over other rings
- Finite Krull dimension (supremum of length of chains of prime ideals)
  - In particular, $\dim \ZZ_k = 1$ since nonzero prime ideals are maximal in a Dedekind domain 
- Regular (nonsingularity condition, can be interpreted in scheme-theoretic language)

> Note: schemes will have "local charts" given by commutative rings, analogous to building a manifold from Euclidean $n\dash$space.
> General philosophy (Grothendieck): Every commutative ring is the ring of functions on some space, so we should study the category of commutative rings as a whole (i.e. let the rings be arbitrary).
>
> This does not hold for non-commutative rings! I.e. we can't necessarily associate a geometric space to every non-commutative ring.
> A common interesting example: $k[G]$, the group ring of an arbitrary group.
> Good references: Lam, 'Lectures on Modules and Rings'.

:::{.example}
Let $X$ be a topological space and $C(X)$ be the continuous functions $f: X \to \RR$.
This is a ring under pointwise addition/multiplication.
(This generally holds for the hom set into any commutative ring.)
:::


*Example:*
Take $X = [0, 1]$ and $C(X)$ as a ring.


:::{.exercise}
\hfill

1. Show that $C(X)$ is a not a domain.

  > Hint: find two nonzero functions whose product is identically zero, e.g. bump functions. 
  > Note that they are not analytic/holomorphic.

2. Show that it is not Noetherian (i.e. there is an ideal that is *not* finitely generated).

3. Fix a point $x\in [0, 1]$ and show that the ideal $\mathfrak{m}_x = \theset{f \suchthat f(x) = 0}$ is maximal.

4. Are all maximal ideals of this form?

> Hint: See textbook chapter 5, or Gilman and Jerison 'Rings of Continuous Functions'.
:::

The following is a theorem about topological vector bundles over $C([0, 1])$, see textbook.

:::{.theorem title="Swan"}
There is a categorical equivalence between vector bundles on a compact space and f.g. projective modules over this ring.
:::


So commutative algebra has something to say about other branches of Mathematics!

:::{.definition title="Boolean Spaces"}
A topological space is called *boolean* (or a *Stone space*) iff it is compact, hausdorff, and totally disconnected.
:::


:::{.example}
A projective variety over $p\dash$adics with $\QQ_p$ points plugged in.
:::


:::{.definition title="Boolean Rings"}
A ring is boolean if every element is idempotent, i.e. $x\in R \implies x^2 = x$.
:::

:::{.exercise title="Boolean Domains are $\FF_2$"}
If $R$ is a boolean domain, then it is isomorphic to the field with 2 elements.
:::

Lemma
: There is a categorical equivalence between Boolean spaces, Boolean rings, and so-called "Boolean algebras".