\newpage
# I: Varieties

:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
    - E.g. $V(x) \intersect V(y) = V(\gens{x} + \gens{y})= V(x, y) = \ts{0}$, the origin.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
    - E.g. $V(x) \union V(y) = V(\gens{x}\gens{y}) = V(xy)$, the union of coordinate axes.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_2) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n/\CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties $\star$

:::{.remark}
Summary:

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if \(k = \bar{k}, \mfa \in \Id(\kxn)\), and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when \(k=\bar{k}\): the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\bar{k}$, and that this fails for $k\neq \bar{k}$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \bar{k}$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} Y
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::