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created: 2023-03-26T11:58
updated: 2024-05-12T10:48
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# AG Notes

- What does $f(x,y) = y^2 - \prod (x-i)$ trace out?
	- For any $x\in \CC$ there exist two square roots $y=\pm \sqrt{\prod x-i}$, so one might guess that $X$ is two copies of $\CC$ glued along 6 points.
	- Problem: monodromy. Letting $x = re^{it}$ one has $\sqrt x = \sqrt r e^{it\over 2}$, but $t=0 \implies \sqrt x = \sqrt r$ and $t=2\pi \implies \sqrt{x} = -\sqrt{r}$.
	- Thus one needs to introduce branch cuts to avoid loops around the roots, and $X$ is two copies of $\CC$ glued along the branch cuts.
	- ![](2024-05-12.png)
- $I(X) = \ts{f\in \kxn \st f(x) = 0 \,\forall x\in X}$
- $V(I) = \ts{x\in \AA^n \st f(x) = 0 \,\forall f\in I}$.
- Any set of the form $V(I)$ is said to be an algebraic set.
- Recall that $R$ is Noetherian iff ACC iff every ideal is finitely generated.
- **Theorem** (Hilbert basis): if $R$ is Noetherian, then $R[x]$ is Noetherian.
- **Corollary**: $\kxn$ is Noetherian.
- **Corollary**: algebraic sets are generated by finitely many elements (polynomials).
- We thus get a correspondence $\ts{\text{Algebraic sets } X \subseteq \AA^n} \mapstofrom_{I, V} \Id(\kxn)$.
	- Here $\mspec(\kxn)$ corresponds to minimal algebraic sets, i.e. points, when $k=\kbar$.
	- Problem when $k\neq \kbar$: over $k=\RR$, note that $I\da \gens{x_1-a_1,\cdots, x_n-a_n}$ yields $\RR[x_1,\cdots, x_n]/I \cong k\in \Field$, so $I\in \mspec\kxn$. However, $J\da \gens{x^2+1}$ yields $V(J) = \emptyset$ (which is not a point), while $R[x]/J \cong \CC\in\Field$.
- **Theorem** (Nullstellensatz): if $k=\kbar$ then $\mspec \kxn = \ts{ \gens{x_1 - a_1, \cdots, x_n-a_n} \st a_i\in k}$.
- **Corollary**: for $k=\kbar$ there is a bijection of sets $\AA^n\cong \mspec \kxn$.
	- So minimal algebraic sets (points) correspond to maximal ideals.
- Recall $\sqrt{I} = \ts{f\in R\st f^n\in I}$. Say $I$ is a radical ideal if $\sqrt I = I$, so $f^n\in I \implies f\in I$.
- One can show that $I(V(J)) = \sqrt{J}$ and $V(I(X)) = X$.
- Thus the correspondence restricts to a bijection $\ts{\text{Algebraic sets } X\subseteq \AA^n} \mapstofrom_{V, I} \ts{J \in \Id \kxn \st \sqrt J = J}$.
- Define an **affine variety** to be an irreducible algebraic set.
- **Theorem**: affine varieties correspond to prime ideals, so $I(X) \in \spec \kxn \iff X$ is an affine variety.
- Show $\AA^n$ is irreducible: check $I(\AA^n) = \gens{0}$, and $\kxn/\gens{0} = \kxn$ is a domain, so $\gens{0}$ is prime.
- Show $V(f)$ is irreducible when $f$ is an irreducible polynomial: check that prime $\implies$ irreducible for any ring, and irreducible $\implies$ prime for UFDs. So $\gens{f}$ is prime.
- Define Noetherian spaces: DCC on closed subsets.
- If $X$ is Noetherian, there is a unique decomposition into irreducible components.
- Define the dimension of $X$ as $\dim X = n$ where $n$ is the sup over all lengths of chains of irreducible closed subsets $\emptyset \neq X_0 \subsetneq X_1 \cdots \subsetneq X_n$ when $X$ is irreducible, and as the sup of $\dim X_j$ over all irreducible components $X_j$ when $X$ is reducible.

# Toward Sheaf Theory

- $A(X) \da \kxn/I(X)$ the **coordinate ring**; when $I(X)$ is prime then $A(X)$ is a domain. Regard as polynomial functions on $X$.
- $K(X) = \ff A(X)$, the field of **rational functions**.
- $\OO_X$ is the **structure sheaf**.
- $\OO_{X, p} = \ts{f/g\in K(X) \st g(p) \neq 0}$, the **local ring** at $p$. Polynomial functions well-defined near $p$.
- $\mfm_{X, p} = \ts{f\in A(X) \st f(p) = 0} \leq \OO_{X, p}$ its **maximal ideal**.
- $\ev_{X, p}: A(X) \to \CC$ where $f\mapsto f(p)$ defines a SES 
  $$\mfm_{X, p} \injects A(X) \surjects \CC$$
- $\OO_{X}(U) = \Intersect_{p\in U} \OO_{X, p}$ the polynomial functions on $U$; **sections** of $\OO_X$.
- $D(f) \da \ts{p\in X \st f(p)\neq 0}$, the **distinguished open set** associated to $f$.
- $\OO_X(D(f)) = A(X)_f = \ts{g/f^n \st g\in A(X)}$ is a **localization** at $f$.
- $\OO_X(X) = A(X)$.
- **Theorem** (identity principle): if $f_1, f_2: X\to \CC$ are regular functions with $\ro{f_1}U \equiv \ro{f_2}U$ on an open $U$, then $f_1\equiv f_2$ on $X$.
- **Definition** (presheaf): an assignment $\mcf$ such that
	- $\forall U \subseteq X$, $\mcf(U) \in \CRing$ (sections)
	- $U \subseteq V \implies \exists \psi_{UV}: \mcf(V) \to \mcf(U)$, written $f\mapsto \ro f U$ (restrictions)
	- $\mcf(\emptyset) = 0$
	- $\psi_{UU} = \id_U$.
	- $U\subseteq V\subseteq W \implies \ro f U = \ro{(\ro f V)}{W}$ for all $f\in \mcf(W)$, i.e. $\psi_{VU} \circ \psi_{WV} = \psi_{WU}$.
- **Definition** (sheaf): a presheaf $\mcf$ satisfying gluing properties
	- Existence: if $\mcu \covers X$ and $f_1, f_2\in \mcf(X)$, if $\ro{f_1}{U_i} = \ro{f_2}{U_i}$ for all $i$, then $f_1\equiv f_2$.
	- Uniqueness: if $f_1 \in \mcf(U_1)$ and $f_2 \in \mcf(U_2)$ with $\ro{f_1}{U_{12}}=\ro{f_2}{U_{12}}$, then there exists $f\in \mcf(U_1\union U_2)$ such that $\ro{f}{U_1} = f_1$ and $\ro{f}{U_2} = f_2$.
- Examples: $\OO_X^\cts, \OO_X^{\hol}$, locally constant sheaves $\ul{R}$ where $\ul{R}(U)$ are the *locally* constant continuous $R\dash$valued functions on $U$. So $\ul{R}(\wait) = \Top(\wait, R)$.
- Write $\Gamma(\mcf) = \mcf(X) = H^0(\mcf)$ for the global sections.