I: Varieties

Some useful basic properties:

• Properties of \(V\):
  • \(\cap_{i\in I} V({\mathfrak{a}}_i) = V\qty{\sum_{i\in I} {\mathfrak{a}}_i}\).
  • \(\cup_{i\leq n} V({\mathfrak{a}}_i) = V\qty{\prod_{i\leq n} {\mathfrak{a}}_i}\).
  • \(V({\mathfrak{a}})^c = \cup_{f\in {\mathfrak{a}}} D(f)\)
  • \(V({\mathfrak{a}}_1) \subseteq V({\mathfrak{a}}_1) \iff \sqrt{{\mathfrak{a}}_1}\supseteq\sqrt{{\mathfrak{a}}_2}\).
• Properties of \(I\):
  • \(I(V({\mathfrak{a}})) = \sqrt{\mathfrak{a}}\) and \(V(I(Y)) = { \operatorname{cl}} _{{\mathbf{A}}^n}(Y)\). The containment correspondence is contravariant in both directions.
  • \(I(\cup_i Y_i) = \cap_i I(Y_i)\).
• If \(F\) is a sheaf taking values in subsets of a giant ambient set, then \(F(\cup U_i) = \cap F(U_i)\). For ${\mathbf{A}}^n_{/ {{\mathbf{C}}}} $, take \({\mathbf{C}}(x_1,\cdots, x_n)\), the field of rational functions, to be the ambient set.
• Distinguished open \(D(f) \coloneqq\left\{{p\in X {~\mathrel{\Big\vert}~}f(p) \neq 0}\right\}\):
  • \({\mathcal{O}}_X(D(f)) = A(X){ \left[ { \scriptstyle \frac{1}{f} } \right] } = \left\{{{g\over f^k} {~\mathrel{\Big\vert}~}g\in A(X), k\geq 0}\right\}\), and taking \(f=1\) shows \({\mathcal{O}}_X(X) = A(X)\), i.e. global regular functions are polynomial.
  • Generally \(D(fg) = D(f) \cap D(g)\)
  • For affines: \begin{align*} {\mathcal{O}}_{\operatorname{Spec}R}(D(f)) = R{ \left[ { \scriptstyle \frac{1}{f} } \right] } .\end{align*}
  • For \({\mathbf{C}}^n\), \begin{align*} {\mathcal{O}}_{{\mathbf{C}}^n}(D(f)) = k[x_1, \cdots, x_{n}] { \left[ \scriptstyle {1/f} \right] } \implies {\mathcal{O}}_{{\mathbf{C}}^n}(V({\mathfrak{a}})^c) = \cap_{f\in {\mathfrak{a}}} {\mathcal{O}}_{{\mathbf{C}}^n}(D(f)) .\end{align*}

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• \({\mathbf{A}}^n_{/ {k}} = \left\{{{\left[ {a_1,\cdots, a_n} \right]} {~\mathrel{\Big\vert}~}a_i \in k}\right\}\), and elements \(f\in A \coloneqq k[x_1, \cdots, x_{n}]\) are functions on it.
• \(Z(f) \coloneqq\left\{{p\in {\mathbf{A}}^n {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\), and for any \(T \subseteq A\) we set \(Z(T) \coloneqq\cap_{f\in T} Z(f)\).
  • Note that \(Z(T) = Z(\left\langle{T}\right\rangle_A) = Z(\left\langle{f_1,\cdots, f_r}\right\rangle)\) 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: \(Y \subseteq {\mathbf{A}}^n\) is algebraic iff \(Y = Z(T)\) for some \(T \subseteq A\).
• The Zariski topology is generated by open sets of the form \(Z(T)^c\).
• \({\mathbf{A}}^1\) is a non-Hausdorff space with the cofinite topology.
• Irreducible: \(Y\) is reducible iff \(Y = Y_1 \cup Y_2\) 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 \({ \operatorname{cl}} _X(Y) \subseteq X\) is again irreducible.
• Affine (algebraic) varieties: irreducible closed subsets of \({\mathbf{A}}^n\).
• Quasi-affine varieties: open subsets of affine varieties.
• The ideal of a subset: \(I(Y) \coloneqq\left\{{f\in A {~\mathrel{\Big\vert}~}f(p) = 0 \,\, \forall p\in Y}\right\}\).
• Nullstellensatz: if \(k = { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } , {\mathfrak{a}}\in \operatorname{Id}(k[x_1, \cdots, x_{n}])\), and \(f\in k[x_1, \cdots, x_{n}]\) with \(f(p) = 0\) for all \(p\in V({\mathfrak{a}})\), then \(f^r \in {\mathfrak{a}}\) for some \(r>0\), so \(f\in \sqrt{\mathfrak{a}}\). Thus there is a contravariant correspondence between radical ideals of \(k[x_1, \cdots, x_{n}]\) and algebraic sets in ${\mathbf{A}}^n_{/ {k}} $.
• Irreducibility criterion: \(Y\) is irreducible iff \(I(Y) \in \operatorname{Spec}k[x_1, \cdots, x_{n}]\) (i.e. it is prime).
• Affine curves: if \(f\in k[x,y]^{\mathrm{irr}}\) then \(\left\langle{f}\right\rangle \in \operatorname{Spec}k[x,y]\) (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 \(f\in k[x_1, \cdots, x_{n}]^{\mathrm{irr}}\) defines a surface.
• Coordinate rings: \(A(Y) \coloneqq k[x_1, \cdots, x_{n}]/I(Y)\).
• Noetherian spaces: \(X\in {\mathsf{Top}}\) is Noetherian iff the DCC on closed subsets holds.
• Unique decomposition into irreducible components: if \(X\in {\mathsf{Top}}\) is Noetherian then every closed nonempty \(Y \subseteq X\) is of the form \(Y = \cup_{i=1}^r Y_i\) 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 \(X\in {\mathsf{Top}}\), the dimension is \(\dim X \coloneqq\sup \left\{{n {~\mathrel{\Big\vert}~}\exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}\right\}\) 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 \({\mathfrak{p}}\in\operatorname{Spec}A\) define \(\operatorname{ht}({\mathfrak{p}}) \coloneqq\sup\left\{{n{~\mathrel{\Big\vert}~}\exists {\mathfrak{p}}_0 \subset {\mathfrak{p}}_1 \subset \cdots \subset {\mathfrak{p}}_n = {\mathfrak{p}}}\right\}\) with \({\mathfrak{p}}_i \in \operatorname{Spec}A\) distinct prime ideals.
• Krull dimension: define \(\operatorname{krulldim}A \coloneqq\sup_{{\mathfrak{p}}\in \operatorname{Spec}A}\operatorname{ht}({\mathfrak{p}})\), the supremum of heights of prime ideals.