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Mckernan AG 1

Lecture 1

• Prove that the Zariski topology is a topology
• Why is every finite subset in the Zariski topology closed?
• Is the Zariski topology Hausdorff?
• What does it mean to be in general position?
• Show that if \(V \subset \mathbb{A}_{K}^{n}\) is any set of \(d \leq 2 n\) points in linear general position then \(V\) is defined by quadratic polynomials.
• Show that \(I(V)\) is an ideal.
• Show that \(V\) and \(I\) are inclusion-reversing.
• \(I(V_1 \cup V_2) = ?\)
• Show that \(V(I(V)) = { \operatorname{cl}} ^{\mathsf{Top}}_{{\mathbb{A}}^n_{/ {k}} } V\).
• Show that \(\sqrt{I}\) is an ideal.
• Prove Hilbert’s Nullstellensatz: \(I(V(I)) = \sqrt{I}\).
• Show that if \(X\) is irreducible then every nonempty open subset is dense.
• Show that \(X\subseteq {\mathbb{A}}^n\) is irreducible iff \(I(X) {~\trianglelefteq~}k[x_1, \cdots, x_{n}]\) is prime.
• What are the equations for the nodal and cuspidal cubics?
• Show that the nodal and cuspidal cubics are irreducible.
• Show that every affine variety is Noetherian.
• What is Noetherian induction?
• Show the uniqueness of decomposition into irreducibles.

Lecture 2

• Show that if \(k\) is a field and \(f\in k[x]\) then the localization \(k[x] \left[ { \scriptstyle { {f}^{-1}} } \right]\) is not a field.
• For \(R\subseteq F\) a subring, what does it mean for \(a\in F\) to be integral over \(R\)?
  • What is the integral closure of \(R\) in \(F\)?
• Show that TFAE: \(a\) is integral over \(F\), \(F[a]\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\), there is an intermediate ring \(F[a] \subseteq L \subseteq F\) where \(L\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fg}}\).
• Show that if \(R\subseteq F\) and \(\left\{{r_i}\right\} \subseteq F\) are all integral over \(R\), then \(R[r_1,\cdots, r_n]\) is integral over \(R\).
• Show that the integral closure of \(R\) in \(F\) is a ring.
• Show that if a field \(k\) is integral over a subring \(R\subseteq k\), then \(R\) must be a field.
• If \(L/k\) is a field extension with \(L\) finitely generated as a \(k{\hbox{-}}\)algbra, then the extension is algebraic.
• Prove the weak Nullstellensatz:
  • In terms of \(\operatorname{mSpec}k[x_1, \cdots, x_{n}]\)
  • In terms of sequences of elements \(\left\{{f_i}\right\} \in k[x_1, \cdots, x_{n}]\).
• Use the Rabinowitsch trick to prove the strong Nullstellensatz in terms of sequences of polynomials.

Lecture 3

• What is a locally small category?
• What is a full subcategory?
• What is an equivalence of categories?
  • What is a full functor?
  • What is an essentially surjective functor?
• How is the Yoneda embedding \(h: \mathsf{C} \to {\mathsf{Fun}}({ {C}^{\operatorname{op}}}, {\mathsf{Set}})\) defined?
  • Prove that \(h\) is fully faithful.
• Show that if \(F: \mathsf{C}\to {\mathsf{Set}}\) is representable by \(Y\) then \(Y\) is unique up to unique isomorphism.
• Show that \(V(y-x^2) \cong {\mathbb{A}}^1\) as affine varieties.
• Show that \(V(y-x^2, z-x^3) \cong {\mathbb{A}}^2\).
• Show that the nodal cubic \(V(y^2 - x^3 - x^2)\) admits a parameterization \(t\mapsto (t^2-1, t^3-t)\).
• What is a quasi-affine variety?
• What is the Jacobian conjecture?

Lecture 4

• Show that if \(X\) is an affine variety then there is a correspondence \({\left\lvert {X} \right\rvert} { \, \xrightarrow{\sim}\, }\operatorname{mSpec}A(X)\) where \(A(X) = k[X]/I(X)\) is the coordinate ring.
• Show that \(A(X) \cong {\mathcal{O}}_X(X)\).
• Show that \(A(X)\) is reduced.
• Describe the image of the functor \(F: {\mathsf{Aff}}{\mathsf{Var}}_{/ {k}} \to \mathsf{CRing}\) given by \(F(X) \coloneqq{\mathcal{O}}_X(X)\).
  • Show that \(F\) restricted to its image yields an equivalence of categories.

Lecture 5 and 6

• Show that for a complex manifold \(X\), the stalks \(\mathop{\mathrm{Hol}}(X; {\mathbb{C}})_p \cong {\mathbb{C}}{\left[\left[ z_1,\cdots, z_n \right]\right] }\).
• Show that for a real manifold \(X\), there is a nontrivial kernel \(C^\infty(X; {\mathbb{R}})_p \to {\mathbb{R}}{\left[\left[ x_1,\cdots, x_n \right]\right] }\).
• Show that for an affine variety, \({\mathcal{O}}_{X, p} \cong A(X) \left[ { \scriptstyle { {{\mathfrak{m}}_p}^{-1}} } \right]\).
• Show that a morphism of sheaves \(F: {\mathcal{F}}\to {\mathcal{G}}\) is an isomorphism iff \(F_p: {\mathcal{F}}_p \to {\mathcal{G}}_p\) is an isomorphism on stalks.
• Produce a morphism of sheaves that is surjective on stalks but not surjective on global sections.
• What is the pushforward sheaf?
• What is the inverse image sheaf?
• What is a morphism of local rings?
• What is a morphism of locally ringed spaces?
• What is an \({\mathcal{O}}_X{\hbox{-}}\)module?
• How are the sheaf homs and tensors of \({\mathcal{O}}_X{\hbox{-}}\)modules defined?
• How are the kernel, image, and cokernel sheaves defined?
• Show that the image presheaf is not a sheaf.
• How is the pullback of a sheaf of \({\mathcal{O}}_X{\hbox{-}}\)modules defined?

Lecture 7

• Let \(X = \operatorname{Spec}k[x,y]\) and \(f\in k[x,y]\), what is the value of \(f\) at \(p\in X\)?
  • Let \(X = \operatorname{Spec}{\mathbb{Z}}\) and \(n \in {\mathbb{Z}}\), what is the value of \(n\) at a prime \(p\)?
  • Let \(n=60\), thought of as a function on \(\operatorname{Spec}{\mathbb{Z}}\). What are its zeros?
• Find an example of an affine scheme \(X = \operatorname{Spec}R\) and \(f\neq 0 \in R\) where the value of \(f\) is everywhere zero.
• What is a basis for the Zariski topology on \(\operatorname{Spec}R\)?
• Show that the Zariski topology is not \(T_1\).
• When is zero a prime ideal?
• What are the proper closed subsets of \(\operatorname{Spec}k[x]\)?
  • What is the generic point?
• Describe all of the prime ideals of \(k[x,y]\).
• Show that \(\sqrt{I}\) is the intersection of all prime ideals containing \(I\).
• Show that if \(D_f = \cup D_{g_i}\) then for some \(n\), \(f^n = \sum b_i g_i\).

Lecture 8

• Describe a section in \({\mathcal{O}}_X(U)\) for \(X = \operatorname{Spec}R\) in terms of functions to \(\displaystyle\coprod_{p\in U} R \left[ { \scriptstyle { {p}^{-1}} } \right]\).
• Show that for \(X = \operatorname{Spec}R \in {\mathsf{Aff}}{\mathsf{Sch}}\), the stalks are \({\mathcal{O}}_{X, p} = R \left[ { \scriptstyle { {p}^{-1}} } \right]\).
  • What are the sections over a distinguished open \(D(f)\)?
  • Show that \({\mathcal{O}}_X(X) = R\).
• Describe a morphism of locally ringed spaces.
• Show that there is an equivalence of categories \({\mathsf{Aff}}{\mathsf{Sch}} { \, \xrightarrow{\sim}\, }\mathsf{CRing}\) (all rings assumed unital).

Lecture 9

• What is a linear subspace of \({\mathbb{P}}^n\)?
  • How is the dimension of a linear subspace computed?
• Show that if \(V_1, V_2\) are subspaces of \({\mathbb{P}}^n\) then \(\dim (V_1 \cap V_2) \geq \dim V_1 + \dim V_2 - n\).
  • Equivalently, \(\operatorname{codim}(V_1 \cap V_2) \leq \operatorname{codim}V_1 + \operatorname{codim}V_2\).

Lecture 10

• Give a precise definition of what it means for a polynomial to be homogeneous (in terms of graded rings).
• Show that \(f\in k[x_1, \cdots, x_{n}]\) is homogeneous of degree \(n\) iff \(f(\lambda \mathbf{x}) = \lambda^n f(\mathbf{x})\) when \(k\) is an infinite field.
• Give a precise definition of a projective variety.
• Describe how to turn an affine variety into a projective variety.
  • Do this with \(V(x^2+y^2-1) \subseteq {\mathbb{A}}^2\).
    • Spoiler: \(V(X^2 + Y^2- ^2)\).
  • Do this with \(V(y=x^3) \subseteq {\mathbb{A}}^2\).
    • Spoiler: \(V(YZ^2-X)\).
• Describe in coordinates the line at infinity in \({\mathbb{P}}^2\).
• Find the intersection between the projectivizations of \(V(y)\) and \(V(y-tx-1)\) in \({\mathbb{P}}^2\) when \(t=0\).
• Show that if \(f\in {\mathbb{R}}[x,y]^{\deg =2}\) and \({\sharp}V_{\mathbb{R}}(f) \geq 1\), then \(V(f) = S^1\) iff its homgenization \(F\) satisfies \(V_{{\mathbb{P}}^2_{/ {{\mathbb{R}}}} }(F) \supseteq\left\{{{\left[ {1:\pm i: 0} \right]}}\right\}\).
• Find the moduli space of conics in \({\mathbb{P}}^2\) and compute its dimension.
  • Give equations for the locus \(H_p\) of conics passing through a fixed point \(p\).
• Show that if \(k\) is infinite, the locus of conics passing through 5 fixed points is either a single point or infinite.
• Show that there is a unique conic passing though 5 points in linear general position.
• Show that there is a unique circle passing through any 3 non-collinear points in \({\mathbb{R}}^2\).

Lecture 11

• Give a precise definition of a morphism between projective varieties.
• Describe the image of \(f: {\mathbb{P}}^1\to {\mathbb{P}}^2\) given by \([x:y]\mapsto [x^2: xy: y^2]\).
• Give equations for a (projective) parameterization of the twisted cubic.
• Give equations for the Veronese embedding.
  • Describe the Veronese surface (e.g. what \({\mathbb{P}}^N\) is it embedded in?)
• What is a determinental variety?

Lecture 12

Note: revisit to practice very geometric proofs.

• Fit \(\operatorname{PGL}_n\) into an exact sequence to describe it as a quotient.
• Let \(W\leq {\mathbb{P}}^n\) be a linear subspace of dimension \(k\), and show that the subset of \({\operatorname{Gr}}_{k+1}({\mathbb{P}}^n)\) of \((k+1){\hbox{-}}\)dimensional subspaces containing \(W\) is isomorphism to \({\mathbb{P}}^m\) where \(m=n-k-1\).
• Show that if \(\left\{{p_i}\right\}, \left\{{q_i}\right\}\) are two sets of \(n+2\) points in \({\mathbb{P}}^n\) in linear general position then there exists a unique \(M\in \operatorname{PGL}_n(k)\) with \(M(p_i) = q_i\).
• For \({\mathbb{P}}^n = {\mathbb{P}}(V)\), show that the dual projective space \({\mathbb{P}}(V {}^{ \vee })\) is a moduli of hyperplanes in \({\mathbb{P}}^n\).
• Show that if \(\left\{{p_i}\right\}\) are \(n+3\) points in linear general position in \({\mathbb{P}}^n\) then there is a unique rational normal curve passing through them. (Warning: tricky/involved proof.)

Lecture 13

• Derive the rational parameterization of \(a^2+b^2 = 1\) as a map \({\mathbb{A}}^1 \to {\mathbb{A}}^2\) and its inverse.
  • What are the projectivizations of these maps?
• What is a quasi-projective variety?
• What is a morphism of quasi-projective varieties?
• What is a quasi-affine variety?
• Show that if \(X\) is projective and \(F\) is a homogeneous polynomial of degree \(d\) then \(\left\{{x\in X {~\mathrel{\Big\vert}~}F(x) \neq 0}\right\}\) is an open affine subset of \(X\).

Lecture 14

• Define an inverse limit.
• Describe initial and terminal objects as limits.
• Determine whether a product is a limit or a colimit (resp. a coproduct).
• Show that a pushout in \(\mathsf{CRing}\) is the tensor product.

Lecture 15

• Give equations for the Segre embedding.
• Show that the categorical product of \(X\subseteq {\mathbb{P}}^n\) and \(Y\subseteq {\mathbb{P}}^m\) is the Segre variety.
  • (Warning: involved proof.)
• What is the graph \(\Gamma_f\) of a morphism \(f: X\to B\) of quasi-projective varieties?

Lecture 16

• What is a valuation?
• What is the valuation ring of a field?
• What is a discrete valuation?
• What is a DVR?
• What is a uniformizer?
• Show that \(R\) is a DVR iff \(R\) is a local PID.
• Is \(\mathop{\mathrm{Hol}}(\Delta, {\mathbb{C}})_p\) for \(p=0\) is a local DVR with uniformizer \(z\).
• Let \(R\) be the local ring of a curve over $k= {

  \mkern 

  1.5mu

  \overline{\mkern-1.5muk\mkern-1.5mu}

  \mkern 

  1.5mu } $ and identify the closed and generic points of \(\operatorname{Spec}R\) and their residue fields.
  • Describe some set-theoretic maps between \(\operatorname{Spec}R\) and \(\operatorname{Spec}\operatorname{ff}(R)\) in this case. Which are morphisms of schemes?
• Let \(k = {\mathbb{Q}}(\sqrt 3)\) and describe the cover \(\operatorname{Spec}{\mathcal{O}}_k \to \operatorname{Spec}{\mathbb{Z}}\) arising from \({\mathbb{Z}}\to {\mathcal{O}}_k\). What are the fibers over primes? What are the splitting/inert/ramification behaviors?

Lecture 17

• What is an irreducible scheme?
• What is a reduced scheme?
  • Show that a scheme is reduced iff it has reduced stalks.
• What is an integral scheme?
• Show that a scheme is integral iff irreducible and reduced.
• What is a locally Noetherian scheme?
• What is a Noetherian scheme?
• Find an example of a scheme \(X\) such that \({\left\lvert {X} \right\rvert}\) is a Noetherian topological space but \(X\) is not a Noetherian scheme.
• Show that \(X\) is locally Noetherian iff for every affine open \(U_i = \operatorname{Spec}R_i\) subset, \(R_i\) is a Noetherian ring.
• Show that if \(A\) is a ring and \(\left\langle{f_i}\right\rangle = \left\langle{1}\right\rangle\) with \(A \left[ { \scriptstyle { {f_i}^{-1}} } \right]\) a Noetherian ring for each \(i\), then \(A\) is Noetherian.
• What is a morphism that is locally of finite type?
• Describe the localization \(R = k[x]_x\).
• What is a finite type morphism?
• What is a finite morphism?
• Describe a morphism that is finite type but not finite.
  • Spoiler: that \(R\) is a finite \(k[x]{\hbox{-}}\)algebra, that the canonical map \(k[x]\to R\) is finite type, but \(k[x]\) is not a finitely generated \(k[x]{\hbox{-}}\)module so that \(f\) is not finite.
• What is an open subscheme?
• What is an open immersion?
• What is a closed subscheme?
• What is a closed immersion?
• Show that \(\operatorname{Spec}R/I\) is a closed subscheme of \(\operatorname{Spec}R\) for any ideal \(I\in \operatorname{Id}(R)\).
• Find an example of an irreducible scheme that is not reduced.
• Find two distinct schemes with the same support.
• Show that to give a morphism of local rings is the same as to give an inclusion of the quotients of the maximal ideals.
• What is the \(n\)th infinitesimal neighborhood of an ideal \(p\in \operatorname{Spec}A\)?

Lecture 18

• Show that if \(\left\{{X_i}\right\} \rightrightarrows X\) and \({\mathcal{F}}_i \in {\mathsf{Sh}}(X_i; {\mathsf{Grp}})\) are sheaves over the opens, given the data of isomorphism \(\phi_{ij}: { \left.{{{\mathcal{F}}_i}} \right|_{{X_{ij}}} } \to { \left.{{{\mathcal{F}}_j}} \right|_{{X_{ij}}} }\) satisfying the cocycle condition \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on every \(X_{ijk}\) then there is a unique sheaf \({\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Grp}})\) with isomorphism \(\psi_{i}: { \left.{{{\mathcal{F}}}} \right|_{{X_i}} } \to {\mathcal{F}}_i\) such that \(\psi_j = \phi_{ij} \circ \psi_i\).
• For \(X\) a scheme and \(Y \subseteq X\) a closed subset, define the reduced integral subscheme structure.
• Given schemes \(U_{i}\), and subschemes \(U_{i j} \subset U_{i}\), together with isomorphisms,
• $$

• \phi

  {i j}: U{i j}

  \longrightarrow

  U_{j i},

• \quad

  \text{ such that }

  \quad

• \phi

  {i k} = {

  \phi

  {j k}

  \circ

  \phi

  {i j}}

  \quad

  \text{on }

  U{ij}

  \cap

  U_{ik},
• $$
• show that there is a scheme \(X\) and open immersions \(U_i \to X\) whose images cover \(X\) such that \({ \left.{{\psi_i}} \right|_{{U_{ij}}} } = \psi_i \circ \phi_{ij}: U_{ij} \to X\).
• Show that if $X, Y

  \in {\mathsf{Sch}}

  _{/ {S}} $ are schemes, \(\left\{{X_i}\right\}\rightrightarrows X\) is an open cover, and the local fiber products \(X_i { \underset{\scriptscriptstyle {S} }{\times} } Y\) exists, then the global fiber product \(X{ \underset{\scriptscriptstyle {S} }{\times} } Y\) exists.