Tags: #study-guides

Schemes

Define each item both in ${\mathsf{Sch}}{/ {k}} $ and in ${\mathsf{Alg}}{/ {k}} $.

Definitions

Properties of Schemes

• What is a scheme?
  • What is an affine scheme?
• What is an affine subscheme? An arbitrary subscheme?
  • What is an open subscheme?
  • What is a closed subscheme?
• What is a normal scheme?
• What is the Unsorted/normalization of a scheme?
• What is a reduced scheme? Irreducible?
  • What is the reduced subscheme of a scheme?
• What is an integral scheme?
• What is a quasicompact scheme?
• What is a Noetherian scheme?
• What is a separated scheme?
  • What is a quasiseparated scheme?
• What is a locally factorial scheme?
• What is a formal scheme?
• Singularities:
  • What is a smooth scheme?
  • What is a Gorenstein scheme?
  • What is a local complete intersection?
  • What is a Cohen-Macaulay scheme?
• What is analogous to the open unit disc? The punctured unit disc?
  • \({\mathbb{D}}\approx \operatorname{Spec}k{\left[\left[ t \right]\right] }\)
  • \({\mathbb{D}}\setminus\left\{{0}\right\}\approx \operatorname{Spec}k{\left(\left( t \right)\right) }\)
• What is a scheme-theoretic intersection?
  • For relative schemes \(X_i\to Y\), the intersection of \(X_1\) and \(X_2\) is the fiber product \(X_1 { \underset{\scriptscriptstyle {Y} }{\times} } X_2\). For affines \(X_1 = R/J_1\) and \(X_2 = R/J_2\), this is realized by \(\operatorname{Spec}(R/J_1 \otimes_R R/J_2) \cong \operatorname{Spec}R/(J_1 + J_2)\). If \(X_i \hookrightarrow Y\) are closed subschemes, then the intersection is again a closed subscheme of \(Y\)..

Morphisms

See properties of morphisms.

• What is a closed immersion of schemes?
• What is a closed embedding of schemes?
• What is an open immersion of schemes?
• What is a smooth morphism?
• What is a proper morphism? How is this related to Hausdorff + compactness properties?
• What is an etale morphism?
• What is a flat morphism?
• Finiteness:
  • What is a finite morphism?
  • What is a quasifinite morphism?
  • What is a finite type morphism?
    • What is a morphism locally of finite type?
  • What is a finitely presented morphism?
    • What is a locally finitely presented morphism?
• What is a closed morphism? A universally closed morphism?
• What is a projective morphism?
• What is a faithfully flat morphism?
• What is a rational morphism?
• What is a ramified morphism?
• What is the derivative of a morphism?
• What is a birational map?
• What is the degree of a morphism?
• Separatedness
  • What is a separated morphism? How is this related to Hausdorff properties?
  • What is a quasiseparated morphism?
  • What is qcqs? (Quasi-compact and quaiseparated)
• What is an affine morphism?
  • What is a quasiaffine morphism?
• What is a normal morphism?
• What is a regular morphism?
• What is a GIT quotient?

\({\mathcal{O}}_X{\hbox{-}}\)Modules

• What is an O_X module?
• What is a sheaf of O_x modules?
• What is a quasicoherent O_x module?
• What is a locally free \({\mathcal{O}}_X\) module?
• What does it mean for an \({\mathcal{O}}_X\) module to be globally generated?
• What is the Euler characteristic of an \({\mathcal{O}}_X\) module?
• What is the Hilbert polynomial of an \({\mathcal{O}}_X\) module?
• What is an elliptic curve over a scheme?

Scheme-Theoretic Constructions

• What is the scheme-theoretic definition of an affine variety?
• What is the scheme associated to a variety?
  • There is a bijection between varieties over \(k\) and separated reduced schemes of finite-type over \(k\).
• What is the scheme-theoretic definition of a curve?
• What is the fiber product of schemes?
• What is the scheme theoretic fiber \(X_y\) for a morphism \(f:X\to Y\)?
• What is the Proj construction?
• What is ${\mathbb{A}}^n_{/ {k}} $ as a scheme?
• How is ${\mathbb{P}}^n_{/ {{\mathbb{Z}}}} $ defined as a scheme? ${\mathbb{P}}^n_{/ {k}} $?
• What is the reduced structure on a closed subset of a scheme?
• What is the scheme-theoretic image?
• What is the hensilization?
• What is an \(n{\hbox{-}}\)fold point over \(k\)?
  • A scheme \(\operatorname{Spec}R\to \operatorname{Spec}k\) with one point such that \(R\) is a \(k{\hbox{-}}\)algebra of dimension \(n\).

Misc

• What is the residue field of a point?
  • What does it mean to evaluate a prime ideal \({\mathfrak{p}}{~\trianglelefteq~}R\) at an element \(f\in R\)?
    • \(f({\mathfrak{p}}) \coloneqq(R\to R/{\mathfrak{p}}\to\kappa({\mathfrak{p}}) \coloneqq\operatorname{ff}(R/{\mathfrak{p}}))(f)\).
• What is a morphism of locally ringed spaces?
• For \(X\to S\) a scheme over \(S\) and \(T\to S\) a ring morphism, what is a \(T{\hbox{-}}\)valued point of \(X\)?
• What is a generic point
• What is a special point?
  • What is a closed point?
  • What is a rational point?
  • What are the singular points of the scheme?
  • What is a specialization?
• What is base change?
  • What does it mean to be stable under base change?
  • What does it mean to be local on the base?
  • What is smooth base change?
  • What is proper base change?
• What does it mean to be regular in codimension one?
• What is the dimension of a scheme?
• What is the tangent space of a scheme?
• What is the blowup of a map of schemes?
• What is a resolution of singularities?
• What is a prime divisor on a scheme?
• What is a constructible set?
• What is a geometrically connected scheme?
• What is a special fiber?
• What is a generic fiber?
• What is a central fiber?

Results

• Valuative criteria:
  • What is the valuative criterion of properness?
  • What is the valuative criterion of separatedness?
• What is the criterion for a morphism to be projective?
• What is the Jacobi criterion?
• What is the Leray acyclicity theorem?
• What is Grothendieck vanishing?
• What is Kodaira vanishing?
• What is Serre vanishing?
• What is Serre’s projection formula?
• What is Serre’s finiteness theorem?
• What is the Riemann-Roch theorem?
• What is GAGA?
• What is the adjunction formula?
• What is Serre duality?
• What is Chevalley’s finiteness theorem?
• What is Hensel’s lemma?
• What is the Birkhoff-Grothendieck theorem?
• What are the weak and hard Lefschetz theorems?

Problems

• What is an example of an unramified morphism?
• When is the canonical sheaf dualizing?
• What is the cohomology of \(O_X\) for \(X = {\mathbb{P}}^n\)
• What is an example of a scheme that is not a variety?
• Show that the affine line with two origins is not separated.
• Show that \(X\) is a projective variety over \(\operatorname{Spec}k\) iff \(X\) is a closed subscheme of ${\mathbb{P}}^n_{/ {k}} $.
• Show that \({\mathsf{R}{\hbox{-}}\mathsf{Mod}} { \, \xrightarrow{\sim}\, }{\mathsf{QCoh}}(\operatorname{Spec}R)\), and that \({\mathsf{Coh}}(\operatorname{Spec}R)\) corresponds to finitely-generated modules.
• Show that a locally free sheaf is always quasicoherent, and is coherent if it has finite rank.
• Show that if \({\mathcal{L}}\) is a line bundle, then \({\mathcal{L}}\otimes_{{\mathcal{O}}_X} {\mathcal{L}} {}^{ \vee } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X\).
• Show that ${\mathbb{A}}^1_{/ {{\mathbb{Z}}}} $ is flat over \(\operatorname{Spec}{\mathbb{Z}}\).
• Show that \(\left\{{f_i}\right\}\) generate the unit ideal in \(R\) iff \(D(f_i) \rightrightarrows\operatorname{Spec}R\).
• Show that \(M\) is a faithfully flat module iff \({-}\otimes_R M\) is a faithful functor.
• Show that a finite morphism between smooth varieties is flat.
• Show that a regular function on an affine scheme is not necessarily determined by its values.