Define each item both in ${\mathsf{Sch}}{/ {k}} $ and in ${\mathsf{Alg}}{/ {k}} $. DefinitionsProperties of Schemes What is a scheme? What is an affine scheme? What is an open subscheme**? A **closed subscheme? What is a normal scheme? What is the 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? ### 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? 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]? 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 ${\mathcal{O}}_X$ 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? 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? Misc What is the residue field of a point? 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 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.