---
created: 2024-05-17T15:56
updated: 2024-06-01T17:06
flashcard: Review::Schemes
---
From the enumerative geometry book, see tablet.

# Chapter 3

## 3.1

### 3.1.1 Schemes and their basic properties

- [x] What is the functorial definition of $\Gamma(\OO_X)$? ✅ 2024-05-17
	- [ ] $\Sch(X\to \spec \ZZ[t]) \cong \CRing(\ZZ[t]\to \Gamma(X)) \cong \Gamma(X)$.
- [ ] Describe $H^0(f_* \OO_X)$.
- [ ] Describe $f^{\sharp}$.
- [x] What is a scheme of pure dimension? ✅ 2024-05-17
	- [ ] All irreducible components have the same dimension
- [x] What is a quasicompact morphism of arbitrary schemes? ✅ 2024-05-17
	- [ ] Preimage of every affine open is quasicompact.
- [x] What is a quasicompact topological space? A compact space.✅ 2024-05-17
	- [ ] Every open cover has a finite subcover.
	- [ ] Compact: Hausdorff and quasicompact.
- [x] What is a morphism that is **locally** of finite type? ✅ 2024-05-17
	- [ ] $f:X\to S$ where $\forall x\in X$ there exist open neighborhoods $x\in \spec A, f(x) \in \spec B$ with $f(\spec A)\subseteq \spec B$ and $B\to A$ is a finite type ring morphism.
- [x] What is a finite type ring morphism? ✅ 2024-05-17
	- [ ] $f: B\to A$ where $A\cong B[x_1,\cdots, x_n]/I$ as a $B\dash$algebra.
- [x] How does a ring morphism induce an algebra structure? ✅ 2024-05-17
	- [ ] $f: B\to A$ makes $A\in \Alg\slice B$ via $b.a \da f(b)a$.
- [x] What is a finite type morphism? ✅ 2024-05-17
	- [ ] Locally of finite type and quasicompact.
- [x] What is a Noetherian scheme? ✅ 2024-05-17
	- [ ] Locally Noetherian and quasicompact.
- [x] What is a locally Noetherian scheme? ✅ 2024-05-17
	- [ ] Covered by $\spec R_i$ Noetherian rings.
- [x] What is a Noetherian ring? ✅ 2024-05-17
	- [ ] ACC on ideals.
	- [ ] Every ideal is finitely generated.
- [x] Give an example of a non-Noetherian ring. ✅ 2024-05-17
	- [ ] $\ZZ[x_1, x_2, \cdots,]$, a polynomial ring on infinitely many variables.
- [x] What is the most important property of Noetherian schemes? ✅ 2024-05-17
	- [ ] Finitely many irreducible components and finitely many associated points.
- [ ] What is an associated point of a scheme?
- [ ] Show that if $f:X\to S$ of finite type with $S$ Noetherian $\implies X$ is Noetherian.
- [ ] Show that a morphism $f:X\to S$ with $X$ Noetherian is quasicompact.
- [x] Give an example of a Noetherian scheme. ✅ 2024-05-17
	- [ ] Any scheme of finite type over a field.
- [x] Give an example of an affine Noetherian scheme whose structure morphism is not locally of finite type. Why is this interesting? ✅ 2024-05-17
	- [ ] $\spec R\fps{x_1,\cdots, x_n} \to \spec R$ over any Noetherian $R$.
	- [ ] Interesting because finite type over a field (Noetherian) implies Noetherian, but Noetherian doesn't imply finite type.
- [x] What is a closed immersion? Open immersion? ✅ 2024-05-17
	- [ ] $f: X\to S$ where $X \to f(X)$ is a homeomorphism onto a closed (resp. open) subset, **and** $f_x^\sharp: \OO_{S. f(x)} \to \OO_{X, x}$ is surjective, (resp. an isomorphism) so regular germs on the image lift. 
- [x] What is a closed (resp. open) subscheme? ✅ 2024-05-17
	- [ ] The image of a closed (resp. open) immersion.
- [x] What is an immersion of schemes? ✅ 2024-05-17
	- [ ] $f:X\to S$ which factors as $X\to Y\to S$ with $X\to Y$ an open immersion and $Y\to S$ a closed immersion.
- [x] What is a locally closed immersion? ✅ 2024-05-17
	- [ ] $f:X\to S$ which factors as $X\to Y\to S$ with $X\to Y$ a closed immersion and $Y\to S$ an open immersion.
- [x] When is a locally closed immersion an immersion? ✅ 2024-05-17
	- [ ] $X\to S$ with $S$ locally Noetherian; also implies quasicompact.
- [x] Give an example of a quasicompact morphism? ✅ 2024-05-17
	- [ ] $X\to S$ a locally closed immersion with $S$ locally Noetherian.
- [x] Is an open subset of a Noetherian scheme Noetherian? ✅ 2024-05-17
	- [ ] Generally only *affine* open subsets.
- [x] What is a reduced ring? ✅ 2024-05-17
	- [ ] No nonzero nilpotents?
- [x] What is a reduced scheme? ✅ 2024-05-17
	- [ ] Reduced local rings $\OO_{X, x}$.
- [x] What is an integral scheme? ✅ 2024-05-17
	- [ ] Reduced and irreducible.
- [x] Give a trivial and a nontrivial example of a non-reduced scheme. ✅ 2024-05-17
	- [ ] Trivial example: $D_n \da \spec \CC[t]/t^n$, a single non-reduced point.
	- [ ] Nontrivial: $\CC[x, y]/(xy, y^2)$ is non-reduced at $(x, y)$.
- [x] Why are quasicompact reduced schemes important? ✅ 2024-05-17
	- [ ] Regular functions are determined by values on points.
	- [ ] Counterexample: $0\neq t\in D_n$ is not the zero function but vanishes at the unique point of $D_n$.
- [x] What is the fibre of $f: X\to S$ at $s\in S$? ✅ 2024-05-17
	- [ ] $X\fiberprod{S} k(s)$.
- [x] What is a separated (resp. quasiseparated) morphism $f$? ✅ 2024-05-17
	- [ ] $\Delta_f$ is a closed immersion (resp. quasicompact).
- [x] What is a separated scheme? ✅ 2024-05-17
	- [ ] Separated structure morphism.
- [x] What is an affine morphism? ✅ 2024-05-17
	- [ ] Preimages of affines are affine.
- [x] Give a characterization of affine morphisms. ✅ 2024-05-17
	- [ ] $f: X\to S$ is affine iff $X\cong \spec_{\OO_S} A$ where $A$ is a quasicoherent sheaf of $\OO_S\dash$algebras.
	- [ ] Recover $A = f_* \OO_X$.
- [ ] What is the global spec construction?
      
### 3.1.2 Varieties, fat points, morphisms

- [x] What is the scheme-theoretic definition of a variety? ✅ 2024-05-17
	- [ ] Separated scheme of finite type over $\spec k$ for $k=\kbar$.
	- [ ] Integral separated scheme of finite type over a field.
- [x] What is the scheme theoretic definition of an affine variety? ✅ 2024-05-17
	- [ ] $X = \spec A$ where $A = k[x_1,\cdots, x_n]/I$
- [x] When is an affine variety reduced? ✅ 2024-05-17
	- [ ] $\sqrt I = I$ for $A = \kxn/I$.
- [x] When is a separated scheme an algebraic variety? ✅ 2024-05-17
	- [ ] If it admits a finite covering by affine varieties.
- [x] Are affine varieties algebraic varieties? ✅ 2024-05-17
	- [ ] Yes, since affine morphisms are separated.
- [x] What is a projective algebraic variety? ✅ 2024-05-17
	- [ ] An algebraic variety admitting a locally closed immersion in some $\PP^n\slice k$.
- [x] What is a complete linear series? ✅ 2024-05-17
	- [ ] $\abs{D}$ is the set of all effective divisors equivalent to $D$, i.e. $D'$ such that $D-D' = (f)$ for some rational function $f$.
- [x] What is the rational normal curve of degree $d$? ✅ 2024-05-17
	- [ ] $[x:y]\mapsto [x^d: x^{d-1}y : \cdots : xy^{d-1}: y^d] \in \PP^d$, the $d$th Veronese embedding determined by $\OO_{\PP^1}(d)$.
- [x] What is the twisted cubic? ✅ 2024-05-17
	- [ ] The rational normal curve of degree $3$ in $\PP^3$.
- [x] What is a finite variety? ✅ 2024-05-17
	- [ ] $h^0(\OO_X) < \infty$.
- [x] What is the length of a variety? ✅ 2024-05-17
	- [ ] $h^0(\OO_X) \da \dim_k \OO_X(X)$.
- [x] What is a fat point of length $\ell$? ✅ 2024-05-17
	- [ ] $X = \spec A$ over $\spec k$ for $A$ a local Artinian $k\dash$algebra with residue field $k$ such that $\dim_k A = \ell$.
	- [ ] Alternatively, the composition $(X_\red \injects X\to \spec k) = \id$.
- [x] Define $T_x X$. ✅ 2024-05-17
	- [ ] $(\mfm_x/\mfm_x^2)\dual$
	- [ ] $\Hom_x(D_2, X) = \ts{f: D_2\to X \st h(\pt) = x}$ where $\pt$ is the unique closed point of $D_2$.
- [x] What is a closed point of a scheme? ✅ 2024-05-17
	- [ ] A point $x\in X$ such that $\Cl_X(x) = x$.
	- [ ] Corresponds to a maximal ideal in some affine open neighborhood.
- [x] Give an example of a family of schemes with reduced general fiber and a non-reduced central fiber. ✅ 2024-05-17
	- [ ] $X_t \da \spec \RR[x,y]/(y-x^2, y-t)$. Note $X_0 \cong D_2$, while $X_t = (x\pm \sqrt t, y-t)$ is two reduced points for $t\neq 0$.
- [x] What is a proper morphism? ✅ 2024-05-17
	- [ ] Separated, of finite type, universally closed.
- [x] What is a universally closed morphism? ✅ 2024-05-17
	- [ ] $f:X\to S$ where for every base change $T\to S$, the induced map $T\fiberprod{S} X \to T$ is topologically closed.
- [x] What is the valuated criterion for properness? ✅ 2024-05-17
	- [ ] Take $X\to S$, complete square to $\spec \ff(A) \injects \spec A$ for $A$ a DVR, require $\exists ! \eta: \spec A\to X$.
- [x] What is a finite morphism? ✅ 2024-05-17
	- [ ] Proper and affine.
- [x] What is a quasifinite morphism? ✅ 2024-05-17
	- [ ] Locally of finite type, quasicompact, finite fibers.
- [x] How can one show a morphism is finite? ✅ 2024-05-17
	- [ ] Sufficient: proper quasifinite morphism of Noetherian schemes.
- [x] What is a flat morphism? ✅ 2024-05-17
	- [ ] $f_x^\sharp: \OO_{S, f(x)}\to \OO_{X, x}$ is a flat ring morphism $\forall x\in X$.
- [x] What is a flat ring morphism? Faithfully flat? ✅ 2024-05-17
	- [ ] $f:A\to B$ where $(\wait)\tensor_A B$ is exact, resp. faithful and exact.
	- [ ] Equivalently, faithfully flat $\iff$ flat and $\spec B\surjects \spec A$.
- [x] What is a faithfully flat morphism? ✅ 2024-05-17
	- [ ] Flat and surjective.
- [x] How is (faithful) flatness related to openness? ✅ 2024-05-17
	- [ ] Flat morphisms are open maps and have open image.
	- [ ] Faithfully flat morphisms are epimorphisms.
- [x] Define an etale morphism. ✅ 2024-05-17
	- [ ] Smooth and unramified.
- [x] Define unramified, smooth, and etale morphisms. ✅ 2024-05-17
	- [ ] Take $X\to S$ locally finite type, $\bar{A} \to A$ a square zero extension of fat points, pull to $\spec A \injects \spec \bar A$ and complete square. Require $\leq 1$, respect $\geq 1$ and $=1$, morphisms $\spec \bar A\to X$.
- [x] What is a normal scheme? ✅ 2024-05-17
	- [ ] For quasicompact integral schemes, $\OO_{X, x}$ is normal.
- [x] What is a normal ring? ✅ 2024-05-17
	- [ ] Integrally closed in its field of fractions.
	- [ ] I.e. if $x\in \ff(A)$ is a root of a monic polynomial in $A[x]$, then $x\in A$.
- [ ] What is a square zero extension?
- [x] What is a dominant morphism? ✅ 2024-05-17
	- [ ] Dense image.
- [ ] When is a dominant morphism surjective?
- [x] Define the normalization of a scheme. ✅ 2024-05-17
	- [ ] For $X$, a pair $(\tilde X, \mu: \tilde X \to X)$ which is initial among dominant morphisms from normal schemes.
- [x] Characterize normalizations. ✅ 2024-05-17
	- [ ] $f: Y\to X$ is the normalization iff $Y$ is normal and $f$ is birational and integral.
- [x] When is the normalization a finite morphism? ✅ 2024-05-17
	- [ ] Sufficient: $X$ is a variety.
- [x] When is a finite morphism flat? ✅ 2024-05-17
	- [ ] $f:X\to S$ with $S$ locally Noetherian where $f$ is affine and $f_* \OO_X$ is locally free.
- [x] When is a morphism smooth of relative dimension $d$? ✅ 2024-05-17
	- [ ] For $f: X\to S$ finite type $k\dash$schemes, flat and smooth fibers of pure dimension $d$.
	- [ ] For $f:X\to S$ smooth $k\dash$varieties, $T_{X, x}\surjects T_{Y, f(x)}$.
- [x] What is miracle flatness? ✅ 2024-05-17
	- [ ] $f: X\to S$ of finite type $k\dash$schemes, $X$ Cohen-Macaulay, $S$ smooth, equidimensional fibers $\implies$ $f$ is flat.
- [x] When is a proper morphism a closed immersion? ✅ 2024-05-17
	- [ ] For varieties: injective on points and tangent spaces.
	- [ ] For $f$ finite type between locally Noetherian schemes, $\iff$ proper monomorphism
- [x] When is $f_* \OO_X \cong \OO_S$? ✅ 2024-05-18
	- [ ] Sufficient: $X\to S$ proper birational with $X$ integral and $S$ normal and locally Noetherian.
- [x] How can one show a morphism is an open immersion? ✅ 2024-05-18
	- [ ] $f: X\to S$ separated, quasifinite, birational, of finite type, with $S$ normal locally Noetherian (Zariski's main theorem)
	- [ ] Any etale injective morphism.
- [x] How can one show a morphism is etale? ✅ 2024-05-18
	- [ ] $f: X\to S$ with $X, S$ smooth $\CC\dash$varieties inducing an isomorphism on tangent spaces.
- [x] How can one show a morphism is an isomorphism? ✅ 2024-05-18
	- [ ] Etale and bijective.
	- [ ] $f: X\to S$ between Noetherian integral schemes with $S$ normal where $f$ is **bijective, birational, proper**.
	- [ ] $f:X\to S$ finite birational with $S$ normal.
	- [ ] Bijective between smooth $\CC\dash$varieties of the same dimension.
- [x] Let $f:X\to S$ with $S$ normal. How can one show $f$ is an isomorphism? ✅ 2024-05-18
	- [ ] $f$ finite, birational.
	- [ ] $f$ bijective, birational, proper.
	- [ ] $X, S$ smooth, $f$ bijective, $\dim X = \dim S$ (doesn't require normality)
- [x] Give the four main consequences of Zariski's main theorem. ✅ 2024-05-18
	- [ ] $X\to S$ proper birational of Noetherian integral schemes with $S$ normal has connected fibers.
	- [ ] $f: X\to S$ between Noetherian integral schemes with $S$ normal where $f$ is bijective, birational, proper is an isomorphism.
	- [ ] A finite (or integral) birational morphism $f:X\to S$ with $S$ normal is an isomorphism. 
	- [ ] Bijection between smooth $\CC\dash$varieties of the same dimension is an isomorphism.
- [x] Is an integral morphism finite? ✅ 2024-05-18
	- [ ] Integral + locally of finite type = finite.

### 3.1.3 Schemes with embedded points

- [x] What is an associated point? ✅ 2024-05-18
	- [ ] $p\in \spec R$ is associated to $M\in \rmod$ if $p = \Ann_R(m)$ for some $m\in M$.
	- [ ] Equivalently, $p$ is associated to $M$ iff $R/p \leq M$ is an $R\dash$submodule.
- [x] What are isolated primes of $M$? ✅ 2024-05-18
	- [ ] Minimal primes $p\in \spec R$ such that $p \contains \Ann_R(M)$.
	- [ ] All isolated primes are contained in the set of associated points.
- [x] What is an embedded prime? ✅ 2024-05-18
	- [ ] An associated point which is non-isolated.
- [x] Why care about associated points? ✅ 2024-05-18
	- [ ] $M_i / M_{i-1} \cong R/p_i$ in a composition series and the $p_i$ are exactly the associated primes of $M$.
- [x] What is a primary ideal? ✅ 2024-05-18
	- [ ] $xy\in q \implies x\in q$ or $y^n\in q$ for some $n$.
	- [ ] Equivalently, $q$ such that every zero divisor in $R/q$ is nilpotent.
- [x] Why care about primary ideals? ✅ 2024-05-18
	- [ ] Any $I\in \Id(R)$ has a primary decomposition $I = \Intersect q_i$ with $q_i$ primary.
	- [ ] If $q$ is primary, $p \da \sqrt{q}$ is prime.
- [x] What does it mean for $q$ to be $p\dash$primary? ✅ 2024-05-18
	- [ ] $\sqrt{q} = p$.
- [x] Let $M = k[x,y]/(xy, y^2)$. What are the associated primes? ✅ 2024-05-18
	- [ ] $(y)$ and $(x, y)$.
- [x] Describe embedded and associated primes of a reduced affine scheme. ✅ 2024-05-18
	- [ ] No embedded primes, no embedded points, associated primes are isolated and minimal and correspond to irreducible components.
- [x] Why care about associated primes of $I\normal R$? ✅ 2024-05-18
	- [ ] Minimal primes containing $I$ correspond to irreducible components of the closed subscheme $\spec (R/I) \leq \spec R$
	- [ ] Every embedded prime $p$ contains a minimal prime $p' \subseteq p$, so $p$ determines an embedded component $V(p)$ embedded in an irreducible component $V(p')$.
- [x] What is an embedded point? ✅ 2024-05-18
	- [ ] A maximal embedded prime.
- [x] When does a curve have no embedded points? ✅ 2024-05-18
	- [ ] Iff Cohen-Macaulay.
- [ ] What is a Cohen-Macaulay scheme?
- [ ] What is a Cohen-Macaulay ring?
- [x] Give a primary decomposition of $I = (xy, y^2)$. ✅ 2024-05-18
	- [ ] $I = (x,y)^2 \intersect (y)$.
- [x] Discuss the subscheme theoretic issue with embedded components. ✅ 2024-05-18
	- [ ] If $p = \sqrt{q}$ for some $q$ in the decomposition of $I$, then $V(p) \embeds V(p')$ for some $p'\in R/I$, but $V(q)$ is not a subscheme of $V(p')$ because of nilpotent fuzziness.
	- [ ] E.g. $I = (xy, y^2) = (x,y)^2 \intersect (y)$, take $p = (x,y) = \sqrt{q_0}$ and $p' = (y)$, then $V(p) \subset V(p')$ but $V(q) = \spec k[x,y]/(x,y)^2$ is not scheme-theoretically contained in $V(p') = \spec k[x,y]/y$ since $y\not\in (x,y)^2$.
- [ ] When does a subscheme $Z\leq Y$ generally have an embedded component?
	- [ ] $\exists U\subseteq Y$ open dense with $Z \intersect U$ dense in $Z$ but $\Cl_Z(Z\intersect U) \neq Z$ scheme theoretically.
- [x] What does it mean for $p\in Y$ to support an embedded point of a closed subscheme $Z$? ✅ 2024-05-18
	- [ ] $Y$ irreducible, $\Cl_Z( Z \intersect (Y\sm p)) \neq Z$.
	- [ ] E.g. $Y = \AA^2$ and $Z = (xy, y^2)$ at $p=0$.

## 3.2 Sheaves and Supports

### 3.2.1 $\Coh(X)$ and projective morphisms

- [x] What is a free sheaf? ✅ 2024-05-19
	- [ ] $\mcf \cong \OO_X^{\oplus I}$ for some (possibly infinite) indexing set $I$.
- [x] What is the rank of a sheaf? ✅ 2024-05-19
	- [ ] A free sheaf with $\mcf \cong \OO_X^{\oplus r}$ and $r < \infty$.
- [x] What is a locally free sheaf? ✅ 2024-05-19
	- [ ] $\exists \mcu\covers X$ with $\ro\mcf{U_i}$ free of constant rank $r$ for all $i$.
- [x] What is a quasicoherent sheaf? ✅ 2024-05-19
	- [ ] Finitely presented: every $x\in X$ admits a neighborhood $U$ such that there is an exact sequence $\ro{\OO_X^{\oplus I}}U \to \ro{\OO_X^{\oplus J}} U \surjects \ro \mcf U$.
- [x] What is a coherent sheaf? ✅ 2024-05-19
	- [ ] Quasiocherent, finitely generated, and for any open, any $n$, and any $s: \ro{\OO_X^{\oplus n}}U \to \ro \mcf U$, the kernel is finitely generated.
- [x] What is a finitely generated sheaf? ✅ 2024-05-19
	- [ ] Every point $x\in X$ admits $U$ and a surjective morphism $\ro{\OO_X\sumpower n}U\surjects \ro\mcf U$ for some $n$.
- [x] When is $\OO_X$ coherent? ✅ 2024-05-19
	- [ ] Always quasicoherent, and coherent when $X$ is locally Noetherian.
- [x] When is $\mcf\in \QCoh(X)$ on a locally Noetherian scheme coherent? ✅ 2024-05-19
	- [ ] Iff finitely generated
	- [ ] iff $F(U) \in \mods{\OO_X(U)}^\fg$ for all $U$ affine open.
- [x] What is the relative cotangent sheaf? ✅ 2024-05-19
	- [ ] For $f:X\to S$, factor $\Delta_f: X\to X\fiberpower{S}{2}$ (a locally closed immersion) into $X\injects U\injects X\fiberpower{S}{2}$ with $\iota: X\injects U$ a closed immersion into an open subscheme.
	- [ ] Let $\mcj \leq \OO_U$ be the ideal sheaf and $\iota:
	- [ ] Set $\Omega_f = \iota^* (\mcj/\mcj^2) = \iota^* \mcc_{X/U}$.
- [x] When is $\Omega_f$ coherent? ✅ 2024-05-19
	- [ ] When $f$ is of finite type and $f:X\to S$ with $S$ Noetherian.
- [ ] Describe $\sheafhom(\mcf, \mcg)$.
- [x] Describe the correspondence between locally free sheaves and algebraic vector bundles. ✅ 2024-05-19
	- [ ] Send $\mcf$ to $\mathbb{V}(\mcf) \da \Spec_{\OO_X} \Sym \mcf\dual$.
	- [ ] $H^0(\VV(\mcf)) \mapstofrom \Hom_{\OO_X}(\OO_X, \mcf)$, so $\mcf$ is the sheaf of sections of $\VV(\mcf)$.
- [ ] What is an algebraic vector bundle?
- [x] What is the sheaf-theoretic fiber? ✅ 2024-05-19
	- [ ] $F(x) = F_x \tensor_{\OO_{X, x}} k(x)$.
- [x] Define the tangent sheaf. ✅ 2024-05-19
	- [ ] $\mct_f = \Hom_{\OO_X}(\Omega_f, \OO_X)$.
- [ ] Describe the $\Sym \mcf$ construction.
- [x] What is a projective morphism of schemes? ✅ 2024-05-19
	- [ ] $f:X\to S$ which factors as $X\injectsvia{\iota} \Proj \Sym \mcf \to S$ where $\iota$ is a closed immersion for some $\mcf\in \QCoh(S)$.
- [x] What is a quasiprojective morphism of schemes? ✅ 2024-05-19
	- [ ] $f:X\to S$ which factors as $X\injects Y \to S$ where $X\injects Y$ is an open immersion and $Y\to S$ is projective.
- [x] What is an ample sheaf? ✅ 2024-05-19
	- [ ] Invertible sheaf $\mcl$ on quasicompact $X$ where every point $x\in X$ has an open neighborhood fo the form $X_s \da \ts{x\in X\st s_x \neq 0}$ with $s\in H^0(\mcl\tensorpower{}{n})$ for some $n$.
- [x] What is an $f\dash$ample sheaf for $f:X\to S$? ✅ 2024-05-19
	- [ ] $f: X\to S$ with $\mcl/X$ invertible where $f$ is quasicompact and $\ro\mcl{f\inv(U)}$ is ample for every affine open $U\subset S$.
- [x] What is an $f\dash$very ample sheaf for $f:X\to S$? ✅ 2024-05-19
	- [ ] $f:X\to S$ with $\mcl/X$ and $\mcf\in \mods{\OO_S}$ and a locally closed immersion $\iota: X\injects \PP(\mcf) \da \Proj \Sym \mcf$ over $S$ suh that $\mcl \cong \iota^* \OO_{\PP(F)}(1)$, the universal quotient bundle on $\PP(\mcf)$.
- [x] How can one check if a morphism $f:X\to S$ is projective? ✅ 2024-05-19
	- [ ] Proper and $X$ admits an $f\dash$very ample sheaf.

### 3.2.2 Torsionfree, pure, reflexive, flat

- [x] What is a torsionfree sheaf? ✅ 2024-05-19
	- [ ] Stalks $F_x$ are torsionfree as $\OO_{X, x}\dash$modules (multiplication is injective)
	- [ ] Equivalently works on opens.
- [x] What is the torsion subsheaf? ✅ 2024-05-19
	- [ ] $\ker(\mcf \mapsto \mcf\dual\dual)$.
- [x] What are the sections of the torsion subsheaf of $\mcf$? ✅ 2024-05-19
	- [ ] Elements $s\in \mcf(U)$ such that there exists a nonzero $a\in \OO_X(U)$ with $as = 0 \in \mcf(U)$.
- [x] What is the singularity set of $\mcf$? ✅ 2024-05-19
	- [ ] $\Sing(\mcf) \da \ts{x\in X\st \mcf_x \text{ is not free}} \subset X$.
- [x] When is a stalk free? ✅ 2024-05-19
	- [ ] Iff $F_x$ has homological dimension zero, i.e. $\tau_{\geq 1}\Ext_{\OO_{X, x}}(F, N)$ for all fg $\OO_{X, x}\dash$modules.
- [x] When is a sheaf locally free on a curve? On a surface? ✅ 2024-05-19
	- [ ] Torsionfree on a smooth curve
	- [ ] Reflexive on a smooth surface
- [x] What is the determinant of a sheaf? ✅ 2024-05-19
	- [ ] $\det \mcf = \qty{\Lambda^{\rank \mcf} \mcf}\dual\dual$.
- [x] What is the ideal sheaf? ✅ 2024-05-19
	- [ ] For $\iota: Z\injects X$ a closed subscheme, $\mci_Z \da \ker(\OO_X \to \iota_* \OO_Z)$.
- [x] Describe the relationship between Cartier divisors and line bundles. ✅ 2024-05-19
	- [ ] Effective Cartier divisors biject with iso classes of $(\mcl, s)$ with $s$ a regular section.
	- [ ] Map $D\mapsto (\OO_X(D), s_D)$ and $s_D$ is the image of 1 under $\OO_X\to \OO_X(D)$.
	- [ ] Map $(\mcl, s)$ to $Z(s)$.
- [x] What is $\OO_X(D)$? ✅ 2024-05-19
	- [ ] $\mci_D\dual$.
- [x] What is a regular section? ✅ 2024-05-19
	- [ ] $s: \OO_X \to \mcl$ which is injective.
- [x] What is the scheme-theoretic image of $f:X\to Y$? ✅ 2024-05-19
	- [ ] Smalled closed subscheme of $Y$ through which $f$ factors
	- [ ] Equivalently, $\mci_{\im f} \da \ker(\OO_Y \to f_* \OO_X)$.
- [x] What is the scheme theoretic preimage? ✅ 2024-05-19
	- [ ] For $Z\subset Y$ defined by $\mci_Z \leq \OO_Y$, set $f\inv(Z) = Z\fiberprod{Y} X$.
	- [ ] Defined by $\mci_{f\inv Z} = f\inv \mci_Z \cdot \OO_X \leq \OO_X$.
- [x] For $f: X\to S$, what is an $S\dash$flat sheaf? ✅ 2024-05-19
	- [ ] $\forall x\in X$ with image $s$, $F_x$ is flat over $\OO_{S, s}$ via $f_x^\sharp: \OO_{S, s}\to \OO_{X, x}$.
- [x] What is a flat sheaf? ✅ 2024-05-19
	- [ ] Flat over the identity.
- [x] What is a flat morphism in terms of sheaves? ✅ 2024-05-19
	- [ ] $\OO_X$ is $S\dash$flat iff $X\to S$ is a flat morphism.
- [x] What is generic flatness? ✅ 2024-05-20
	- [ ] If $f:X\to S$ finite type with $S$ reduced and $F\in \Coh(X)$ then $\exists U\subseteq S$ with $f\inv U \to U$ flat and $\ro{F}{f\inv U}$ flat over $U$.
- [ ] What is the support of a sheaf?
- [x] What is the dimension of a sheaf? ✅ 2024-05-20
	- [ ] $\dim F \da \dim \mathrm{Supp} F$.


### 3.2.3 Fitting ideals


### 3.2.4 Derived categories

- [ ] What does it mean for a complex to have (quasi)coherent cohomology?
- [x] What is a perfect complex? ✅ 2024-05-20
	- [ ] Locally quasi-isomorphic to a bounded complex of locally free $\oxmods$s of finite type.

### 3.2.5 Dualising complexes, Cohen-Macaulay and Gorenstein schemes

- [ ] What is Grothendieck duality?
- [ ] Describe $f^!$.
- [ ] What is the relative dualising **complex** of $f: X\to Y$?
- [x] What is a Cohen-Macaulay scheme? ✅ 2024-05-20
	- [ ] $\cocomplex\omega_X$ has one nonvanishing cohomology sheaf.
- [x] What is a Gorenstein scheme? ✅ 2024-05-20
	- [ ] Cohen-Macaulay and the nonvanishing sheaf is a line bundle.
- [x] What is a Cohen-Macaulay/Gorenstein morphism? ✅ 2024-05-20
	- [ ] $f: Y\to S$ with locally notherian fibers which is flat, where the fibers are Cohen-Macaulay (resp. Gorenstein) schemes.
- [x] What is the relative dualising **sheaf** of $f: Y\to S$? ✅ 2024-05-20
	- [ ] For $f$ Cohen-Macaulay of relative dimension $d$, so $d\dash$dimensional fibers, $\omega_f \da h^{-d}(f^! \OO_S) = h^{-d}(\cocomplex \omega_f)$.
- [x] When is the relative dualising sheaf a line bundle? ✅ 2024-05-20
	- [ ] $f:Y\to S$ smooth and proper with $S$ Noetherian, in which case $\omega_f = \Omega_f^d$.


## 3.5 Representable functors

- [x] What is a fully faithful functor? ✅ 2024-05-20
	- [ ] For $f:\cat C\to \cat D$, there are bijections $\cat C(x, y) \to \cat D(fx, fy) \, \forall x,y$.
- [x] What is an essentially surjective functor? ✅ 2024-05-20
	- [ ] Every $\forall y\in \cat{D}, \exists x\in \cat C$ such that $fx \cong y$.
- [x] What is the essential image of a functor? ✅ 2024-05-20
	- [ ] All objects $y\in \cat D$ *isomorphic* to an object of the form $fx$ for $x\in \cat C$.
- [x] What is a universal object for a functor? Why care? ✅ 2024-05-20
	- [ ] For $f: \opcat{\cat C} \to \Set$, a pair $(x, \xi)$ where $\xi\in fx$ such that for every pair $(u, \sigma\in fu)$, $\exists ! \alpha: u\to x$ such that $f\alpha: fx\to fu$ sends $\xi \mapsto \sigma$.
	- [ ] $f$ is representable $\iff$ it admits a universal object?

## 3.6 Representability and GIT

### 3.6.1 Fine moduli spaces and automorphisms

- [x] What is a fine moduli space? ✅ 2024-05-20
	- [ ] $f: \opcat\Sch \to \Set$ where $\B f\in \Sch$ represents $f$; call $\B f \to S$ the fine moduli space.
	- [ ] Unique, carries a universal object $\xi \in f(\B f\to S)$.
	- [ ] Yields an natural transformation $f \cong \Sch(\wait, \B f)$.

- [x] What is an isotrivial family? ✅ 2024-05-20
	- [ ] Becomes a trivial family after an etale base change.
- [x] Explain why automorphisms don't preclude existence of a universal family. ✅ 2024-05-20
	- [ ] Take $f: \opcat\Set\to \Set$ isomorphism classes of finite sets; it is representable by a set.
	- [ ] Prevents representability when one can construct an isotrivial family which is not globally trivial.
- [x] Why can't the functor of smooth curves of genus $g$ be representable? ✅ 2024-05-20
	- [ ] Take an isotrivial family $\mcx\to U$ to get $U\to \Mg$ with general fibre $C$; this has to be constant by continuity, but this also happens for the trivial family $C\times U\to U$.

### 3.6.2 More moduli spaces

- [x] What is a moduli space? ✅ 2024-05-20
	- [ ] For $\mcm: \opcat\Sch\to \Set$, a pair $(M, \eta: \mcm\to h_M )$ which is universal: for any other such pair $(M', \eta')$ there is a morphism $f: M\to M'$ with $h_f: h_M\to h_{M'}$ and $h_f \circ \eta = \eta'$.
	- [ ] Implies $M$ corepresents $\mcm$.
- [ ] What is a coarse moduli space?
	- [ ] A pair $(M, \eta: \mcm \to h_M)$ where $\eta$ is only a bijection of sets.
	- [ ] Still implies $M$ corepresents $\mcm$.


### 3.6.3 Quotients

- [ ] What is a $G\dash$scheme?
	- [ ] A pair $(X, \sigma: G\times X\to X)$ plus conditions. 
- [ ] What is a $G\dash$equivariant morphism between $G\dash$schemes?
- [ ] What is a categorical quotient?
- [ ] What is the functor of orbits?
	- [ ] For a $G\dash$scheme $(X, \sigma)$, a  functor $\opcat\Sch\to \Set$ where $S\mapsto X(S)/ G(S)$.
- [ ] How is representability related to categorical quotients?
	- [ ] A $G\dash$scheme $(X, \sigma: G\times X\to X)$ admits a categorical quotient iff the functor of orbits is corepresentable.
- [ ] What is a reductive algebraic group?
- [ ] What is a good quotient?
- [x] What is a geometric quotient? ✅ 2024-05-20
	- [ ] A good quotient whose geometric fibers are orbits of geometric points of $X$.
- [ ] What is the semistable locus?
- [ ] What is the stable locus?


End of the chapter, oh my god.