What is algebraic geometry

 What is algebraic geometry?

I am just giving a summary of the usual pipeline of algebraic geometry. We start from the scheme theory and then develop the quasi-coherent sheaf theory and then study various type of scheme and their morphism. Then we can study the blows up and pullback etc. We also study some of the projective geometry and projective scheme.

Spec versus Proj constructions:

• (Affine schemes) Given a ring $A$ and an $A$-module $M$:
  We get an affine scheme $X$ = Spec ⁡ A and a quasi-coherent $O$ X-module $M$ ~. For any $p$ ∈ Spec ⁡ A  and any $f$ ∈ A we have $M$ ~ p ≅ M p and $\Gamma$ ( D ( f ) , M ~ ) ≅ M f. In particular $\Gamma$ ( X , M ~ ) ≅ M. Conversely, for any quasi-coherent sheaf $F$ on $X$ = Spec ⁡ A  there is an $A$-module $M$ such that $F$ ≅ M ~. 
  
  
• (Affine morphisms) Given a scheme $Y$, a quasi-coherent $O$ Y-algebra $A$, and a quasi-coherent $A$-module $M$ on $Y$: 
  We get a scheme $X$ = Spec ⁡ A together with an affine morphism $\pi$ : Spec ⁡ A → Y and a quasi-coherent $O$ X-module $M$ ~. Conversely, any quasi-coherent sheaf on $X$ = Spec ⁡ A is of this form. This construction specializes to the previous item if we take $Y$ = Spec ⁡ Z.
  
  
• Given a graded ring $S$ and a graded $S$-module $M$:
  We get a scheme $X$ = Proj ⁡ S and a quasi-coherent $O$ X-module $M$ ~. For any  $f$ ∈ S d with $d$ > 0 there is an open affine $D$ + ( f ) ≅ Spec ⁡ S ( f ) ≅ Spec ⁡ S ( d ) ( f − 1 )in $X$, and we have $\Gamma$ ( D + ( f ) , M ~ ) ≅ M ( f ) ≅ M ( d ) ( f − 1 ) M ( d ). For any $p$ ∈ Proj ⁡ S we have $M$ ~ p ≅ M ( p ). Conversely, if $S$ is generated by finitely many elements in $S$ 1 then any quasi-coherent $O$ X-module $F$ is of this form.
  
  
• Given a scheme $Y$, a quasi-coherent graded $O$ Y-algebra $S$, and a quasi-coherent graded $S$-module $M$ on $Y$: 
  We get a scheme $X$ = Proj ⁡ S together with a morphism $\pi$ : Proj ⁡ S → Y and a quasi-coherent $O$ X-module $M$ ~.  For any  $f$ ∈ Γ ( Y , S d ) with $d$ > 0 there is an open  $X$ f ≅ Spec ⁡ S ( d ) ( f − 1 )in $X$, and we have $\Gamma$ ( X f , M ~ ) ≅ M ( d ) ( f − 1 ) M ( d ). Conversely, if $S$ 1is finite type and generates $S$ then any quasi-coherent $O$ X-module is of this form. This construction specializes to the previous item if we take $Y$ = Spec ⁡ Z.
  
  
• We will use the last two items to define projective schemes and projective morphisms in the next few lectures.

Functoriality:

• If $Y$ is a scheme and $\varphi$ : A ′ → A is a homomorphism of quasi-coherent $O$ Y-algebras we get a unique $Y$-morphism $\Phi$ : Spec ⁡ A → Spec ⁡ A ′ and any morphism $\Phi$ arises this way. If $\varphi$ is surjective then $\Phi$ is a closed immersion. 
  If $Y$ = Spec ⁡ Z, this specializes to the usual correspondence between the set of ring homomorphisms $\varphi$ : A ′ → A and the morphisms of affine schemes $\Phi$ : Spec ⁡ A → Spec ⁡ A ′.

 

• If $Y$ is a scheme and $\varphi$ : S ′ → S is a graded homomorphism of quasi-coherent graded $O$ Y-algebras we get a $Y$-morphism $\Phi$ : U → Proj ⁡ S ′, where $U$ ⊂ Proj ⁡ S is an open subset depending on $\varphi$. Unlike the previous item, different $\varphi$ ′ s may give the same $\Phi$, and not all $\Phi$ arise this way. If $\varphi$ is TN-surjective (i.e. it is surjective after certain degrees) then $U$ = Proj ⁡ S and $\Phi$ is a closed immersion. 
  If $Y$ = Spec ⁡ Z, this specializes to the fact that to a given graded homomorphism $\varphi$ : S ′ → S of graded rings is associated an affine morphism  $\Phi$ : ⋃ f ′ ∈ S d ′ , d > 0 D + ( ϕ ( f ′ ) ) → Proj ⁡ S ′, such that the restriction $\Phi$ | D + ( ϕ ( f ′ ) ) : D + ( ϕ ( f ′ ) ) ≅ Spec ⁡ S ( ϕ ( f ′ ) ) → Spec ⁡ S ( f ′ ) ′ ≅ D + ( f ′ )
  corresponds to the homomorphism $\varphi$ ( f ′ ) : S ( f ′ ) ′ → S ( ϕ ( f ′ ) ) induced by $\varphi$.

Closed subschemes:

Let $Y$ be a scheme and $A$ (resp. $S$) be a quasi-coherent $O$ Y-algebra (resp. quasi-coherent graded $O$ Y-algebra).

• For any closed subscheme $Z$ ⊂ Spec ⁡ A there is a unique quasi-coherent ideal $I$ < A such that $Z$ is the image of the closed immersion $Spec$ ⁡ A / I ↪ Spec ⁡ A corresponding to the canonical surjection $A$ → A / I. 

 

• If $S$ 0 = O Y and  $S$ 1 is finite type as $O$ Y-module and generates $S$ then for any closed subscheme $Z$ ⊂ Proj ⁡ S there is a (not necessarily unique) quasi-coherent graded ideal $I$ < S such that $Z$ is the image of the closed immersion $Proj$ ⁡ S / I ↪ Proj ⁡ S associated to the canonical surjection 

Affine versus projective cones: (Different references use slightly different names for these.)

This is an important special case of Spec/Proj constructions. Let $Y$ be a scheme and $E$ be a quasi-coherent $O$ Y-module. Define $V$ ( E ) := Spec ⁡ ( Sym ⁡ E ) → p Y , P ( E ) := Proj ⁡ ( Sym ⁡ E ) → π Y .
Both Spec and Proj constructions commute with base change. In particular, if $Spec$ ⁡ k ( y ) → i Y is the canonical morphism corresponding to a point $y$ ∈ Y then
$V$ ( i ∗ E ) ≅ Spec ⁡ k ( y ) × Y V ( E ) = p − 1 ( y ) , P ( i ∗ E ) ≅ Spec ⁡ k ( y ) × Y P ( E ) = π − 1 ( y )
are the fibers over $y$.

Note that $i$ ∗ E ≅ E y m y E y is the fiber of $E$ over $y$, so in the case that $E$ is of finite type $i$ ∗ E is a finite dimensional vector space over $k$ ( y ), say of dimension $n$,  and so $Sym$ ⁡ i ∗ E ≅ k ( y ) [ x 1 , … , x n ] and hence
$p$ − 1 ( y ) ≅ A k ( y ) n , π − 1 ( y ) ≅ P k ( y ) n − 1 .
For this we can think of $V$ ( E ) (resp. $P$ ( E )) as a fibration over $Y$ fibered by affine spaces (resp. projective spaces).
Note that $n$ may vary by $y$ here, but say if $E$ is locally free of rank $n$ then $n$ is independent of $y$.

Morphisms to affine/projective cones: Let $Y$ be a scheme, $\mathcal{E}$ be a quasi-coherent ${\mathcal{O}}_{}$ Y-module, and $q$ : X → Y be a $Y$-scheme.

• To give a $Y$-morphism $X$ → V ( E ) it is equivalent to give an ${\mathcal{O}}_{}$ X-module homomorphism $q^{}$ ∗ E → O X. In particular (when $q$ is the identity), the $Y$-sections of  $\mathbb{V}$ ( E ) are in bijection with the global sections of ${\mathcal{E}}^{}$ ∨. 
  
  
• To give a $Y$-morphism $X$ → P ( E ) it is equivalent to give a sub-${\mathcal{O}}_{}$ X-module $\mathcal{F}$ ⊂ q ∗ E such that ${\displaystyle \frac{q^{}}{}}$ ∗ E F is invertible. In particular (when $q$ is the identity), the $Y$-sections of  $\mathbb{P}$ ( E ) are in bijection with sub-${\mathcal{O}}_{}$ Y-module $\mathcal{F}$ ⊂ E such that ${\displaystyle \frac{\mathcal{E}}{}}$ F is invertible. 

Segre embedding: $Y$ is any scheme and $E$ , F quasi-coherent $O$ Y-modules. 

• $V$ ( E ) × Y V ( F ) ≅ V ( E ⊗ O Y F ). If $Y$ = Spec ⁡ k and $E$ = E ~ , F = F ~ where $E$ , F are $k$-vector spaces of dimensions $m$ , n then this specializes to $A$ k m × A k n ≅ A k m + n.
• There is a closed immersion $P$ ( E ⊗ O Y F ) ↪ P ( E ) × Y P ( F ) called Segre embedding. If $Y$ = Spec ⁡ k and $E$ = E ~ , F = F ~ where $E$ , F are $k$-vector spaces of dimensions $m$ , n then this specializes to a closed immersion $P$ k m × P k n ↪ P k m n + m + n .

For example, if $m$ = 2 = n, then the Segre embedding $P$ k 1 × P k 1 ↪ P k 3can be described in open affine charts using the calculation of page 113 of the notes. Take $R$ = Sym ⁡ ( E ) = k [ x , y ] , S = Sym ⁡ ( F ) = k [ u , v ] then $T$ = Sym ⁡ ( E ⊗ k F ) = k [ x u , x v , y u , y v ]. If we take $e$ = x , f = v , then $R$ ( x ) = k [ y / x ] , S ( v ) = k [ u / v ] and so $B$ = R ( x ) ⊗ k S ( v ) = k [ y / x ] ⊗ k k [ u / v ] = k [ y / x , u / v ], and $T$ ( e ⊗ f ) = k [ x u , x v , y u , y v ] ( x v ) = k [ x u x v , y u x v , y v x v ]. The restriction of Segre embedding to D_+(x)\times D_+(v) is therefore the closed immersion $A$ k 1 × k A k 1 = Spec ⁡ k [ y / x ] × k Spec ⁡ k [ u / v ] ↪ Spec ⁡ k [ x u x v , y u x v , y v x v ] = A k 3

induced by the surjective $k$-algebra homomorphism $k$ [ x u x v , y u x v , y v x v ] → k [ y x , u v ]

that sends $x$ u x v ↦ u / v , y u x v ↦ y u / x v , y v x v ↦ y / x.

Very ample invertible sheaves: Let $q$ : X → Y be a quasi-compact and separated morphism of schemes and $L$ be an invertible $O$ X-module. We say $L$ is very ample over $Y$ (or with respect to $q$) if one of these two equivalent conditions holds:

1. There is a $Y$-immersion $i$ : X ↪ P ( E ) for some quasi-coherent $O$ Y-module $E$ such that $L$ = i ∗ O P ( E ) ( 1 ).
2. The canonical homomorphism $\sigma$ : q ∗ q ∗ L → L is surjective and the associated morphism $r$ L , σ : X → P ( q ∗ L ) is an immersion.

 

• If $q$ is of finite type then $E$ can be chosen to be of finite type. 

As a special case, if $X$ is a separated finite type scheme over a field $k$ and $L$ is a very ample invertible $O$ X-module (over $k$) there there is an immersion (over $k$) $i$ : X ↪ P k N for some $N$ such that $L$ = i ∗ O P k N ( 1 ). So, in particular $X$ can be regarded as a subscheme of some projective space.

Ample invertible sheaves: Let $X$ be a quasi-compact and separated scheme, and $L$ be an invertible $O$ X-module. Roughly speaking, ampleness of $L$ means that high powers of $L$ have plenty of global sections. We give several equivalent conditions for ampleness. Three of them are as follows:

• $\{$ X f ∣ f ∈ Γ ( X , L n )  for some  n ≥ 0 } form a basis for the topology of $X$.
• If $S$ = ⊕ ≥ 0 Γ ( X , L n ) the identity endomorphism $S$ → S induces a dominant open immersion $X$ ↪ Proj ⁡ S. 
• For quasi-coherent $O$ X-module $F$ of finite type there exists an $n$ 0 such that for any $n$ ≥ n 0 F ( n ) is generated by its global sections. 

Example: $n$ > 0 , X = P A n = Proj ⁡ A [ x 0 , … , x n ] , L = O ( d ).
If $d$ < 0 then $S$ = 0, so item 1 above fails.
If $d$ = 0 then $S$ ≅ A [ t ] and $Proj$ ⁡ S ≅ Spec ⁡ A, so item 2 above fails (also item 3 fails say for $F$ = O ( − 1 )). Therefore, for $d$ ≤ 0 O ( d ) is not ample. 
For $d$ > 0 S = A [ x 0 , … , x n ] ( d ) and the morphism in item 2 is the canonical isomorphism  isomorphism $P$ A n → ≅ Proj ⁡ A [ x 0 , … , x n ] ( d ). Therefore, for $d$ > 0 O ( d ) is ample.

Ample / Very Ample Invertible sheaves: Let $X$ be a quasi-compact separated scheme over a ring $A$, and $L$ be an invertible $O$ X-module. 

• If $L$ is very ample over $A$ then $L$ is ample. 
• If $X$ is of finite type over $A$ and $L$ is ample then there is a $k$ 0 such that $L$ k is very ample over $A$ for all $k$ ≥ k 0.

Example: If $i$ : X ⊂ P A n is a closed subscheme then $i$ ∗ O ( d ) for any $d$ > 0 is very ample over $A$, and hence is an ample $O$ X-module. 

Projective/Proper morphisms:

• A morphism $f$ : X → Y is proper if it is finite type, separated and universally closed.
• A morphism $f$ : X → Y in which $Y$ is quasi-compact and separated is projective if $X$ is $Y$-isomorphic to $Proj$ ⁡ S, where $S$ is a graded quasi-coherent $O$ Y-algebra in which $S$ 1 is finite type and generates $S$. Equivalently, if $X$ is $Y$-isomorphic to a closed subscheme of $P$ ( E ) for some quasi-coherent $O$ Y-module of finite type $E$. Equivalently, if $f$ is quasi-projective and proper.
• For a ring $A$, any closed subscheme of a projective space $P$ A n is projective over $A$, so in particular they are proper over $A$. It is not easy to find examples of schemes which are proper but not projective. See Remark 10.3.6 in Vakil or II.4.10.2 in Hartshorne. We may discuss some of these constructions in MATH 607, when we develop more tools. 

Chow's Lemma (simplified version): Let $X$ be an irreducible separated scheme of finite type. There exists an irreducible quasi-projective scheme $X$ ′ and a projective surjective morphism $f$ : X ′ → X such that for an open $U$ ⊂ X the restriction  $f$ − 1 ( U ) → f U is an isomorphism.

• $X$ proper $\iff$ X ′ projective
• As stated in the notes, a more general statement works over a quasi-compact and separated base scheme and allows $X$ to have finitely many irreducible components.
• Chow's lemma is used to possibly extend a property of projective schemes to proper schemes. An instance of this is Grothendieck’s Coherence Theorem that will be discussed in MATh607.

Finite morphisms: These morphisms always have finite and discrete fibers and provided that the base scheme is quasi-compact and separated they are always projective (=> proper).

Some nice and simple picture of finite morphisms are give on pg 211 of Vakil's book. Here is an example. For a field $k$, let $f$ = a 0 + ⋯ + a n x n ∈ k [ x ] be a nonzero polynomial of degree $n$ ≥ 1. Sending $t$ to $f$ defines a $k$-algebra homomorphism $k$ [ t ] → k [ x ] and hence a morphism $\varphi$ : A k 1 → A k 1. This morphism is finite because $k$ [ x ] is a $k$ [ t ]-module of finite type.  For any $b$ ∈ k, the fiber (as a set) over the closed point $($ t − b ) ∈ A k 1 is $V$ ( a 0 + ⋯ + a n x n − b ), which consists of the roots of the polynomial $a$ 0 + ⋯ + a n x n − b. If $k$ is algebraically closed then, counted with multiplicities, each fiber over a closed point as exactly $n$ points. 

Rational maps

• If $X$ is a scheme with finitely many irreducible components $X$ 1 , … , X n with the generic points $x$ 1 , … , x n, respectively. The the ring of rational functions $R$ ( X ) ≅ ∏ i = 1 n O X i , x i. Each factor is a local ring of dimension 0. In particular, if X is an integral scheme, then $R$ ( X ) ≅ Q ( A ) is the faction field of $A$ for any nonempty open affine $Spec$ ⁡ A ⊂ X.
• If $X$ is reduced and $Y$ is separated any rational map from $X$ to $Y$ can be represented by a morphism from its domain of definition to $Y$. This follows from the fact that if $f$ , g : X → Y are two morphisms that agree on an open dense subset of $X$ then necessarily $f$ = g.

This last claim is not true if one of these two conditions are dropped, for example, if $X$ = A 1 and $Y$ is the affine line with two origins then we can write two open immersions from $X$ into $Y$ each containing one of the origins. These two morphisms are not the same but they agree on the open dense $A$ 1 ∖ 0 ⊂ X. On the other hand, if $X$ = Spec ⁡ Z [ x , y ] / ( x 2 , x y ) and $Y$ = A 1 and $f$ = y , g = x + y ∈ Z [ x , y ] / ( x 2 , x y ) define two different maps to $Y$ but their restrictions to the open dense $D$ ( y ) ≅ Spec ⁡ Z [ y , y − 1 ] ⊂ X are the same as $y$.

Blow up: Let $Y$ be an integral scheme and $I$ < O Y a non-zero finite type quasi-coherent ideal (or more generally a fractional ideal). The blow up of $Y$ relative to $I$ is $\pi$ : X : = Proj ⁡ ⨁ k ≥ 0 I k → Y, which is a birational surjective and projective morphism.  
We will see in MATH 607 that any projective birational morphism to $Y$ must be the structure morphism of a blow up relative to a non-zero finite type quasi-coherent ideal sheaf.

Simple example to have in mind: $Y$ = A k 2 = Spec ⁡ k [ x , y ] , I = ⟨ x , y ⟩ ~. Then $X$ = Proj ⁡ A [ s , t ] ⟨ s x − t y ⟩ where $A$ = k [ x , y ]. One sees that $\pi$ − 1 ( 0 ) = Proj ⁡ k [ s , t ] ≅ P k 1, where $\pi$ | X ∖ π − 1 ( 0 ) is an isomorphism onto $A$ k 2 ∖ 0.

Also,  $X$ is covered by the following open affine$D$ + ( s ) ≅ Spec ⁡ k [ y , t / s ] ≅ A k 2 , D + ( t ) ≅ Spec ⁡ k [ x , s / t ] ≅ A k 2.

Universal property Blow up: Let  be a noetherian scheme and  be a coherent sheaf of ideals corresponding to the closed subscheme . Let  be the blow up of  relative to . Then   is an invertible sheaf and very ample with respect to . It is the ideal corresponding to the closed subscheme . In the example of the blow up of  at the origin in Summary 15    ${}^{}$ + ( s ) ≅ Spec ⁡ k [ y , t / s ] ≅ A k 2 , D + ( t ) ≅ Spec ⁡ k [ x , s / t ] ≅ A k 2.

Conversely, if  is any morphism such that  is invertible then there exists a unique lift of  to .

In the example above, if  is any line passing the origin then  is the ideal of a point (the origin) on  and hence is invertible. So the closed immersion  can be lifted to (a necessarily closed immersion) .