Morphism of schemes

• Let R be a field or ${\displaystyle \mathbb {Z} .}$ For each R-algebra A, to specify an element of A, say f in A, is to give a R-algebra homomorphism ${\displaystyle R[t]\to A}$ such that ${\displaystyle t\mapsto f}$. Thus, ${\displaystyle A=\operatorname {Hom} _{R-{\text{alg}}}(R[t],A)}$. If X is a scheme over S = Spec R, then taking ${\displaystyle A=\Gamma (X,{\mathcal {O}}_{X})}$ and using the fact Spec is a right adjoint to the global section functor, we get
  $${\displaystyle \Gamma (X,{\mathcal {O}}_{X})=\operatorname {Mor} _{S}(X,\mathbb {A} _{S}^{1})}$$

  
  where ${\displaystyle \mathbb {A} _{S}^{1}=\operatorname {Spec} (R[t])}$. Note the equality is that of rings.
• Similarly, for any S-scheme X, there is the identification of the multiplicative groups:
  $${\displaystyle \Gamma (X,{\mathcal {O}}_{X}^{*})=\operatorname {Mor} _{S}(X,\mathbb {G} _{m})}$$

  
  where ${\displaystyle \mathbb {G} _{m}=\operatorname {Spec} (R[t,t^{-1}])}$ is the multiplicative group scheme.
• Many examples of morphisms come from families parameterized by some base space. For example,
  $${\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y][a,b,c]}{(ax^{2}+bxy+cy^{2})}}\right)\to {\text{Proj}}(\mathbb {C} [a,b,c])=\mathbb {P} _{a,b,c}^{2}}$$

  
  is a projective morphism of projective varieties where the base space parameterizes quadrics in ${\displaystyle \mathbb {P} ^{1}}$.

Given a morphism of schemes ${\displaystyle f:X\to Y}$ over a scheme S, the morphism ${\displaystyle X\to X\times _{S}Y}$ induced by the identity ${\displaystyle 1_{X}:X\to X}$ and f is called the graph morphism of f. The graph morphism of the identity is called the diagonal morphism.

Morphisms of finite type are one of the basic tools for constructing families of varieties. A morphism ${\displaystyle f:X\to S}$ is of finite type if there exists a cover ${\displaystyle \operatorname {Spec} (A_{i})\to S}$ such that the fibers ${\displaystyle X\times _{S}\operatorname {Spec} (A_{i})}$ can be covered by finitely many affine schemes ${\displaystyle \operatorname {Spec} (B_{ij})}$ making the induced ring morphisms ${\displaystyle A_{i}\to B_{ij}}$ into finite-type morphisms. A typical example of a finite-type morphism is a family of schemes. For example,

${\displaystyle \operatorname {Spec} \left({\frac {\mathbb {Z} [x,y,z]}{(x^{n}+zy^{n},z^{5}-1)}}\right)\to \operatorname {Spec} \left({\frac {\mathbb {Z} [z]}{(z^{5}-1)}}\right)}$

is a morphism of finite type. A simple non-example of a morphism of finite-type is ${\displaystyle \operatorname {Spec} (k[x_{1},x_{2},x_{3},\ldots ]))\to \operatorname {Spec} (k)}$ where ${\displaystyle k}$ is a field. Another is an infinite disjoint union

${\displaystyle \coprod ^{\infty }X\to X}$

A morphism of schemes ${\displaystyle i:Z\to X}$ is a closed immersion if the following conditions hold:

1. ${\displaystyle i}$ defines a homeomorphism of ${\displaystyle Z}$ onto its image
2. ${\displaystyle i^{\#}:{\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}}$ is surjective

This condition is equivalent to the following: given an affine open ${\displaystyle \operatorname {Spec} (R)=U\subseteq X}$ there exists an ideal ${\displaystyle I\subseteq R}$ such that ${\displaystyle i^{-1}(U)=\operatorname {Spec} (R/I)}$

Of course, any (graded) quotient ${\displaystyle R/I}$ defines a subscheme of ${\displaystyle \operatorname {Spec} (R)}$ (${\displaystyle \operatorname {Proj} (R)}$). Consider the quasi-affine scheme ${\displaystyle \mathbb {A} ^{2}-\{0\}}$ and the subset of the ${\displaystyle x}$-axis contained in ${\displaystyle X}$. Then if we take the open subset ${\displaystyle \operatorname {Spec} (k[x,y,y^{-1}])}$ the ideal sheaf is ${\displaystyle (x)}$ while on the affine open ${\displaystyle \operatorname {Spec} (k[x,y,x^{-1}])}$ there is no ideal since the subset does not intersect this chart.

Separated morphisms define families of schemes which are "Hausdorff". For example, given a separated morphism ${\displaystyle X\to S}$ in ${\displaystyle {\text{Sch}}/\mathbb {C} }$ the associated analytic spaces ${\displaystyle X(\mathbb {C} )^{an}\to S(\mathbb {C} )^{an}}$ are both Hausdorff. We say a morphism of scheme ${\displaystyle f:X\to S}$ is separated if the diagonal morphism ${\displaystyle \Delta _{X/S}:X\to X\times _{S}X}$ is a closed immersion. In topology, an analogous condition for a space ${\displaystyle X}$ to be Hausdorff is if the diagonal set

${\displaystyle \Delta =\{(x,x)\in X\times X\}}$

is a closed subset of ${\displaystyle X\times X}$. Nevertheless, most schemes are not Hausdorff as topological spaces, as the Zariski topology is in general highly non-Hausdorff.

Most morphisms encountered in scheme theory will be separated. For example, consider the affine scheme

${\displaystyle X=\operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{(f)}}\right)}$

over ${\displaystyle \operatorname {Spec} (\mathbb {C} ).}$ Since the product scheme is

${\displaystyle X\times _{\mathbb {C} }X=\operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{(f)}}\otimes _{\mathbb {C} }{\frac {\mathbb {C} [x,y]}{(f)}}\right)}$

the ideal defining the diagonal is generated by

${\displaystyle x\otimes 1-1\otimes x,y\otimes 1-1\otimes y}$

showing the diagonal scheme is affine and closed. This same computation can be used to show that projective schemes are separated as well.

The only time care must be taken is when you are gluing together a family of schemes. For example, if we take the diagram of inclusions

${\displaystyle {\begin{matrix}\operatorname {Spec} (R[x,x^{-1}])&&\\&\searrow &\\&&\operatorname {Spec} (R[x])\\&\nearrow &\\\operatorname {Spec} (R[x,x^{-1}])&&\end{matrix}}}$

then we get the scheme-theoretic analogue of the classical line with two-origins.

A morphism ${\displaystyle f:X\to S}$ is called proper if

1. it is separated
2. of finite-type
3. universally closed

The last condition means that given a morphism ${\displaystyle S'\to S}$ the base change morphism ${\displaystyle S'\times _{S}X}$ is a closed immersion. Most known examples of proper morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry.

Projective morphisms define families of projective varieties over a fixed base scheme. Note that there are two definitions: Hartshornes which states that a morphism ${\displaystyle f:X\to S}$ is called projective if there exists a closed immersion ${\displaystyle X\to \mathbb {P} _{S}^{n}=\mathbb {P} ^{n}\times S}$ and the EGA definition which states that a scheme ${\displaystyle X\in {\text{Sch}}/S}$ is projective if there is a quasi-coherent ${\displaystyle {\mathcal {O}}_{S}}$-module of finite type such that there is a closed immersion ${\displaystyle X\to \mathbb {P} _{S}({\mathcal {E}})}$. The second definition is useful because an exact sequence of ${\displaystyle {\mathcal {O}}_{S}}$ modules can be used to define projective morphisms.

A projective morphism ${\displaystyle f:X\to \{*\}}$ defines a projective scheme. For example,

${\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{n}+y^{n}-z^{n})}}\right)\to \operatorname {Spec} (\mathbb {C} )}$

defines a projective curve of genus ${\displaystyle (n-1)(n-1)/2}$ over ${\displaystyle \mathbb {C} }$.

If we let ${\displaystyle S=\mathbb {A} _{t}^{1}}$ then the projective morphism

${\displaystyle {\underline {\operatorname {Proj} }}_{S}\left({\frac {{\mathcal {O}}_{S}[x_{0},x_{1},x_{2},x_{3},x_{4}]}{\left(x_{0}^{5}+\cdots +x_{4}^{5}-tx_{0}x_{1}x_{2}x_{3}x_{4}\right)}}\right)\to S}$

defines a family of Calabi-Yau manifolds which degenerate.

Another useful class of examples of projective morphisms are Lefschetz Pencils: they are projective morphisms ${\displaystyle \pi :X\to \mathbb {P} _{k}^{1}=\operatorname {Proj} (k[s,t])}$ over some field ${\displaystyle k}$. For example, given smooth hypersurfaces ${\displaystyle X_{1},X_{2}\subseteq \mathbb {P} _{k}^{n}}$ defined by the homogeneous polynomials ${\displaystyle f_{1},f_{2}}$ there is a projective morphism

${\displaystyle {\underline {\operatorname {Proj} }}_{\mathbb {P} ^{1}}\left({\frac {{\mathcal {O}}_{\mathbb {P} ^{1}}[x_{0},\ldots ,x_{n}]}{(sf_{1}+tf_{2})}}\right)\to \mathbb {P} ^{1}}$

giving the pencil.

A nice classical example of a projective scheme is by constructing projective morphisms which factor through rational scrolls. For example, take ${\displaystyle S=\mathbb {P} ^{1}}$ and the vector bundle ${\displaystyle {\mathcal {E}}={\mathcal {O}}_{S}\oplus {\mathcal {O}}_{S}\oplus {\mathcal {O}}_{S}(3)}$. This can be used to construct a ${\displaystyle \mathbb {P} ^{2}}$-bundle ${\displaystyle \mathbb {P} _{S}({\mathcal {E}})}$ over ${\displaystyle S}$. If we want to construct a projective morphism using this sheaf we can take an exact sequence, such as

${\displaystyle {\mathcal {O}}_{S}(-d)\oplus {\mathcal {O}}_{S}(-e)\to {\mathcal {E}}\to {\mathcal {O}}_{X}\to 0}$

which defines the structure sheaf of the projective scheme ${\displaystyle X}$ in ${\displaystyle \mathbb {P} _{S}({\mathcal {E}}).}$

Flat morphisms have an algebraic definition but have a very concrete geometric interpretation: flat families correspond to families of varieties which vary "continuously". For example,

${\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [x,y,t]}{(xy-t))}}\right)\to \operatorname {Spec} (\mathbb {C} [t])}$

is a family of smooth affine quadric curves which degenerate to the normal crossing divisor

${\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{(xy)}}\right)}$

at the origin.

One important property that a flat morphism must satisfy is that the dimensions of the fibers should be the same. A simple non-example of a flat morphism then is a blowup since the fibers are either points or copies of some ${\displaystyle \mathbb {P} ^{n}}$.

Let ${\displaystyle f:X\to S}$ be a morphism of schemes. We say that ${\displaystyle f}$ is flat at a point ${\displaystyle x\in X}$ if the induced morphism ${\displaystyle {\mathcal {O}}_{f(x)}\to {\mathcal {O}}_{x}}$ yields an exact functor ${\displaystyle -\otimes _{{\mathcal {O}}_{f(x)}}{\mathcal {O}}_{x}.}$ Then, ${\displaystyle f}$ is flat if it is flat at every point of ${\displaystyle X}$. It is also faithfully flat if it is a surjective morphism.

Using our geometric intuition it obvious that

${\displaystyle f:\operatorname {Spec} (\mathbb {C} [x,y]/(xy))\to \operatorname {Spec} (\mathbb {C} [x])}$

is not flat since the fiber over ${\displaystyle 0}$ is ${\displaystyle \mathbb {A} ^{1}}$ with the rest of the fibers are just a point. But, we can also check this using the definition with local algebra: Consider the ideal ${\displaystyle {\mathfrak {p}}=(x)\in \operatorname {Spec} (\mathbb {C} [x,y]/(xy)).}$ Since ${\displaystyle f({\mathfrak {p}})=(x)\in \operatorname {Spec} (\mathbb {C} [x])}$ we get a local algebra morphism

${\displaystyle f_{\mathfrak {p}}:\left(\mathbb {C} [x]\right)_{(x)}\to \left(\mathbb {C} [x,y]/(xy)\right)_{(x)}}$

If we tensor

${\displaystyle 0\to \mathbb {C} [x]_{(x)}{\overset {\cdot x}{\longrightarrow }}\mathbb {C} [x]_{(x)}}$

with ${\displaystyle (\mathbb {C} [x,y]/(xy))_{(x)}}$, the map

${\displaystyle (\mathbb {C} [x,y]/(xy)))_{(x)}{\xrightarrow {\cdot x}}(\mathbb {C} [x,y]/(xy))_{(x)}}$

has a non-zero kernel due the vanishing of ${\displaystyle xy}$. This shows that the morphism is not flat.

A morphism ${\displaystyle f:X\to Y}$ of affine schemes is unramified if ${\displaystyle \Omega _{X/Y}=0}$. We can use this for the general case of a morphism of schemes ${\displaystyle f:X\to Y}$. We say that ${\displaystyle f}$ is unramified at ${\displaystyle x\in X}$ if there is an affine open neighborhood ${\displaystyle x\in U}$ and an affine open ${\displaystyle V\subseteq Y}$ such that ${\displaystyle f(U)\subseteq V}$ and ${\displaystyle \Omega _{U/V}=0.}$ Then, the morphism is unramified if it is unramified at every point in ${\displaystyle X}$.

One example of a morphism which is flat and generically unramified, except for at a point, is

${\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [t,x]}{(x^{n}-t)}}\right)\to \operatorname {Spec} (\mathbb {C} [t])}$

We can compute the relative differentials using the sequence

${\displaystyle {\frac {\mathbb {C} [t,x]}{(x^{n}-t)}}\otimes _{\mathbb {C} [t]}\mathbb {C} [t]dt\to \left({\frac {\mathbb {C} [t,x]}{(x^{n}-t)}}dt\oplus {\frac {\mathbb {C} [t,x]}{(x^{n}-t)}}dx\right)/(nx^{n-1}dx-dt)\to \Omega _{X/Y}\to 0}$

showing

${\displaystyle \Omega _{X/Y}\cong \left({\frac {\mathbb {C} [t,x]}{(x^{n}-t)}}dx\right)/(x^{n-1}dx)\neq 0}$

if we take the fiber ${\displaystyle t=0}$, then the morphism is ramified since

${\displaystyle \Omega _{X_{0}/\mathbb {C} }=\left({\frac {\mathbb {C} [x]}{x^{n}}}dx\right)/(x^{n-1}dx)}$

otherwise we have

${\displaystyle \Omega _{X_{\alpha }/\mathbb {C} }=\left({\frac {\mathbb {C} [x]}{(x^{n}-\alpha )}}dx\right)/(x^{n-1}dx)\cong {\frac {\mathbb {C} [x]}{(\alpha )}}dx\cong 0}$

showing that it is unramified everywhere else.

A morphism of schemes ${\displaystyle f:X\to Y}$ is called étale if it is flat and unramfied. These are the algebro-geometric analogue of covering spaces. The two main examples to think of are covering spaces and finite separable field extensions. Examples in the first case can be constructed by looking at branched coverings and restricting to the unramified locus.