Morphism of schemes

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

More generally, morphisms p:X →S with various schemes X but fixed scheme S form the category of schemes over S (the slice category of the category of schemes with the base object S.) An object in the category is called an S-scheme and a morphism in the category an S-morphism; explicitly, an S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ.

Definition

By definition, a morphism of schemes is just a morphism of locally ringed spaces.

A scheme, by definition, has an open affine chart and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties).^[1] Let ƒ:X→Y be a morphism of schemes. If x is a point of X, since ƒ is continuous, there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊂ V. Then ƒ: U → V is a morphism of affine schemes and thus is induced by some ring homomorphism B → A (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:X→Y is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts.

• Note: It would not be desirable to define a morphism of schemes as a morphism of ringed spaces. One trivial reason is that there is an example of a ringed-space morphism between affine schemes that is not induced by a ring homomorphism (for example,^[2] a morphism of ringed spaces:

  $\operatorname {Spec} k(x)\to \operatorname {Spec} k[y]_{(y)}=\{\eta =(0),s=(y)\}$

that sends the unique point to s and that comes with $k[y]_{(y)}\to k(x),\,y\mapsto x$.) More conceptually, the definition of a morphism of schemes needs to capture "Zariski-local nature" or localization of rings;^[3] this point of view (i.e., a local-ringed space) is essential for a generalization (topos).

Let ƒ:X→Y be a morphism of schemes with $\phi :{\mathcal {O}}_{Y}\to f_{*}{\mathcal {O}}_{X}$. Then, for each point x of X, the homomorphisms on the stalks:

$\phi :{\mathcal {O}}_{Y,f(x)}\to {\mathcal {O}}_{X,x}$

is a local ring homomorphism: i.e., $\phi ({\mathfrak {m}}_{f(x)})\subset {\mathfrak {m}}_{x}$ and so induces an injective homomorphism of residue fields

$\phi :k(f(x))\hookrightarrow k(x)$.

(In fact, φ maps th n-th power of a maximal ideal to the n-th power of the maximal ideal and thus induces the map between the (Zariski) cotangent spaces.)

For each scheme X, there is a natural morphism

$\theta :X\to \operatorname {Spec} \Gamma (X,{\mathcal {O}}_{X}),$

which is an isomorphism if and only if X is affine; θ is obtained by gluing U → target which come from restrictions to open affine subsets U of X. This fact can also be stated as follows: for any scheme X and a ring A, there is a natural bijection:

$\operatorname {Mor} (X,\operatorname {Spec} (A))\cong \operatorname {Hom} (A,\Gamma (X,{\mathcal {O}}_{X})).$

(Proof: The map $\phi \mapsto \operatorname {Spec} (\phi )\circ \theta$ from the right to the left is the required bijection. In short, θ is an adjunction.)

Moreover, this fact (adjoint relation) can be used to characterize an affine scheme: a scheme X is affine if and only if for each scheme S, the natural map

$\operatorname {Mor} (S,X)\to \operatorname {Hom} (\Gamma (X,{\mathcal {O}}_{X}),\Gamma (S,{\mathcal {O}}_{S}))$

is bijective.^[4] (Proof: if the maps are bijective, then $\operatorname {Mor} (-,X)\simeq \operatorname {Mor} (-,\operatorname {Spec} \Gamma (X,{\mathcal {O}}_{X}))$ and X is isomorphic to $\operatorname {Spec} \Gamma (X,{\mathcal {O}}_{X})$ by Yoneda's lemma; the converse is clear.)

Affine case

Let $\phi :B\to A$ be a ring homomorphism and let $\phi ^{a}:\operatorname {Spec} A\to \operatorname {Spec} B,\,{\mathfrak {p}}\mapsto \phi ^{-1}({\mathfrak {p}})$ be the induced map.

• φ^a is continuous.^[5]
• If φ is surjective, then φ^a is a homeomorphism onto its image.^[6]
• For every ideal I of A, ${\overline {\phi ^{a}(V(I))}}=V(\phi ^{-1}(I)).$^[7]
• φ^a has dense image if and only if the kernel of φ consists of nilpotent elements. (Proof: the preceding formula with I = 0.) In particular, when B is reduced, φ^a has dense image if and only if φ is injective.

Let ƒ: Spec A → Spec B be a morphism of schemes between affine schemes with the pullback map φ: B → A. That it is a morphism of locally ringed spaces translates to the following statement: if $x={\mathfrak {p}}_{x}$ is a point of Spec A,

${\mathfrak {p}}_{f(x)}=\phi ^{-1}({\mathfrak {p}}_{x})$.

(Proof: In general, ${\mathfrak {p}}_{x}$ consists of g in A that has zero image in the residue field k(x); that is, it has the image in the maximal ideal ${\mathfrak {m}}_{x}$. Thus, working in the local rings, $g(f(x))=0\Rightarrow \phi (g)\in \phi ({\mathfrak {m}}_{f(x))})\subset {\mathfrak {m}}_{x}\Rightarrow g\in \phi ^{-1}({\mathfrak {m}}_{x})$. If $g(f(x))\neq 0$, then g is a unit element and so φ(g) is a unit element.)

Hence, each ring homomorphism B → A defines a morphism of schemes Spec A → Spec B and, conversely, all morphisms between them arise this fashion.

Morphisms as points

By definition, if X, S are schemes (over some base scheme or ring B), then a morphism from S to X (over B) is an S-point of X and one writes:

$X(S)=\{S\to X{\text{ over }}B\}$

for the set of all S-points. This notion generalizes the notion of solutions to a system of polynomial equations in classical algebraic geometry. Indeed, let X = Spec(A) with $A=B[t_{1},\dots ,t_{n}]/(f_{1},\dots ,f_{m})$. For a B-algebra R, to give an R-point of X is to give an algebra homomorphism A →R, which in turn amounts to giving a homomorphism

$B[t_{1},\dots ,t_{n}]\to R,\,t_{i}\mapsto r_{i}$

that kills f_i's. Thus, there is a natural identification:

${\displaystyle X(\operatorname {Spec} R)=\{(r_{1},\dots ,r_{n})\in R^{n}|f_{1}(r_{1},\dots ,r_{n})=\cdots =f_{m}(r_{1},\dots ,r_{n})=0\}.}$

Example: If X is an S-scheme with structure map π: X → S, then an S-point of X (over S) is the same thing as a section of π.

In the category theory, Yoneda's lemma says that, given a category C, the contravariant functor

$C\to {\mathcal {P}}(C)=\operatorname {Fct} (C^{\text{op}},\mathbf {Sets} ),\,X\mapsto \operatorname {Mor} (-,X)$

is fully faithful (where ${\mathcal {P}}(C)$ means the category of presheaves on C). Applying the lemma to C = the category of schemes over B, this says that a scheme over B is determined by its various points.

It turns out that in fact it is enough to consider S-points with only affine schemes S, precisely because schemes and morphisms between them are obtained by gluing affine schemes and morphisms between them. Because of this, one usually writes X(R) = X(Spec R) and view X as a functor from the category of commutative B-algebras to Sets.

Example: Given S-schemes X, Y with structure maps p, q,

$(X\times _{S}Y)(R)=X(R)\times _{S(R)}Y(R)=\{(x,y)\in X(R)\times Y(R)|p(x)=q(y)\}$.

Example: With B still denoting a ring or scheme, for each B-scheme X, there is a natural bijection

$\mathbf {P} _{B}^{n}(X)=$ { the isomorphism classes of line bundles L on X together with n + 1 global sections generating L. };

in fact, the sections s_i of L define a morphism $X\to \mathbf {P} _{B}^{n},\,x\mapsto (s_{0}(x):\dots :s_{n}(x))$. (See also Proj construction#Global Proj.)

Remark: The above point of view (which goes under the name functor of points and is due to Grothendieck) has had a significant impact on the foundations of algebraic geometry. For example, working with a category-valued (pseudo-)fuctor instead of a set-valued functor leads to the notion of a stack, which allows one to keep track of morphisms between points.

Rational map

A rational map of schemes is defined in the same way for varieties. Thus, a rational map from a reduced scheme X to a separated scheme Y is an equivalence class of a pair $(U,f_{U})$ consisting of an open dense subset U of X and a morphism $f_{U}:U\to Y$. If X is irreducible, a rational function on X is, by definition, a rational map from X to the affine line A¹ or the projective line P¹.

A rational map is dominant if and only if it sends the generic point to the generic point.^[8]

A ring homomorphism between function fields need not induce a dominant rational map (even just a rational map).^[9] For example, Spec k[x] and Spec k(x) and have the same function field (namely, k(x)) but there is no rational map from the former to the latter. However, it is true that any inclusion of function fields of algebraic varieties induces a dominant rational map (see morphism of algebraic varieties#Properties.)

Types of morphisms

For now, see glossary of algebraic geometry.

See also

• closed immersion
• regular embedding
• Constructible set (topology)

Notes

1. Vakil, Exercise 6.3.C.
2. Vakil, Exercise 6.2.E.
3. http://www.math.harvard.edu/~lurie/papers/DAG-V.pdf, § 1.
4. EGA I, Ch. I, Corollarie 1.6.4.
5. Proof: ${\phi ^{a}}^{-1}(D(f))=D(\phi (f))$ for all f in A.
6. EGA I, Ch. I, Corollaire 1.2.4.
7. EGA I, Ch. I, 1.2.2.3.
8. Vakil, Exercise 6.5.A
9. Vakil, A paragraph after Exercise 6.5.B

References

• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Milne, Review of Algebraic Geometry at Algebraic Groups: The theory of group schemes of finite type over a field.
• Vakil, Foundations of Algebraic Geometry