Math 256B. Answer to Hartshorne II Ex. 3.11d

Proposition. Let f: Z → X be a morphism, with Z noetherian. Then there is a unique closed subscheme Y of X with the following property: the morphism f factors through Y , and if Y′ is any other closed subscheme of X thorough which f factors, then Y → X factors through Y′ also. We call Y the scheme-theoretic image of f . If Z is a reduced scheme, then Y is just the reduced induced structure on the closure of the image f(Z) .

Proof. First consider the special case in which X is affine, say X = Spec A. Let 𝔞 be the kernel of the corresponding ring homomorphism A → Γ(Z, 𝒪_{Z}) (Ex. 2.4), and let Y = Spec A/𝔞, viewed as a closed subscheme of X. Then, by Ex. 2.4, f factors (uniquely) through Y . Also, if Y′ is another such closed subscheme, then by part (b) it is of the form Y′ = Spec A/𝔞′, and the map A → Γ(Z, 𝒪_{Z}) must factor through A/𝔞′, so 𝔞′ ⊆ 𝔞. Therefore A → A/𝔞 factors through A/𝔞′, so Y → X factors through Y′ (uniquely).

Now let U = D(a) be a principal open affine subset of X, let Z_{1} = f^{−1}(U) , and let 𝔞_{1} be the kernel of the map A_{a} → Γ(Z_{1}, 𝒪_{Z}) . By the commutativity of the diagram

A  ⟶  A_{a}
↓  ↓
Γ(Z, 𝒪_{Z}) ⟶ Γ(Z_{1}, 𝒪_{Z}) ,

we have 𝔞_{1} ⊇ 𝔞A_{a}.

We claim that the opposite inclusion also holds. Let x ∈ 𝔞_{1} . We may assume that x lies in the image of A, and by abuse of notation shall regard x as an element of A. Then f^{#}(x) ∈ Γ(Z, 𝒪_{Z}) vanishes on Z_{1} , so by Ex. 2.16b, a^{n}x ∈ 𝔞 for some n ∈ ℕ. Thus x ∈ 𝔞A_{a}, so 𝔞A_{a} = 𝔞_{1} . This shows that taking the scheme-theoretic intersection is preserved under passing to a principal open affine. (Note that Ex. 2.16b can be applied since Z was assumed to be noetherian.)

Now let X be an arbitrary scheme. If U is an open affine in X, let Y_{U} be the scheme-theoretic image of f^{−1}(U) → U . If U′ is a principal open affine in U , then we have shown that Y_{U′} = Y_{U} ∩ U′.

We can glue the X-schemes Y_{U} , as follows. Given two open affines, cover their intersection with common principal open affines. This shows that the sheaves of ideals of Y_{U} and Y_{V} coincide on that intersection, hence we get a well-defined closed subscheme Y of X coinciding with Y_{U} over each open affine U .

The fact that f factors (uniquely) through Y follows by glueing the morphisms g_{U}: f^{−1}(U) → Y_{U} for all U , again by the same procedure. This can also be shown by first proving uniqueness on an intersection of open affines and then glueing in the usual way.

If Y′ is another closed subscheme of X through which f factors, then for all open affines U ⊆ X, we get unique maps Y ∩ U → Y′ ∩ U ; by uniqueness this set is compatible with passing to principal open subsets of U , so they glue to give a unique map Y′ → Y through which Y → X factors.

The uniqueness of Y with the above properties follows by the existence and uniqueness of the above map Y → Y′. Indeed, if Y and Y′ both satisfy the above properties, then there exist X-morphisms Y → Y′ and Y′ → Y which must be inverses of each other, by uniqueness of the maps Y → Y and Y′ → Y′ in the above context.

Finally, suppose that Z is reduced. Then, for every open affine U ⊆ X, the affine ring of Y ∩ U will be a subring of the ring Γ(f^{−1}(U), 𝒪_{Z}) , which has no nilpotents, so Y is a reduced scheme. Moreover, by construction, the space of Y is equal to the closure of the image of f (from the first paragraph, on open affines). Therefore Y is the closure of the image f(Z) , with reduced induced subscheme structure.