Let \(\phi: S \to T\) be a surjective homomorphism of graded rings, preserving degrees. Show that the open set \(U\) of Hartshorne II.2.14 is equal to \(\Proj T\), and that the morphism \(f: \Proj T \to \Proj S\) is a closed immersion.
If \(I \subseteq S\) is a homogeneous ideal, take \(T = S/I\) and let \(Y\) be the closed subscheme of \(X = \Proj S\) defined as the image of the closed immersion \(\Proj S/I \to X\). Show that different homogeneous ideals can give rise to the same closed subscheme. For example, let \(d_0\) be an integer and let \(I' = \bigoplus_{d \geq d_0} I_d\). Show that \(I\) and \(I'\) determine the same closed subscheme.
Hartshorne II.5.16 shows that every closed subscheme of \(X\) comes from a homogeneous ideal \(I\) of \(S\), at least when \(S\) is a polynomial ring over \(S_0\).
Solution
1. If\[ \phi:S\twoheadrightarrow T \]is a surjective degree-preserving homomorphism, then\[ \phi(S_+)=T_+. \]
For every \(d>0\), surjectivity as a graded homomorphism means \[ \phi(S_d)=T_d. \] Taking the direct sum over positive degrees gives \[ \phi(S_+)=\bigoplus_{d>0}T_d=T_+. \]
2. The open set\[ U=\{\mathfrak p\in\Proj T:\mathfrak p\not\supseteq\phi(S_+)\} \]from Hartshorne II.2.14 is all of \(\Proj T\).
By 1, \[ \phi(S_+)=T_+. \] By definition, a point of \(\Proj T\) is a homogeneous prime ideal which does not contain \(T_+\). Hence every point belongs to \(U\).
3. For every homogeneous \(s\in S_+\), the induced ring map\[ S_{(s)}\longrightarrow T_{(\phi(s))} \]is surjective.
Localizing the surjection \(S\twoheadrightarrow T\) at the powers of \(s\) gives a surjection of graded rings \[ S_s\twoheadrightarrow T_{\phi(s)}. \] If a degree-zero element of the target is represented by \[ \frac{t}{\phi(s)^n}, \qquad \deg t=n\deg s, \] choose a homogeneous lift \(u\in S_{n\deg s}\) of \(t\). Then \[ \frac{u}{s^n}\in S_{(s)} \] maps to the given element. Thus the degree-zero map is surjective.
4. On the standard affine chart \(D_+(s)\subseteq\Proj S\), the morphism\[ f:\Proj T\longrightarrow\Proj S \]restricts to the closed immersion\[ D_+(\phi(s)) = \Spec T_{(\phi(s))} \longrightarrow \Spec S_{(s)} = D_+(s). \]
3. A surjective ring homomorphism induces a closed immersion of affine spectra.
5. The morphism\[ \boxed{f:\Proj T\longrightarrow\Proj S} \]is a closed immersion.
The standard opens \(D_+(s)\) for homogeneous \(s\in S_+\) cover \(\Proj S\). By 4, the inverse image over every such member is a closed subscheme and the restricted morphism is a closed immersion.
Being a closed immersion is local on the target. Hence \(f\) is a closed immersion globally.
6. Now let \(I\subseteq S\) be homogeneous and put\[ I'=\bigoplus_{d\ge d_0}I_d. \]Then \(I'\subseteq I\) is a homogeneous ideal, and the quotient map induces a graded surjection\[ S/I'\twoheadrightarrow S/I. \]
Because the grading is nonnegative, multiplying an element of \(I_d\) with \(d\ge d_0\) by a homogeneous element of nonnegative degree stays in a component of degree at least \(d_0\). Thus \(I'\) is an ideal, and it is homogeneous by construction.
The inclusion \(I'\subseteq I\) gives the displayed quotient homomorphism.
7. For every \(d\ge d_0\), the degree-\(d\) component map\[ (S/I')_d\longrightarrow(S/I)_d \]is an isomorphism.
For \(d\ge d_0\), \[ I'_d=I_d. \] Therefore \[ (S/I')_d=S_d/I'_d=S_d/I_d=(S/I)_d. \]
8. The morphism\[ \Proj(S/I)\longrightarrow\Proj(S/I') \]induced by 6 is an isomorphism.
By 7, the graded homomorphism \[ S/I'\longrightarrow S/I \] is an isomorphism in every sufficiently large degree. Hartshorne II.2.14(c) states that such a graded map induces an isomorphism on \(\Proj\).
9. The homogeneous ideals \(I\) and \(I'\) define the same closed subscheme of\[ X=\Proj S. \]
8. Both composites are induced by the same quotient map from \(S\), so this isomorphism identifies the two closed immersions over \(X\). Therefore their scheme-theoretic images in \(X\) are the same closed subscheme.
10. Thus homogeneous ideals are not determined by the closed subschemes they define on \(\Proj S\).
Whenever \(I\) has a nonzero component in some degree below \(d_0\), one has \[ I'\ne I, \] while 9 shows that the associated closed subschemes are identical.
11. Q.E.D.
Steps <1>1–<1>5 prove part (a), and steps <1>6–<1>10 prove part (b).