Let \(R\) be a discrete valuation ring containing its residue field \(k\). Let \(X = \Spec R[t]\) be the affine line over \(\Spec R\). Show that statements (a), (d), and (e) of Hartshorne II.3.20 are false for \(X\).
Solution
Let \[ \pi\in R \] be a uniformizer and let \[ K=\operatorname{Frac}(R). \]
1. The scheme\[ X=\Spec R[t] \]has dimension \(2\).
The DVR \(R\) is a noetherian ring of dimension \(1\). For a noetherian ring, \[ \dim R[t]=\dim R+1=2. \] Concretely, the chain \[ (0)\subsetneq(\pi)\subsetneq(\pi,t) \] already shows that the dimension is at least \(2\).
2. The ideal\[ \mathfrak m=(\pi t-1)\subseteq R[t] \]is maximal.
In the quotient, the relation \[ \pi t=1 \] makes \(\pi\) invertible. Hence \[ R[t]/(\pi t-1) \cong R[\pi^{-1}] =K, \] which is a field. Therefore \(\mathfrak m\) is maximal and the corresponding point \[ P\in X \] is closed.
3. The maximal ideal \(\mathfrak m=(\pi t-1)\) has height \(1\).
The ring \(R[t]\) is a noetherian domain and \(\pi t-1\) is a nonzero nonunit. Since the principal ideal it generates is prime by 2, Krull’s principal ideal theorem gives \[ \operatorname{ht}(\mathfrak m)\le1. \] Because \[ (0)\subsetneq\mathfrak m, \] its height is at least \(1\). Hence \[ \operatorname{ht}(\mathfrak m)=1. \]
4. Hartshorne II.3.20(a) fails for \(X\).
1. Thus the equality asserted in II.3.20(a) is false here.
5. Hartshorne II.3.20(d) also fails for \(X\).
Take the closed subset \[ Y=\{P\}=V(\mathfrak m). \] Since \(Y\) is a single point, \[ \dim Y=0. \] Its codimension in the integral affine scheme \(X\) is the height of \(\mathfrak m\): \[ \codim(Y,X)=1. \] Thus \[ \dim Y+\codim(Y,X)=0+1=1\ne2=\dim X. \]
6. The nonempty open subset\[ D(\pi)\subseteq X \]has dimension \(1\).
Localizing at \(\pi\) gives \[ D(\pi) \cong \Spec R[t]_\pi \cong \Spec K[t]. \] The polynomial ring in one variable over the field \(K\) has dimension \(1\).
7. Hartshorne II.3.20(e) fails for \(X\).
The open \(D(\pi)\) is nonempty, but by 6 \[ \dim D(\pi)=1, \] while by 1 \[ \dim X=2. \] Thus a nonempty open subset need not have the same dimension once the base is no longer a field.
8. Q.E.D.
Step 4 disproves II.3.20(a), 5 disproves II.3.20(d), and 7 disproves II.3.20(e).