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I'm working on exercise II.6.1 in Hartshorne and I'm stuck on the following step. I would appreciate some help.

Let $ X $ be a Noetherian integral scheme. Let $ Z $ be a closed subscheme of $\mathbb{P}^n_{\mathbb{Z}} $ of codimension $1$. Then $X\times Z $ is of codimension $1$ in $\mathbb{P}^n_{X}$.

My intuition is to reduce to the affine case by taking an affine open cover of the projective space, but I can't set this up right.

PeterM
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Locally $D$ is given by the vanishing of $f \in \mathbb{Z}[x_0, \ldots, x_n]$. Let $Spec\ A$ be an affine open of $X$, so $A$ in a Noetherian integral domain, then $X \times Z$ is locally given by $f \in A[x_0,\ldots, x_n]$, this latter ring which is a Noetherian domain, and hence $f$ is a non-zero divisor and we can apply Krull's Principal Ideal theorem, which says that in a Noetherian ring every minimal prime over $(f)$ has height 1, i.e locally $f$ cuts out a codimension 1 subscheme.
So we can construct an open cover of $X \times P^n_Z$ and $X \times Z$ has codimension 1 subscheme in this each open cover, so it has codimension 1 globally. For details of this last step, see Krull Dimension of a scheme

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ykm
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  • One thing to note is that there is some commutative algebra behind the assertion that $Z$ is locally principal. – Hoot Jul 18 '15 at 05:22