Question about the dimensions of generic fibers $W_{\eta}$ and $X_{\eta}$, where $W$ is a closed subscheme of $X$

Question

Let $f: X \rightarrow Y$ be a morphism of schemes of finite type. $X$ is reduced, $Y$ is an integral scheme with generic point $\eta$ and $k(\eta)$, the residue field of $\eta$, has characteristic $0$. Let $W$ be a closed subset of $X$, viewed as a reduced closed subscheme of $X$. In an article I am reading it then states "Since $k(\eta)$ is a field of characteristic zero and $X_{\eta}$ is reduced, we have $\dim W_{\eta} < \dim X_{\eta}$." I was wondering how this follows. Any explanation would be appreciated. Thank you.

PS Here $W$ is a proper closed subset of $X$

PPS We assume $X_{\eta} \not = \emptyset$

Answer 1

As pointed out in the comments, things can get a little strange when $X$ is reducible or $f$ isn't dominant. We'll treat the case where $X$ is irreducible and $f$ is dominant while trusting the reader to make the correct generalization of this to the case of $X$ reducible (treat each component $X_{\alpha}$ individually via the composite of the closed immersion $X_\alpha\to X$ and $f$, etc).

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Here's a few results we'll need:

Theorem (Generic flatness; EGA IV2 Theoreme 6.9.1): Let $Y$ be a locally noetherian integral scheme with $f:X\to Y$ a finite type morphism and $\mathcal{F}$ a coherent $\mathcal{O}_X$ module. Then there's a nonempty open subscheme $U\subset Y$ so that $\mathcal{F}$ restricted to $f^{-1}(U)$ is flat over $U$.

Lemma (EGA IV2 Corollaire 6.1.2): Let $X,Y$ be locally noetherian schemes with $f:X\to Y$ flat. Then for all $x\in X$, we have $\dim_x X = \dim_{f(x)} Y + \dim_x X_{f(x)}$.

Fact: If $X$ is a scheme, $\dim X = \sup_{x\in X} \dim_x X$.

Fact: For an irreducible scheme $X$, the local dimension at any point is equal to the global dimension of $X$.

In addition to the hypotheses in the problem statement, we assume $X$ irreducible, $f$ dominant, and $W$ a proper closed subscheme.

By two applications of generic flatness (to $\mathcal{F}=\mathcal{O}_X$ and $\mathcal{F}=\mathcal{O}_W$), we may assume that both $f:X\to Y$ and $f|_W:W\to Y$ are flat by replacing $Y$ with a slightly smaller open subscheme which doesn't affect our calculations at the generic point*. Let $n=\dim X$ and $m=\dim Y$. If there exists a $w\in W$ so that $\dim_w W_{f(w)} \geq n-m$, then we have $\dim_w W \geq n-m+m = n$, which contradicts the fact that $W$ is a proper closed subscheme. So for all $w\in W$, $\dim_w W_{f(w)} < n-m$, so in particular $\dim W_\eta < n-m = \dim X_\eta$ and we're done.

If you're looking for more intuition about what should happen with respect to fibers of a morphism, Stacks has a good section on it here.

*: If this makes $W=\emptyset$, we're already done and we can stop right here - h/t Ben in the comments pointing out I didn't write this down.