The Stacks project

Remark 37.18.5. Here are some cases where the material above, especially Lemma 37.18.4, allows one to conclude that a morphism $f : X \to S$ of schemes has relative dimension $d$ as defined in Morphisms, Definition 29.29.1. For example, this is true if

  1. $X$ is integral with generic point $\xi $,

  2. the transcendence degree of $\kappa (\xi )$ over $\kappa (f(\xi ))$ is $d$,

  3. $f$ is locally of finite type, and

  4. there exists a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ of finite type which is flat over $S$ with $\text{Supp}(\mathcal{F}) = X$.

Another set of hypotheses that work are the following:

  1. $S$ is irreducible with generic point $\eta $,

  2. $X_\eta $ is dense in $X$,

  3. every irreducible component of $X_\eta $ has dimension $d$,

  4. $f$ is locally of finite type, and

  5. there exists a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ of finite type which is flat over $S$ with $\text{Supp}(\mathcal{F}) = X$.

Of course, we can relax the flatness condition on $\mathcal{F}$ and require only that $\mathcal{F}$ is flat over $S$ in codimension $0$, i.e., that $\mathcal{F}$ is flat over $S$ at every generic point of every fibre. If we ever need these results, we will carefully state and prove them here.

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