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The Stacks project

Remark 37.18.6. Here are some cases where the material above, especially Lemma 37.18.5, 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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