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
X is integral with generic point \xi ,
the transcendence degree of \kappa (\xi ) over \kappa (f(\xi )) is d,
f is locally of finite type, and
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:
S is irreducible with generic point \eta ,
X_\eta is dense in X,
every irreducible component of X_\eta has dimension d,
f is locally of finite type, and
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.
Comments (0)