The following are stable under arbitrary base change and under composition: open and closed immersions, affine, finite, integral, quasicompact, locally of finite type, of finite type, quasi-finite, flat, surjective, separated, proper, projective, smooth, unramified, étale.
The length of the list reflects common stability mechanisms under tensor product and fibre product. A property defined by a condition on the ring maps in an affine cover is stable under base change whenever the corresponding module-theoretic condition survives \(\wait \tensor_B B'\), and finite generation, flatness, and surjectivity all do.
The instructive entries are the ones that are not on it. Being an isomorphism onto the image, being a closed map, and being dominant are not stable, and universal closedness is defined by forcing the issue: it is exactly “closed after every base change”, which is why the definition of proper contains the word universally. \(\AA^1_k \to \Spec k\) is closed and not universally closed, and the witness is again the hyperbola in \(\fiberprod{\AA^1}{k}{\AA^1}\).
Stability under base change is what makes a property a statement about fibres, and that is its real use: a proper morphism has proper fibres, a flat morphism has flat base changes, and a smooth morphism stays smooth after any extension of the base field.
Dependencies
D-MORFIBThe fibre of a morphism as a base changeuses
Backlinks
D-MORLOCALClasses of morphisms stable under base change and local on the source or targetrelated-toPR-MORCANCELThe cancellation theorem for properties of morphismsrelated-to