A phenomenon where a natural formation (normalization, formation of fibers, higher direct images, schematic closure, etc.) does not commute with base change along a morphism of schemes, producing unexpected or ill-behaved fibers or breaking properties expected to be preserved under change of base.

Principle

Commutation with base change is governed by finiteness, flatness, properness, and coherence hypotheses. When those hypotheses fail or subtle torsion/embedded-component issues arise, the naive exchange of formation and base change can be pathological: the fiberwise construction differs from the base-changed global construction.

Demonstration

Illustrative example: normalization need not commute with arbitrary base change. One can have a family X→S whose normalization on S, pulled back to a fiber, does not coincide with the normalization of the fiber after base change — branches can fail to separate uniformly. Another frequent instance is higher direct images R^i f_* F whose formation fails to commute with base change when f is not proper or F not coherent, leading to jumps in fiber dimensions of cohomology.

Misapplication

Assuming all reasonable constructions commute with base change without checking hypotheses like flatness or properness; for instance, applying fiberwise arguments to deduce global statements about normalization or cohomology without verifying base-change theorems can lead to incorrect conclusions.

Consequence

Pathological base change produces fibers with different geometry than expected, invalidates descent or deformation arguments, may cause failure of semicontinuity, and breaks compatibility of invariants (Picard groups, cohomology, singularity types) with base extension. It forces additional checks or the use of derived/pushforward-corrected procedures.

Reversal

Good base change occurs when hypotheses (coherence, flatness, properness, finite presentation) hold and theorems guarantee that formation commutes with base change; in that case fiberwise and global constructions align and invariants behave predictably under extension of the base.

Boundary

This concern arises in families of schemes and morphisms of schemes; it is most pronounced outside the class of flat, proper, finite-presentation morphisms with coherent sheaves. Derived and stack-theoretic frameworks can sometimes control pathologies, but the classical notion requires checking standard hypotheses before asserting commutation.

Semantic Tension

The tension is between the formal operation of pulling back and the algebraic/geometric subtleties (torsion, embedded primes, nonflatness) that obstruct commutation. Practitioners may conflate naive set-theoretic fiberwise arguments with genuine base-change properties guaranteed only under precise hypotheses.

Synthesis

Pathological Base Change names the breakdown when formation of geometric or algebraic constructions does not commute with changing the base: it flags the failure of fiberwise reasoning and signals the need for finiteness/flatness/properness checks or for working in derived categories to restore controlled behavior.