An obstruction that prevents local objects or data defined on a covering from gluing to a global object; typically detected by a nontrivial cohomology class or an ineffectivity of descent data in a chosen topology (étale, fppf, fpqc, etc.).

Principle

Descent formalizes when local compatibility data (restrictions and cocycle conditions) are sufficient and effective for reconstructing a global object; failure occurs when compatibility holds up to coherent isomorphism but cannot be realized by a global object in the category.

Demonstration

A classical example is a torsor under a group scheme given by local trivializations and transition functions that represent a nontrivial class in H^1 of the cover: the cocycle is nonzero, so the data define a nontrivial torsor that does not descend to a trivial global object, or more generally descent data that are not effective produce no global object.

Misapplication

Assuming that because objects agree on overlaps they must glue globally without checking higher cocycle obstructions or the required topology (e.g., using Zariski descent where only fppf descent applies), leading to incorrect global constructions.

Consequence

Recognizing failure of descent forces refinement: either change the topology (use a finer cover), pass to stacks or gerbes where descent becomes effective, or identify the cohomological obstruction that must vanish for descent to succeed.

Reversal

Effective descent is the reversal: descent data satisfying the cocycle conditions that indeed arise from a unique (up to isomorphism) global object, typically witnessed by vanishing obstruction classes in appropriate cohomology groups.

Boundary

Pertains to categories fibered in groupoids, sheaves, schemes, and modules over coverings in specified topologies; does not refer to trivial gluing in discrete categories or to formal patching that ignores cocycle coherence.

Semantic Tension

Tension between naive patching (local agreement on pairwise overlaps) and full descent (coherent higher cocycle conditions and effectivity); also tension between representability by schemes versus requiring stacky structures for effective descent.

Synthesis

Failure of descent is the phenomenon where locally defined compatible data fail to realize a global object because of nonvanishing obstruction classes or ineffectivity; resolving it requires adjusting topology, enhancing the category (stacks, gerbes), or removing the cohomological obstructions.