cohomological

Cohomological is an adjective relating to cohomology, a broad framework in mathematics that assigns algebraic invariants to spaces, schemes, or other objects to capture global structure. In general, a cohomological statement or method uses cohomology groups or cohomology theories to detect features such as holes, obstructions, or symmetries. Cohomology groups H^n(X) are contravariant with respect to maps, in contrast to homology, which is covariant; this distinction informs many constructions and results.

Common contexts for cohomological ideas include topology, differential geometry, and algebraic geometry. In topology and geometry, theories such as singular cohomology, de Rham cohomology, Čech cohomology, and sheaf cohomology play central roles. In algebraic geometry, one often uses étale cohomology or algebraic de Rham cohomology, within the broader framework of derived functors on sheaves. Derived categories, Ext groups, and other cohomological tools provide a unifying language for these methods. Cohomological operations, such as Steenrod squares, describe systematic transformations among cohomology classes.

A key numerical invariant is the cohomological dimension, which bounds the largest nonzero cohomology group with given coefficients and reflects the complexity of the object. Cohomological techniques underpin numerous theories and results, from the study of fiber bundles and characteristic classes to questions in algebraic geometry, topology, and number theory, where theories like étale cohomology connect geometric objects to arithmetic information.