A family of canonical pairings between Galois cohomology groups of local or global fields, typically pairing H^i(G_K, M) with H^{2-i}(G_K, M^*) (or their analogues), that identify arithmetic invariants and dualize obstructions for finite Galois modules and related coefficients.

Principle

Cohomological duality arises from a perfect bilinear pairing induced by a local or global trace/reciprocity map together with a choice of Pontryagin or Cartier dual on coefficients, yielding isomorphisms between cohomology and the dual of complementary-degree cohomology.

Demonstration

For a local field K and a finite discrete G_K-module M of prime-to-characteristic order, local Tate duality produces a perfect pairing H^i(G_K,M) × H^{2-i}(G_K,M^*) → Q/Z, which for i=1 identifies the Selmer/obstruction classes with orthogonal complements under local reciprocity.

Misapplication

Assuming Tate duality holds without finiteness, torsion, or appropriate coefficient duals (for example naively pairing infinite-dimensional continuous representations or modules without taking Pontryagin duals), which can lead to false conclusions about vanishing or perfectness of pairings.

Consequence

When applicable, Tate duality converts local and global cohomology problems into dual statements, allowing computation of dimensions, identification of annihilators of classes, and formulation of Poitou–Tate exact sequences that control global arithmetic obstructions.

Reversal

The opposite viewpoint is to ignore dual structures and treat cohomology groups independently; this loses orthogonality relations, making detection of global reciprocity constraints and exact sequences relating local and global data much harder or impossible.

Boundary

Applies primarily to Galois cohomology of local and global fields, finite discrete Galois modules (or compact modules with Pontryagin duals), and in specified degree ranges (classically degrees 0,1,2); it does not extend verbatim to arbitrary infinite coefficient modules or to cohomology theories lacking an appropriate trace/duality.

Semantic Tension

Competes with other dualities (Pontryagin duality for topological abelian groups, Poincaré duality in manifold cohomology); the tension is choosing the correct dual notion (Pontryagin vs. Cartier vs. Hom) and the correct degree shift for arithmetic versus geometric contexts.

Synthesis

Tate duality is the arithmetic cohomology principle that pairs complementary-degree Galois cohomology groups of fields via a canonical trace/reciprocity pairing and a coefficient duality, turning questions about existence and obstructions of classes into orthogonality and exact-sequence statements.