A duality theorem in étale cohomology for arithmetic schemes that generalizes Poincaré duality: it relates the cohomology of a scheme with compact support to the Galois cohomology of its function field and identifies natural perfect pairings between cohomology groups and arithmetic duals such as Tate modules and the Brauer group.

Principle

Cohomological information on an arithmetic scheme pairs with dual arithmetic invariants under a global-local functoriality, producing perfect pairings after appropriate shifts and torsion control; local dualities and global reciprocity combine to yield the global duality statement.

Demonstration

For a regular, proper arithmetic scheme X of dimension d, Artin–Verdier duality produces pairings H^i_c(X_et, F) × H^{2d+2-i}(X_et, F^*) → Q/Z for finite étale sheaves F, and identifies the finite groups and their Pontryagin duals explicitly, reflecting how local Tate duality at places contributes to global pairings.

Misapplication

Treating Artin–Verdier duality as a purely topological Poincaré duality and applying it to schemes without accounting for arithmetic torsion, non-properness, or required Tate twists; this can yield incorrect identifications of pairings or miss necessary restrictions on coefficients.

Consequence

When applied correctly it organizes arithmetic cohomology into dual pairs, constrains possible cohomology classes (for instance controlling the Brauer group), and provides the formal backbone for arithmetic duality results and reciprocity laws used in descent and obstruction theories.

Reversal

The inversion would be attempting to reconstruct local duality statements from a putative global pairing without verifying the compatibility of local contributions or the exactness properties that glue local pairings into a global perfect pairing.

Boundary

Applies to schemes of arithmetic interest with étale topology, finite constructible coefficients, and appropriate finiteness hypotheses (e.g., properness or control of cohomology with compact support); it excludes naive topological analogues, infinite coefficient modules without Tate twists, and settings lacking the required finiteness or dualizing complexes.

Semantic Tension

Competes with classical Poincaré duality language: both assert pairings between cohomology in complementary degrees, but Artin–Verdier incorporates arithmetic phenomena (Galois actions, torsion, reciprocity) that make its statements and required hypotheses distinct from purely topological dualities.

Synthesis

Artin–Verdier duality is the arithmetic analogue of Poincaré duality set in étale cohomology: using local dualities and reciprocity, it produces perfect pairings between cohomology with compact support and arithmetic dual groups, subject to finiteness and twisting conditions that reflect number-theoretic torsion.