Brochard’s duality conjecture

Brochard’s conjecture concerns duality and biduality for proper, flat, finitely presented commutative group stacks with finite flat inertia. The 2014 literature records substantial partial results and a conditional proof, while noting that the restriction on $2$ might be unnecessary.

Known results

• For a proper, flat, finitely presented algebraic commutative group stack $G$, its dual is algebraic and finitely presented with affine diagonal, and its fibers are proper (2014).
• The conjecture was proved under cohomological flatness of $H^0(G)$ (2014).
• Over a regular base where $2$ is invertible, dualizability and preservation of the relevant structural conditions were established (2014).

September 2026 conditional proof

A new paper proves structure and duality theorems for commutative group schemes, algebraic spaces, and group stacks, including Brochard’s conjecture for a substantial class of proper flat commutative group stacks when $2$ is invertible. This is a significant conditional advance, but the unrestricted conjecture is not claimed to be proved, and no independent verification was found.

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Current status (as of September 2026): The conjecture is claimed for the stated class when $2$ is invertible; the unrestricted case remains open.