Kazhdan–Laumon finite cohomological dimension conjecture

Kazhdan and Laumon proposed the conjecture in 1988. It was disproved by Bezrukavnikov and Polishchuk in 2001 for $G=\operatorname{SL}_3$.

Known results

• The simple object corresponding to the constant sheaf has infinite cohomological dimension (Bezrukavnikov and Polishchuk, 2001).
• The weaker localized-$K_0$ statement was established in types $A_n$ with $n\ge 3$ and $B_2$ (Polishchuk, 2001).

Later replacement result, 2023

A 2023 paper proved the localized-$K_0$ replacement in all types and established well-definedness of the Kazhdan–Laumon construction by this alternative route; it does not restore the disproved finite-cohomological-dimension conjecture.

Current status (as of September 2026): The original conjecture is settled negatively by the $\operatorname{SL}_3$ counterexample; the replacement conjecture and well-definedness result are established more broadly.