Cogeneration by microstalk corepresentatives

Determine whether, for every finite collection of corepresentatives of microstalks associated with smooth points of a good Whitney triangulation, the dg algebra End(\bigoplus_i \chi_i) cogenerates its module category.

Background

The paper studies dg algebras formed from finite collections of microstalk corepresentatives in microlocal sheaf theory. These algebras are proper and coconnective, and their degree-zero cohomology is directed and of finite global dimension. Since the paper’s counterexample has similar formal properties but fails to cogenerate, the authors explicitly ask whether the additional geometric origin of the microstalk algebras guarantees cogeneration.

References

Does the dg algebra $\End\left( \bigoplus_i \chi_i \right)$ cogenerates its module category?

— A proper dg algebra which does not cogenerate  (2609.34193 - Hu et al., 28 Sep 2026) in Question at the end of Section \ref{sec: chord}