Extension of the Gorenstein-locus characterization beyond triangulable spaces

Establish whether, for every locally compact Hausdorff space X and field k, the Gorenstein locus of the derived category of k-sheaves on X equals the full subcategory of sheaves supported on the k-homology manifold locus of X, without assuming that X admits a triangulation.

Background

For a dualizable stable category, the paper defines the Gorenstein locus as the full subcategory of objects on which the Serre-functor duality behaves invertibly. For a locally compact triangularizable space X and a field k, the main theorem proves that this locus in Shv(X)[D(k)] is precisely the subcategory of sheaves supported on the k-homology manifold locus of X.

The paper proves in general that every sheaf supported on the homology manifold locus is regular, and that regular sheaves have invertible dualizing-sheaf stalks on their support. The unresolved step is whether the converse characterization remains valid for locally compact Hausdorff spaces that are not triangulable.

References

We do not know whether this description remains true if one drops the assumption that X admits a triangulation, but it is always true that a sheaf supported on the homology manifold locus is regular, as the first proposition of this subsection shows.

— Reconstruction of graphs and surfaces from derived categories  (2610.13488 - Harr, 9 Oct 2026) in Section 1, subsection “Statement of results”; see also Section 5, subsection “The Gorenstein locus of a dualizable category”