We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald–Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.
Algebraic Geometry | Geometry and Topology
Bruner, R., Greenlees, J. & Rognes, J. The local cohomology spectral sequence for topological modular forms. Math. Z. 302, 129–180 (2022). https://doi.org/10.1007/s00209-022-03033-4