On Some Local Cohomology Spectral Sequences

Abstract

The purpose of this report is to introduce a formalism to produce two collections of spectral sequences. On one hand, a collection is made up by spectral sequences which involve in their second page the left derived functors of the colimit on a certain finite poset. On the other hand, the other is made up by spectral sequences which involve in their second page the right derived functors of the limit on a certain finite poset. In both cases, we provide sufficient conditions to ensure their degeneration at the second page. Finally, we see how to use our second collection of spectral sequences to produce a decomposition of local cohomology modules which can be regarded as a generalization of the classical Hochster formula for the local cohomology of a Stanley–Reisner ring.