  We introduce the notion of E-depth of graded modules over polynomial rings to
measure the depth of certain Ext modules. First, we characterize graded modules
over polynomial rings with (sufficiently) large E-depth as those modules whose
(sufficiently) partial general initial submodules preserve the Hilbert function
of local cohomology modules supported at the irrelevant maximal ideal,
extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules.
Second, we describe the cone of local cohomology tables of modules with
sufficiently high E-depth, building on previous work of the second author and
Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by
Kustin and Ulrich.
