We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version quasi-resolutions introduced by Qin-Wang-Zhang. prove that noetherian bounded below and locally finite graded algebra with injective dimension 2 are always Morita equivalent. When we restrict to quadric hypersurfaces, hypersurface, singularity, admits pre-res...

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...