Let R be a standard graded algebra over field k and I homogeneous ideal of R. We study the question whether there is constant c such that Soc(Hmj(R/It))<−ct=0 for all t≥1 variation this question. also draw connection between notion gauge-boundedness in prime characteristic.

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...