3 problems

Let $K$ be a field, let $S=K[x_1,\ldots,x_n]$, and let $\mathbf m=(x_1,\ldots,x_n)$ be its maximal graded ideal. For an integer $t\geq 1$, let $\operatorname{sdepth}(\mathbf m^t)$…

Let $M$ be a finitely generated graded Gorenstein $R$-module, minimally generated by homogeneous elements of degree zero. Let $s=\operatorname{codim} M$ and…

Let $S=k[x_1,\ldots,x_n]$ be a standard graded polynomial ring, and let $M$ be a finitely-generated graded $S$-module of codimension $c$. Let $J$ be a homogeneous ideal contained i…