Categories of Multigraded Local Cohomology Modules: Serre Filtrations and Nakayama Duality

Let $\mathbb{k}$ be a field, let $S=\mathbb{k}[x_1,\dots ,x_n]$ with its standard ${\mathbb{N}}^n$-grading, and let $\mathfrak{m}=(x_1,\dots ,x_n)$. For $0\le i<n$ and $q=n-i$, we identify the category ${\mathcal{H}}_i(\mathbf{t})$ of shifted multigraded local cohomology modules with $$\text{Rep}(U_q(\mathbf{t})),U_q(\mathbf{t})=\{\mathbf{a}\in [\mathbf{0},\mathbf{t}]\mid \mathrm{\mid }\mathrm{supp}(\mathbf{a})\mathrm{\mid }\ge q\}\mathrm{.}$$ This gives the finite and global Serre filtrations and their pure support-rank quotients. We organize the resulting torsion and quotient structures through abelian recollement: an order-ideal decomposition produces a canonical TTF triple, hereditary support torsion pairs, and Gabriel quotients. For finite posets both complementary recollement orientations exist, whereas for the global finite-support categories only the inward-finite orientation is automatic. These recollements admit bounded derived lifts. Under an additional finite-resolution condition the derived finite-support categories have right Serre functors, and derived Kan extensions satisfy a right-Serre exchange. In finite boxes we further construct a functorial rank-layer resolution comparing the left and right Kan sections; Nakayama–Serre duality transforms it into an explicit costandard rank complex. The exceptional top category ${\mathcal{H}}_n(\mathbf{t})$ is treated separately via second cosyzygies.