发表机构
Soochow University（苏州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多分次局部上同调模的范畴结构，通过将其等同于表示范畴，建立Serre滤过、TTF三元组及Nakayama-Serre对偶，并构造秩层分辨率。
AI 中文摘要
设 $\Bbbk$ 为一个域，令 $S=\Bbbk[x_1,\ldots,x_n]$ 带有标准的 $\mathbb N^n$-分次，并令 $\mathfrak m=(x_1,\ldots,x_n)$。对于 $0\le i<n$ 和 $q=n-i$，我们将平移多分次局部上同调模的范畴 $\mathcal H_i(\mathbf t)$ 等同于 $\Rep(U_q(\mathbf t))$，其中 $U_q(\mathbf t)=\{\mathbf a\in[\mathbf0,\mathbf t]\mid |\operatorname{supp}(\mathbf a)|\ge q\}$。这给出了有限和全局Serre滤过及其纯支撑秩商。我们通过阿贝尔再分割来组织所得的挠和商结构：一个序理想分解产生一个典范的TTF三元组、遗传支撑挠对和Gabriel商。对于有限偏序集，两种互补的再分割方向都存在，而对于全局有限支撑范畴，只有内向有限方向是自动的。这些再分割具有有界导出提升。在额外的有限分辨率条件下，导出有限支撑范畴具有右Serre函子，且导出Kan扩张满足右Serre交换。在有限盒子中，我们进一步构造了一个函子性的秩层分辨率，比较左和右Kan截面；Nakayama--Serre对偶将其转化为一个显式的余标准秩复形。例外顶层范畴 $\mathcal H_n(\mathbf t)$ 通过第二余合冲单独处理。
英文摘要
Let $\Bbbk$ be a field, let $S=\Bbbk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading, and let $\mathfrak m=(x_1,\ldots,x_n)$. 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 \[ \Rep(U_q(\mathbf t)),\qquad U_q(\mathbf t)=\{\mathbf a\in[\mathbf0,\mathbf t]\mid |\operatorname{supp}(\mathbf a)|\ge q\}. \] 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.
Comments35 pages comments are welcome
