Veronese子代数的典范迹的双侧界
Two-sided bounds for canonical traces of Veronese subalgebras
- The University of Osaka（大阪大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文推广Goto--Watanabe的Veronese公式至更广的模有限扩张，得到典范迹的双侧界，并刻画了近似Gorenstein性质的条件，应用于Segre积等环类。
中文摘要 AI 辅助
Goto--Watanabe的经典Veronese公式将Veronese子代数的典范模与相应的Veronese模等同起来。我们在模有限扩张下证明了分级对偶的类似相容性，其中上环是等维的，满足Serre条件$(S_2)$，且维数至少为2，并得到了$b$-不变量的公式。这给出了典范迹的显式双侧界：下界由典范迹商的Loewy长度控制，而上界由$a$-和$b$-不变量控制，并针对非混合环给出了$S_2$-化改进。我们还刻画了当$a$-不变量为负时，标准分级level代数的所有足够大的Veronese子代数何时近似Gorenstein。应用包括Segre积、Stanley--Reisner环和行列式环。
英文摘要
The classical Veronese formula of Goto--Watanabe identifies the canonical module of a Veronese subalgebra with the corresponding Veronese module. We prove the analogous compatibility for graded duals under a module-finite extension whose upper ring is equidimensional, satisfies Serre's condition $(S_2)$, and has dimension at least two, and obtain the formula for the $b$-invariant. This yields explicit two-sided bounds for canonical traces: the lower bound is controlled by the Loewy length of the canonical-trace quotient, whereas the upper bound is governed by the $a$- and $b$-invariants, with an $S_2$-ification refinement for unmixed rings. We also characterize when all sufficiently large Veronese subalgebras of standard graded level algebras are nearly Gorenstein when the $a$-invariant is negative. Applications include Segre products, Stanley--Reisner rings and determinantal rings.
