带符号的欧拉-史密斯剖面：分次代数
Signed Euler--Smith Profiles of Graded Algebras
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中文总结 AI 辅助
本文针对分次代数引入带符号的欧拉-史密斯剖面，从无限极小分解构造有理欧拉矩阵，细化完美模极点阶增长，并证明其在等价下不变，且完全负剖面导致指数增长。
中文摘要 AI 辅助
矩阵希尔伯特级数保留了其行列式所丢弃的局部初等因子数据。对于顶点单模属于 $FP_\infty$ 型且矩阵希尔伯特级数为有理数的局部有限正分次初等代数，我们从可能无限的极小分解构造了有理欧拉矩阵及其在 $x=1$ 处的带符号局部史密斯剖面。该剖面细化了完美模的极点阶增长，并在平移兼容的分次莫里塔等价和完美导出等价下不变。每个有限整数剖面都会出现，而完全负的剖面则强制指数角增长。这将扭曲卡拉比-丘代数的非负局部史密斯框架扩展到欧拉矩阵本身可能具有极点的情形。
英文摘要
Matrix Hilbert series retain local elementary-divisor data that their determinants discard. For locally finite positively graded elementary algebras whose vertex simples are of type $FP_\infty$ and whose matrix Hilbert series is rational, we construct from possibly infinite minimal resolutions a rational Euler matrix and its signed local Smith profile at $x=1$. The profile refines pole-order growth for perfect modules and is invariant under shift-compatible graded Morita and perfect-derived equivalences. Every finite integer profile occurs, while a wholly negative profile forces exponential corner growth. This extends the nonnegative local Smith framework for twisted Calabi--Yau algebras to a setting in which the Euler matrix may itself have poles.
发表机构
- Istanbul Technical University（伊斯坦布尔理工大学）
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