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| 摘要: |
| 给定交换整环R，以及Hopf R-代数H，且H是一个有限生成的自由R-模.设A是一个R-代数，并且A是H-余模代数.如果自然映射β：A⊕AcoH A→A ⊕R H的余核是商有限的，则称A/AcoH是一个Hopf稠密伽罗瓦扩张.它是域上Hopf稠密伽罗瓦扩张的推广.本文证明了R上的Hopf稠密伽罗瓦扩张隐含了一个弱化的Auslander定理.此外，假设A是几乎可交换代数，且gr（A）是一个整环.如果A/AcoH是Hopf稠密伽罗瓦扩张，且自然映射β是严格的，本文证明了在此情形下，H在一个包含R的代数闭域上对偶于一个有限群代数. |
| 关键词: Hopf稠密伽罗瓦扩张 局域化 商范畴 滤过代数 |
| DOI： |
| 分类号:O153 |
| 基金项目:Supported by National Natural Science Foundation of China (11571239; 11671351); Natural Science Foundation of Zhejiang Province (LY19A010011). |
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| HOPF DENSE GALOIS EXTENSIONS OVER A RING |
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HE Ji-wei,HU Hai-gang
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| Abstract: |
| Let R be a commutative domain, let H be a Hopf R-algebra which is a finitely generated free R-module, and let A be an R-algebra which is also a H-comodule algebra. We will say that A/AcoH is a Hopf dense Galois extension if the cokernel of the associated canonical map β : A⊕AcoHA→A⊕RH is quotient finite. It is a generalization of Hopf dense Galois extension over a field. This paper shows that a weaker version of Auslander theorem holds for Hopf dense Galois extensions over R. It is also proved that if the algebra A is almost commutative such that gr(A) is a domain, and the canonical map β is strict, then a Hopf dense Galois extension A/AcoH will imply that H is dual to a finite dimensional group algebra over an algebraic closed field containing R. |
| Key words: Hopf dense Galois extension localization quotient category flltered algebra |
