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Pure semisimplicity conjecture and Artin problem for dimension sequences
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2021-03-13 , DOI: 10.1016/j.jpaa.2021.106745
Jan Šaroch

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type.

The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding FMn(G) exists provided that n<5. As a byproduct, we obtain a division ring extension GF such that the bimodule FFG has the right dimension sequence (1,2,2,2,1,4).

Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.



中文翻译:

尺寸序列的纯半单纯性猜想和Artin问题

受何塞·路易斯·加西亚(JoséLuisGarcía)最近发表的论文的启发,我们再次回顾了丹尼尔·西姆森(Daniel Simson)构造纯半简单猜想的反例的尝试。使用紧致性,我们表明这种反例的存在将很容易地从有限表示类型的某些(可数集)世袭阿丁环的存在中得出。

这样的环的存在就被证明等同于特殊类型的嵌入物的存在,我们将其称为紧密环,将其划分为简单的阿蒂尼安环。使用1980年代艾丹·斯科菲尔德(Aidan Schofield)的工具,我们可以证明这种嵌入F中号ñG 存在,只要 ñ<5。作为副产品,我们获得了除法环延伸GF 这样的双模块 FFG 具有正确的尺寸顺序 1个2个2个2个1个4

最后,我们提出猜想A,该猜想断言可以对分隔环进行元素的特定类型的附加,并证明其有效性足以证明总体上存在紧密嵌入,从而证明了纯正嵌入半简单猜想。

更新日期:2021-03-17
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