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AN EXPLICIT DIAGONAL RESOLUTION FOR A NON‐ABELIAN METACYCLIC GROUP
Author(s) -
Remez J. J.
Publication year - 2017
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579317000018
Subject(s) - indecomposable module , mathematics , diagonal , hilbert's syzygy theorem , resolution (logic) , abelian group , pure mathematics , group (periodic table) , diagonally dominant matrix , geometry , computer science , invertible matrix , chemistry , organic chemistry , artificial intelligence
We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups G ( p q ) however admit periodic resolutions. In the particular case of G ( 21 ) it is possible to achieve a fully diagonalized resolution . In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over Λ . Such a list aids the decomposition of the augmentation ideal (the first syzygy ) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.
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