Algorithms for finding the gradings of reduced rings

2019/11/06 by van Gent, Daniël M. H.
#Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1911.02957

Abstract

This work is a Master thesis supervised by Prof. Dr. H.W. Lenstra. Lenstra and Silverberg showed that each reduced order has a universal grading, which can be viewed as the `largest possible grading'. We present an algorithm to compute the universal grading for a given order R, which has runtime n^O(m), where n is the length of the input and m is the size of the minimal spectrum of R. We do this by computing all gradings of the corresponding reduced ℚ-algebra with cyclic abelian groups of prime-power order. We additionally generalize the result of Lenstra and Silverberg that reduced orders have a universal grading to a broader class of rings.

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