Vector space summands of lower syzygies

2024/10/16 by Asgharzadeh, Mohsen, DeBellevue, Michael, Dey, Souvik +2
#13D02 #13E10 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.12227

Abstract

In this paper, we investigate problems concerning when the residue field k of a local ring (R,\frak m, k) appears as a direct summand of syzygy modules, from two perspectives. First, we prove that the following conditions are equivalent: (i) k is a direct summand of second syzygies of all non-free finitely generated R-modules; (ii) k is a direct summand of third syzygies of all non-free finitely generated R-modules; (iii) k is a direct summand of \frak m. We also prove various consequences of these conditions. The second point of this article is to investigate for what artinian local rings R the dual E^*=Hom_R(E_R(k),R) of the injective envelope of the residue field, which is also a second syzygy, is a k-vector space. Using the notion of Eliahou-Kervaire resolution, we introduce a large class of artinian local rings that satisfy this condition.

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