Burch index, summands of syzygies and linearity in resolutions

2022/08/10 by Hailong Dao, David Eisenbud, Dao, Hailong +1 · 1 citation
Mathematics · #13H10. Secondary: 14B99 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary: 13D02 #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2208.05427

openalex publication_date 2022/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Burch index is a new invariant of a local ring R whose positivity implies a kind of linearity in resolutions of R-modules. For example, if R has depth zero and Burch index at least 2, then any non-free 7th R-syzygy contains the residue field as a direct summand. We also compute the Burch index in various cases of interest.

Cited by

• On direct summands of syzygies of the residue field of a local ring

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