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The residue field as an infinite syzygy
Abstract
Let (R,m,k) be a commutative Noetherian ring which is either local or positively graded over the field k. We provide a method based on liftings for constructing free coresolutions of k over R. We apply this method to twisted trivial extensions of the node $k[u,v]/(uv)$ to build a family of positively graded k-algebras of dimension $1$ over which k is an infinite graded syzygy, resolving an open question of Iyengar in the negative. We also prove a formula for alternating sums of lengths of Ext modules when R is positively graded, and use it to show that a graded free coresolution of k cannot dualize to a resolution unless R is Artinian Gorenstein.
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BibTeX
@misc{liu2026residue,
title = {{The residue field as an infinite syzygy}},
author = {Jian Liu and Justin Lyle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38393},
eprint = {2610.38393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38393},
}
RIS
TY - UNPB
TI - The residue field as an infinite syzygy
AU - Jian Liu
AU - Justin Lyle
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38393
UR - https://arxiv.org/abs/2610.38393
AB - Let (R,m,k) be a commutative Noetherian ring which is either local or positively graded over the field k. We provide a method based on liftings for constructing free coresolutions of k over R. We apply this method to twisted trivial extensions of the node $k[u,v]/(uv)$ to build a family of positively graded k-algebras of dimension $1$ over which k is an infinite graded syzygy, resolving an open question of Iyengar in the negative. We also prove a formula for alternating sums of lengths of Ext modules when R is positively graded, and use it to show that a graded free coresolution of k cannot dualize to a resolution unless R is Artinian Gorenstein.
ER -