A new proof of characteristic-free Knörrer periodicity
Abstract
Let R = S/(f) be a complete local hypersurface ring and let A = S[ [u,v] ]/(f+uv). Pfister and Popescu showed that the stable categories of maximal Cohen-Macaulay modules over R and over A are equivalent, with no assumption on the characteristic. We give a short proof. If (Φ,Ψ) is a matrix factorization of h and ∂ is a derivation, then (∂Ψ,∂Φ) is a null-homotopy for multiplication by ∂ h. Since u and v are partial derivatives of f+uv, they act as zero on the stable category of A, so the Koszul complex on u and v splits in the singularity category, and this gives density. For the double branched cover S[ [z] ]/(f+z2) the same argument yields 2z: it recovers Knörrer's lemma up to free summands on the double branched cover when $2$ is a unit, and we show by an example that this lemma, and periodicity for the iterated double branched cover S[ [z,w] ]/(f+z2+w2), fail in characteristic two. We also compute the second syzygy over A of a maximal Cohen-Macaulay R-module by an explicit resolution, and show that the change of basis used there cannot be given by a universal matrix formula.
Related
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BibTeX
@misc{leuschke2026proof,
title = {{A new proof of characteristic-free Knörrer periodicity}},
author = {Graham J. Leuschke},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37839},
eprint = {2609.37839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37839},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A new proof of characteristic-free Knörrer periodicity
AU - Graham J. Leuschke
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.37839
UR - https://arxiv.org/abs/2609.37839
AB - Let R = S/(f) be a complete local hypersurface ring and let A = S[ [u,v] ]/(f+uv). Pfister and Popescu showed that the stable categories of maximal Cohen-Macaulay modules over R and over A are equivalent, with no assumption on the characteristic. We give a short proof. If (Φ,Ψ) is a matrix factorization of h and ∂ is a derivation, then (∂Ψ,∂Φ) is a null-homotopy for multiplication by ∂ h. Since u and v are partial derivatives of f+uv, they act as zero on the stable category of A, so the Koszul complex on u and v splits in the singularity category, and this gives density. For the double branched cover S[ [z] ]/(f+z^2) the same argument yields 2z: it recovers Knörrer's lemma up to free summands on the double branched cover when $2$ is a unit, and we show by an example that this lemma, and periodicity for the iterated double branched cover S[ [z,w] ]/(f+z^2+w^2), fail in characteristic two. We also compute the second syzygy over A of a maximal Cohen-Macaulay R-module by an explicit resolution, and show that the change of basis used there cannot be given by a universal matrix formula.
ER -