Hammock coordinates for silting subcategories of derived-discrete algebras
Abstract
We introduce hammock coordinates to study the combinatorial structure of partial silting configurations in the bounded derived category of a derived-discrete algebra Λ(r,n,m). By extracting exact integer inequalities from the Bobiński--Geiß--Skowroński Hom-hammock formula, we establish a closed-form criterion for pairwise partial silting both within any 𝒵-component and across distinct components. Using this coordinate framework, we establish three primary results. First, we prove the universal upper bound |S| ≤ n+m on the cardinality of any partial silting configuration S of 𝒵-objects ---both globally via Bongartz completion in the silting-discrete setting, and combinatorially within a single component via a Σr-invariant staircase argument. Second, we prove that this bound is universally sharp by constructing an explicit L-shaped partial silting configuration of size n+m for every Λ(r,n,m), alongside specialized diagonal configurations. Third, we show that the configuration lattice carries a faithful action of the cyclic group ℤ/g where g = gcd(r+m, n−r). We conclude by showing a structural obstruction to local determination: pairwise partial silting of consecutive objects in the lattice ordering does not imply global partial silting, due to the strict additivity of a coordinate defect along chains.
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BibTeX
@misc{sardar2026hammock,
title = {{Hammock coordinates for silting subcategories of derived-discrete algebras}},
author = {Shantanu Sardar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03204},
eprint = {2611.03204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03204},
}
RIS
TY - UNPB
TI - Hammock coordinates for silting subcategories of derived-discrete algebras
AU - Shantanu Sardar
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03204
UR - https://arxiv.org/abs/2611.03204
AB - We introduce hammock coordinates to study the combinatorial structure of partial silting configurations in the bounded derived category of a derived-discrete algebra Λ(r,n,m). By extracting exact integer inequalities from the Bobiński--Geiß--Skowroński Hom-hammock formula, we establish a closed-form criterion for pairwise partial silting both within any 𝒵-component and across distinct components. Using this coordinate framework, we establish three primary results. First, we prove the universal upper bound |S| ≤ n+m on the cardinality of any partial silting configuration S of 𝒵-objects ---both globally via Bongartz completion in the silting-discrete setting, and combinatorially within a single component via a Σ^r-invariant staircase argument. Second, we prove that this bound is universally sharp by constructing an explicit L-shaped partial silting configuration of size n+m for every Λ(r,n,m), alongside specialized diagonal configurations. Third, we show that the configuration lattice carries a faithful action of the cyclic group ℤ/g where g = gcd(r+m, n−r). We conclude by showing a structural obstruction to local determination: pairwise partial silting of consecutive objects in the lattice ordering does not imply global partial silting, due to the strict additivity of a coordinate defect along chains.
ER -