Contributions to the Stable Derived Categories of Gorenstein Rings
Abstract
The stable derived category Dsg(R) of a Gorenstein ring R is the Verdier quotient of the bounded derived category Db(mod R) by the thick subcategory of perfect complexes, and was introduced by Buchweitz as a homological measure of the singularities of R. This thesis contributes to its study, centered around representation-theoretic and Koszul duality aspects. In Part I, we complete (over ℂ) the classification of homogeneous complete intersection isolated singularities R for which Dℤsg(R) (respectively, Db(coh X) for X = proj R) contains a tilting object. We then use results of Buchweitz-Iyama-Yamaura to describe the indecomposable objects in Dsgℤ(RY) for RY the coordinate rings of $4$ points on ℙ1 and $4$ points on ℙ2 in general position, and classify their Betti tables. In Part II, for R a Koszul Gorenstein algebra, we study a natural pair of subcategories whose intersection ℋlin(R) ⊆ Dsgℤ(R) consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if R is absolutely Koszul in the sense of Herzog-Iyengar, in which case Db(ℋlin(R)) ≅ Dsgℤ(R). We then relate the heart to modules over the Koszul dual R!. This extends the BGG correspondence to any pair of Koszul dual algebras (R, R!) with R absolutely Koszul Gorenstein, applying e.g. to elliptic normal curves and del Pezzo surfaces of degree ≥ 4. Finally, we relate our results to conjectures of Bondal and Minamoto on the graded coherence of AS-regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimensions ≥ 4.
Related
- On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II
- Hermitian K-theory for stable ∞-categories III: Grothendieck--Witt groups of rings
- Ranks of Verdier quotients of derived categories over local rings
- Almost Gorenstein standard graded rings and their localizations
- A Fitting criterion for quasi-Gorenstein local rings
BibTeX
@misc{gelinas2026contributions,
title = {{Contributions to the Stable Derived Categories of Gorenstein Rings}},
author = {Vincent Gélinas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32074},
eprint = {2610.32074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32074},
}
RIS
TY - UNPB
TI - Contributions to the Stable Derived Categories of Gorenstein Rings
AU - Vincent Gélinas
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32074
UR - https://arxiv.org/abs/2610.32074
AB - The stable derived category D_sg(R) of a Gorenstein ring R is the Verdier quotient of the bounded derived category D^b(mod R) by the thick subcategory of perfect complexes, and was introduced by Buchweitz as a homological measure of the singularities of R. This thesis contributes to its study, centered around representation-theoretic and Koszul duality aspects. In Part I, we complete (over ℂ) the classification of homogeneous complete intersection isolated singularities R for which D^ℤ_sg(R) (respectively, D^b(coh X) for X = proj R) contains a tilting object. We then use results of Buchweitz-Iyama-Yamaura to describe the indecomposable objects in D_sg^ℤ(R_Y) for R_Y the coordinate rings of $4$ points on ℙ^1 and $4$ points on ℙ^2 in general position, and classify their Betti tables. In Part II, for R a Koszul Gorenstein algebra, we study a natural pair of subcategories whose intersection ℋ^lin(R) ⊆ D_sg^ℤ(R) consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if R is absolutely Koszul in the sense of Herzog-Iyengar, in which case D^b(ℋ^lin(R)) ≅ D_sg^ℤ(R). We then relate the heart to modules over the Koszul dual R^!. This extends the BGG correspondence to any pair of Koszul dual algebras (R, R^!) with R absolutely Koszul Gorenstein, applying e.g. to elliptic normal curves and del Pezzo surfaces of degree ≥ 4. Finally, we relate our results to conjectures of Bondal and Minamoto on the graded coherence of AS-regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimensions ≥ 4.
ER -