On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II
Abstract
Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let underlineCM(A) be its stable category of finitely generated maximal Cohen-Macaulay A-modules. Suppose we have an equivalence Φ:underlineCM(A) → underlineCM(A) as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then dim A = dim B. (2) If M is a maximal \CM \ A-module then curvA(M) = curvB (Φ(M)), here curvA(M) = limsupn √[n]ℓ(TorAn(M,k)). (3) A satisfies Serre's condition Ri if and only if B satisfies Ri. (4) A is a complete intersection on the punctured spectrum of A if and only if B is a complete intersection on the punctured spectrum of B. We also show that Φ imposes constraints of residue field of B in terms of residue field of A and vice-versa. Finally if I is an ideal in A such that the extended Rees algebra ℛ(I) = A[It, t−1] is Gorenstein then we construct a triangulated functor Ψ: underlineCMZ(ℛ(I)) → underlineCM(A) where underlineCMZ(ℛ(I)) is the stable category of all graded maximal Cohen-Macaulay ℛ(I)-modules. We show that Ψ induces an equivalence underlineCMZ(ℛ(I))/ker Ψ→ underlineCM(A). We give some applications of this map.
Related
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- Regularity and v-number of syzygies of graded maximal Cohen-Macaulay modules
- Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
- Adjoint test modules along Cohen--Macaulay morphisms
- Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension
BibTeX
@misc{puthenpurakal2026stable,
title = {{On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II}},
author = {Tony J. Puthenpurakal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18626},
eprint = {2609.18626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18626},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II
AU - Tony J. Puthenpurakal
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18626
UR - https://arxiv.org/abs/2609.18626
AB - Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let underlineCM(A) be its stable category of finitely generated maximal Cohen-Macaulay A-modules. Suppose we have an equivalence Φ:underlineCM(A) → underlineCM(A) as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then dim A = dim B. (2) If M is a maximal \CM \ A-module then curv_A(M) = curv_B (Φ(M)), here curv_A(M) = limsup_n √[n]ℓ(Tor^A_n(M,k)). (3) A satisfies Serre's condition R_i if and only if B satisfies R_i. (4) A is a complete intersection on the punctured spectrum of A if and only if B is a complete intersection on the punctured spectrum of B. We also show that Φ imposes constraints of residue field of B in terms of residue field of A and vice-versa. Finally if I is an ideal in A such that the extended Rees algebra ℛ(I) = A[It, t^−1] is Gorenstein then we construct a triangulated functor Ψ: underlineCM^Z(ℛ(I)) → underlineCM(A) where underlineCM^Z(ℛ(I)) is the stable category of all graded maximal Cohen-Macaulay ℛ(I)-modules. We show that Ψ induces an equivalence underlineCM^Z(ℛ(I))/ker Ψ→ underlineCM(A). We give some applications of this map.
ER -