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Singularity categories of representations of algebras over local rings

Volume 161 / 2020

Ming Lu Colloquium Mathematicum 161 (2020), 1-33 MSC: Primary 16E45, 16E65, 18E35. DOI: 10.4064/cm7683-4-2019 Published online: 20 February 2020

Abstract

Let $\Lambda $ be a finite-dimensional algebra with finite global dimension, $R_k=K[X]/(X^k)$ be the $\mathbb Z $-graded local ring with $k\geq 1$, and $\Lambda _k=\Lambda \otimes _K R_k$. We consider the singularity category $\mathcal {D}_{\rm sg}(\operatorname{mod} ^\mathbb Z (\Lambda _k))$ of the graded modules over $\Lambda _k$. It is shown that there is a tilting object in $\mathcal {D}_{\rm sg}(\operatorname{mod} ^\mathbb Z (\Lambda _k))$ whose endomorphism algebra is isomorphic to the triangular matrix algebra $T_{k-1}(\Lambda )$ with coefficients in $\Lambda $ and there is a triangulated equivalence between $\mathcal {D}_{\rm sg}(\operatorname{mod} ^{\mathbb Z /k\mathbb Z }(\Lambda ))$ and the root category of $T_{k-1}(\Lambda )$. Finally, a classification of $\Lambda _k$ up to the Cohen–Macaulay representation type is given.

Authors

  • Ming LuDepartment of Mathematics
    Sichuan University
    Chengdu 610064, P.R. China
    e-mail

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