Singularity categories of representations of algebras over local rings

Abstract

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

27 pages, a dozen of figures. The structure of this paper is modified deeply. arXiv admin note: text overlap with arXiv:1103.3335 and arXiv:1201.5487 by other authors

References in corpus (3)

• On triangulated orbit categories
• Singularity categories of Gorenstein monomial algebras
• Gorenstein projective bimodules via monomorphism categories and filtration categories

Cited by in corpus (4)

• Singularity categories of Gorenstein monomial algebras
• Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras
• On Cohen-Macaulay Auslander algebras
• Singularity categories of derived categories of hereditary algebras are derived categories