Singularity categories of representations of algebras over local rings

Abstract

Let $Λ$ be a finite-dimensional algebra with finite global dimension, $R_k=K[X]/(X^k)$ be the $\mathcal{Z}$-graded local ring with $k\geq1$, and $Λ_k=Λ\otimes_K R_k$. We consider the singularity category $\mathcal{D}_{sg}(\mathrm{mod}^\mathcal{Z}(Λ_k))$ of the graded modules over $Λ_k$. It is showed that there is a tilting object in $\mathcal{D}_{sg}(\mathrm{mod}^\mathcal{Z}(Λ_k))$ such that its endomorphism algebra is isomorphic to the triangular matrix algebra $T_{k-1}(Λ)$ with coefficients in $Λ$ and there is a triangulated equivalence between $\mathcal{D}_{sg}(\mathrm{mod}^{\mathcal{Z}/k\mathcal{Z}}(Λ))$ and the root category of $T_{k-1}(Λ)$. Finally, a classification of $Λ_k$ 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