Singularity categories of representations of algebras over local rings

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 $k\geq1$, 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.