Singularity categories of representations of algebras over local rings
Volume 161 / 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.
