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arXiv · 2003.06782

When the Schur functor induces a triangle-equivalence between Gorenstein defect categories

Huanhuan Li·Jiangsheng Hu·Yuefei Zheng
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Abstract

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$ for any $G\in{\rm GProj} eRe$ and $i$ sufficiently large. Necessary and sufficient conditions are given for the Schur functor $S_e$ to induce a triangle-equivalence $\mathbb{D}_{def}(R)\simeq\mathbb{D}_{def}(eRe)$. Combine this with a result of Psaroudakis-Skartsaterhagen-Solberg [29], we provide necessary and sufficient conditions for the singular equivalence $\mathbb{D}_{sg}(R)\simeq\mathbb{D}_{sg}(eRe)$ to restrict to a triangle-equivalence $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj} eRe}$. Applying these to the triangular matrix algebra $T=\left( \begin{array}{cc} A & M \quad 0 & B \end{array} \right)$, corresponding results between candidate categories of $T$ and $A$ (resp. $B$) are obtained. As a consequence, we infer Gorensteinness and CM-freeness of $T$ from those of $A$ (resp. $B$). Some concrete examples are given to indicate one can realise the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algabras.

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Huanhuan Li, Jiangsheng Hu, Yuefei Zheng. 2020-03-15. When the Schur functor induces a triangle-equivalence between Gorenstein defect categories. https://doi.org/10.1007/s11425-021-1899-3

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