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

Let $R$ be an Artin algebra and $e$ an idempotent of $R$. Assume that ${\rm Tor}_i^{eRe}(Re,G)=0$forany$G\in{\rm GProj} eRe$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 $\underline{{\rm GProj} R}\simeq\underline{{\rm GProj} eRe}$\mathrm{.}Applyingthesetothetriangularmatrixalgebra$T=\left( \begin{array}{cc} A & M \quad 0 & B \end{array} \right)$,correspondingresultsbetweencandidatecategoriesof$T$and$A$(resp\mathrm{.}$B$) 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.