论文标题
当Schur函子诱导Gorenstein缺陷类别之间的三角等效
When the Schur functor induces a triangle-equivalence between Gorenstein defect categories
论文作者
论文摘要
让$ r $为artin代数，$ e $ $ r $。假设$ {\ rm tor} _i^{erere}（re，g）= 0 $对于{\ rm gproj} in {\ rm gproj} erere $和$ i $的任何$ g \。为Schur functor $ s_e $提供了必要和足够的条件，以诱导三角形等效$ \ Mathbb {d} _ {def}（r）\ simeq \ simeq \ mathbb {d} _ {def} _ {def}（erere）$。结合psaroudakis-skartsaterhagen-solberg [29]的结果，我们为奇异等价性提供必要的条件，$ \ mathbb {d} _ {sg} _ {sg}（r）\ simeq \ simeq \ simeq \ mathbb {d} _ {d} _ {sg} _ {sg}（sg}（eere） gproj} r} \ simeq \ lisesline {{\ rm gproj} erere} $。将它们应用于三角矩阵代数$ t = \ left（ \ begin {array} {cc} A＆M \ Quad 0＆b \ end {array} \ right）$，获得$ t $和$ a $（分别$ b $）的候选类别之间的相应结果。结果，我们从$ a $ a $（分别$ b $）的gorensteinness和$ t $的CM f $中。给出了一些具体的例子，以表明人们可以意识到三角基质代数的戈伦斯坦缺陷类别是其角algabras之一的奇异性类别。
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.
