标题： 导出等价，矩阵等价和同调猜想 显示英文标题

标题： Derived equivalences, matrix equivalences, and homological conjectures

摘要： 中心化矩阵代数最初由费迪南德·弗罗贝尼乌斯在1877年左右的《克莱尔杂志》中进行了研究。 通过引入域上所有方阵上的三个新的等价关系，我们分别用这些矩阵等价关系完全描述了中心化矩阵代数之间的Morita等价、导出等价和几乎$\nu$-稳定导出等价。 因此，范畴等价被简化为线性代数中的矩阵等价。 此外，我们证明了置换矩阵的中心化矩阵代数之间的导出等价会同时诱导出$p$-正则部分和$p$-奇异部分的Morita等价和额外的导出等价。 作为应用，我们证明了有限维猜想和中岛猜想对于中心化矩阵代数是成立的。 显示英文摘要

摘要： Centralizer matrix algebras were investigated initially by Georg Ferdinand Frobenius in the Crelle's Journal around 1877. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost $\nu$-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Thus the categorical equivalences are reduced to matrix equivalences in linear algebra. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for $p$-regular parts and for $p$-singular parts. As applications, we show that the finitistic dimension conjecture and Nakayama conjecture are valid for centralizer matrix algebras.