Reduction techniques for homological conjectures
The aim of this article is to show that using the language of triangulated categories certain reduction techniques can be obtained which apply to the Finitistic Dimension Conjecture, the Vanishing Conjecture, and the Generalized Nakayama Conjecture. More precisely, it is shown that if \(A\) is a finite dimensional algebra, \(T\) a tilting module, and \(B = \text{End}_ A T\), then \(fd(A) < \infty\) if and only if \(fd(B) < \infty\). It is also shown that if \(D^ b (A)\), the derived category of bounded complexes, has recollement relative to \(D^ b (A')\) and \(D^ b(A'')\) for some finite dimensional algebras \(A'\), \(A''\), then \(fd(A) < \infty\) if and only if \(fd(A')\) and \(fd(A'') < \infty\).
scientific article; zbMATH DE number 515577