  n-recollements of triangulated categories and n-derived-simple algebras are
introduced. The relations between the n-recollements of derived categories of
algebras and the Cartan determinants, homological smoothness and Gorensteinness
of algebras respectively are clarified. As applications, the Cartan determinant
conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry
conjecture is reduced to 2-derived-simple algebras.
