On derived equivalences and homological dimensions

Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.

ON GRADED SYMMETRIC CELLULAR ALGEBRAS

Let $A=\bigoplus _{i\in \mathbb{Z}}A_{i}$ be a finite-dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that if $d\neq 0$ then $A_{-d}$ contains the Higman ideal $H(A)$ and $\dim H(A)\leq \dim A_{0}$, and provide a semisimplicity criterion for $A$ in terms of the centralizer of $A_{0}$.

On the radical of the group algebra of a symmetric group

Let [Formula: see text] with [Formula: see text] a prime and [Formula: see text] a symmetric group. We prove in this paper that if [Formula: see text], then [Formula: see text], where [Formula: see text] is the nilpotent ideal constructed in [Radicals of symmetric cellular algebras, Collog. Math. 133 (2013) 67–83]. Finally we give two remarks on algebras [Formula: see text] with [Formula: see text].