# Finitistic dimension equal to cofinistic dimension for QF-3?

Given a finite dimensional algebra A such that the regular module embeds into a projective-injective module (such algebras are called QF-3 algebras and generalise Frobenius algebras).

Define the finitistic dimension findim(A) of A as the supremum of projective dimensions of modules having finite projective dimension.

Question:Is the finitistic dimension of $$A$$ equal to the finitistic dimension of the opposite algebra $$A^{op}$$?

I see no reason, but some random tests gave no counterexamples to my surprise.

• If you're considering left $A$-modules, then by duality your co-finitistic dimension should coincide with the right finitisitic dimension of $A$. There are many examples where the left and right finitistic dimension are different, but I don't know yet if any satisfy your additional condition. – Alex Dugas Jan 30 '17 at 21:57