Homological invariants under Frobenius extensions

Let $A/S$ be a Frobenius extension of artin algebras such that $S$ is commutative and $A$ is an $S$-algebra. We prove that if $(C,T)$ is a tilting pair of right $S$-modules, then $(C\mathbin{\otimes_{S}}A,T\mathbin{\otimes_{S}}A)$ is a tilting pair of right $A$-modules; conversely, if $(C,T)$ is a tilting pair of right $A$-modules, then $(C,T)$ is also a tilting pair of right $S$-modules. We also prove that the so-called $(l,n)$-condition and certain classes of algebras are preserved under right-split or separable Frobenius extensions. Finally, we prove that the validity of some homological conjectures is preserved under (separable) Frobenius extensions.