Transfer of Wakamatsu tilting modules along Frobenius extensions

Let $\iota :R\to A$ be a Frobenius extension and let $T$ be a Wakamatsu tilting left $R$-module. We give sufficient conditions for the induced module $A{\otimes }_{R}T$ to remain Wakamatsu tilting and establish an ascent–descent result for split centrally projective Frobenius extensions. We also characterize the tilting case by the vanishing of ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^i(T,A{\otimes }_{R}T)$ for all $i>0$. Moreover, if $A{\otimes }_{R}T\in {\mathrm{a}\mathrm{d}\mathrm{d}}_R(T)$, then the natural map $S={\mathrm{E}\mathrm{n}\mathrm{d}}_R(T)\to B={\mathrm{E}\mathrm{n}\mathrm{d}}_A(A{\otimes }_{R}T)$ of endomorphism rings is a Frobenius extension and $T{\otimes }_{S}B\cong A{\otimes }_{R}T$ as $R$-$B$-bimodules. Applications to Brenner–Butler–Miyashita equivalences and some specific classes of Frobenius extensions are also discussed.