Let \(u:A\rightarrow B\) be a morphism of noetherian rings of prime characteristic \(p>0\) and \(\omega_{B/A}:A^{(p)}\otimes_AB\rightarrow B^{(p)}\) be the induced morphism, where \(A^{(p)}\) means the \(A\)-algebra given by the Frobenius morphism of \(A\). The aim of the paper is to investigate the connections between the injective dimension of \(\omega_{B/A}\) and the regularity of \(u\). It is shown that for a large class of rings (including complete local rings and algebras essentially of finite type over a field) if \(u\) is flat and \(\omega_{B/A}\) has finite injective dimension, then \(u\) is regular.