In their work on the period map and the dualising sheaf for Lubin–Tate space, Gross and the second author wrote down an equivalence between the Spanier–Whitehead and Brown–Comenetz duals of certain type n-complexes in the K(n)-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin–Tate space, two important invertible sheaves become isomorphic modulo p.