Semiampleness for Calabi--Yau surfaces in positive and mixed characteristic

In this note we prove the semiampleness conjecture for klt Calabi–Yau surface pairs over an excellent base ring. As applications we deduce that generalised abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for Calabi–Yau threefolds over a mixed characteristic DVR.