Moduli stacks of Lubin--Tate $(\phi ,\mathrm{\Gamma })$-modules

We define and study stacks which parametrize Lubin–Tate $(\phi ,\mathrm{\Gamma })$-modules. By working at a perfectoid level, we compare these with the Emerton–Gee stacks of cyclotomic $(\phi ,\mathrm{\Gamma })$-modules. As a consequence, we deduce perfectness of the Herr complex in the Lubin–Tate setting.