Classicality of derived Emerton--Gee stack II: generalised reductive groups

We use the Tannakian formalism to define the Emerton–Gee stack for general groups. For a flat algebraic group G over Z_p, we are able to prove the associated Emerton–Gee stack is a formal algebraic stack locally of finite presentation over Spf(Z_p). We also define a derived stack of Laurent F-crystals with G-structure on the absolute prismatic site, whose underlying classical stack is proved to be equivalent to the Emerton–Gee stack. In the case of connected reductive groups, we show that the derived stack of Laurent F-crystals with G-structure is classical in the sense that when restricted to truncated animated rings, it is the étale sheafification of the left Kan extension of the Emerton–Gee stack along the inclusion from classical commutative rings to animated rings. Moreover, when G is a generalised reductive group, the classicality result still holds for a modified version of the Emerton–Gee stack. In particular, this completes the picture that the derived stack of local Langlands parameters for the Langlands dual group of a reductive group is classical.