Group Schemes Flat of Finite Type

Let G be a flat 5-group scheme of finite type. Note that G is faithfully flat over 5. A G-scheme is an 5-scheme with a left G-action by definition. Set Ag to be the category of noetherian G-schemes and G-morphisms separated of finite type. 

Note that (Bg(- )OI(A) is canonically isomorphic to Bg G x X), where G x X is viewed as a principal G-action. Note also that there is an isomorphism from BaiXy to Nerve(p2 G x X X) given by 

Lemma 29.3. Let G and X be as above. Then Bg X) is a simplicial S-groupoid with do(l) and di(l) faithfully flat of finite type. 

We denote the category of (G, Gx)-modules by Mod(G, X). The category of equivariant (resp. locally quasi-coherent, quasi-coherent, coherent) (G, Gx)-modules is denoted by EM(G, X) (resp. Lqc(G, X), Qch(G, X), Coh(G,X)). 

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