Definition S. 4. Let S be a scheme. A single-sorted category scheme consists of: • an S-scheme C; • S-morphisms α, ω : C → C (the source, resp. target morphisms); • an S-morphism ◦ : C ×α,ω C → C (the composition morphism), such that the following diagrams (representing respectively the relations regarding the source and target of the unit, the source and target of the composition, the unit morphism, and associativity) commute.
Definition S. Let S be a scheme, and let a = (ai)s ∈ Seq be a finite sequence of
Definition S. 9. Let a, b ∈ Seq. The functor Modst is the functor Schop → Set that sends a scheme S to the set of OP1 -linear maps OP1 (a) → OP1 (b).
Definition S. S0. Let S be a scheme, let a Seq, and let Seq. A standard Γ-equivariant G-algebra of type a (resp. locally standard Γ-equivariant G-algebra of type in S ) over S is a Γ-equivariant G-algebra over OP1 which is standard of type a (resp. lo- cally standard of type in S ) as an OP1 -module. An equivalent description is the following. Note that an action of Γ on a standard algebra is given by a set ργ of endomor- phisms such that • ρ1 = id; • ργργ′ = ργγ′ for all γ, γ′ ∈ Γ. A morphism ϕ : OP1 (a) → OP1 (b) between standard algebras that have a Γ-action is Γ-equivariant if and only if for all γ ∈ Γ the following diagram commutes. OP1 (a) OP1 (b) ργ ργ OP1 (a) ϕ OP1 (b) Now a Γ-equivariant G-action on a standard Γ-algebra is given by a set rg of endomorphisms such that • r1 = id; • rgrg′ = rgg′ for all g, g′ ∈ G; • rγg = ργrgρ 1 for all γ ∈ Γ, g ∈ G. A Γ-equivariant morphism ϕ : OP1 (a) → OP1 (b) between standard Γ-equivariant G-algebras is G-equivariant if and only if for all g G the following diagram com- mutes. OP1 (a) OP1 (b) rg rg OP1 (a) ϕ OP1 (b)
Definition S. S1. Let Seq. The category scheme Γ, G - Algst of locally standard Γ-equivariant G-algebras of type in S is the functor Schop → Cat senSding a scheme S to the category of locally standard Γ-equivariant G-algebras over S of type in S . As usual, we define the corresponding auxiliary functor in order to show that (Γ, G)- Algst is representable. b,a
Definition S. S2. Let a, b ∈ Seq. The functor (Γ, G)- Algst is the functor from Schop to b,a Set sending a scheme S to the set of (r′g ), (ργ′ ), ϕ, (rg), (ργ) such that ϕ ∈ Algst (S), the tuples (rg) and (ργ) define a Γ-equivariant G-action on the source of ϕ, and the tuples (r′g ) and (ργ′ ) define a Γ-equivariant G-action on the target of ϕ. b,a
Definition S. 60. Let R be a set of non-negative integers. The category scheme Etproj of standard finite e´tale Γ-equivariant G-schemes over X with degree in D is the functor Schop → Cat sending a k-scheme S to the subcategory of Flatproj(S) of finite e´tale morphisms Y → X of standard schemes over S, with degree in D. We show that Etproj is indeed a category scheme.
Definition S. Let S ⊆ Seq. The category scheme Flataff is the functor Schop X,S -scheme S to the category S k → Cat that sends a k
Definition S. Let V = {0, 1, 2...v − 1} be a set of v elements and B = B0, B1, B2...Bb−1 be a set of b blocks, where Bi is a subset of V and Bi = k. For a finite incidence structure σ = (V, B), if σ satisfies the following conditions, then it is a BIBD, which is called a (b, v, r, k, λ)-design.
Definition S. Unless otherwise expressly provided herein, the following words and phrases, when used in this Declaration shall be deemed to have the following meanings.
