Research proof index

If φ : 𝔸_K^f ⊗ V → 𝔸_K^f ⊗ V is 𝔸_K^f-linear and φₚ is its local component at a place p then for all x : 𝔸_K^f ⊗ V we have (evalₚ ⊗ id_V) (φ x) = φₚ ((evalₚ ⊗ id_V) x), or, more colloquiually, (φ x)ₚ = φₚ (xₚ).

The canonical map L ⊗[K] K_v → ∏_{w|v} L_w is surjective.

L ⊗[K] K_v and ∏_{w|v} L_w have equal dimensions

The closure of A in K_v is 𝒪_v.

The closure of A in ∏_{v ∈ s} K_v is ∏_{v ∈ s} 𝒪_v. s may be infinite.

If s is finite then K in dense in ∏_{v ∈ s} K_v.