Gabriel Dimension

Let A be a complete lattice and let (Δ,a) ∈ Qdim(R,A) × A. We define a sequence α_Δ,a ∈ seq(A), called the Gabriel filtration of (Δ,a), as follows:

α_Δ,a (0) = a;

If 0 < i < o(A) is not a limit ordinal then α_Δ,a (i) = α_Δ,a(i-l) ∪ (∪{Δ(M) | M is an ε(Δ,α_Δ,a(i-l))-cocritical left R-module});

If 0 < i < o(A) is a limit ordinal then α_Δ,a (i) = ∪_j<i α_Δ,a(j).