On Σ ¹ ₁-completeness of quasi-orders on

We prove under V = L that the inclusion modulo the non-stationary ideal is a Σ ¹ ₁-complete quasi-order in the generalized Borel-reducibility hierarchy (κ > ω). This improvement to known results in L has many new consequences concerning the Σ ¹ ₁completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in the literature. Additionally the theorem is applied to prove a dichotomy in L: If the isomorphism of a countable first-order theory (not necessarily complete) is not ∆ ¹ ₁, then it is Σ ¹ ₁-complete. We also study the case V 6= L and prove Σ ¹ ₁-completeness results for weakly ineffable and weakly compact κ.