On Sigma(1)(1)-completeness of quasi-orders on kappa(kappa)

We prove under V = L that the inclusion modulo the non-stationary ideal is a Sigma(1)(1)-complete quasi-order in the generalized Borel-reducibility hierarchy (kappa > omega). This improvement to known results in L has many new consequences concerning the Sigma(1)(1)-completeness 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 Delta(1)(1), then it is Sigma(1)(1)-complete.

We also study the case V not equal L and prove Sigma(1)(1)-completeness results for weakly ineffable and weakly compact kappa.