One of the major questions over a field of mixed characteristics is, "Is there a divisor D such that the pair (R, D) is BCM-regular when R is a splinter?" Inspired by this, we introduce compatible systems of divisors on finite covers of R, which have data of divisors on all finite covers of the ring. With this, we will define +-test/adjoint ideals for the system of divisors, and discuss the behavior of the +-test ideals including the inverse of adjunction. This is a joint work with Karl Schwede.
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