We give a description of the prime spectrum of the localization of the Laurent polynomial ring R[X, X^?1] at doubly monic polynomials using its interactions with the rings and respectively the localizations of the rings R[X], R[X^?1],and R[X^?1 + X] at monic polynomials. We also give the analogue of Horrock?s theorem for Laurent polynomial rings. In fact, we provide a process to systematically translate the results related to Serre?s conjecture and modules over R[X₁, ?, X_n] to modules over the multivariate Laurent polynomial ring

localization of polynomial ring, Krull dimension, valuative dimension, projective modules.