About direct summands of projective modules over Laurent polynomial rings

Abstract

Suppose A A is a local ring and R = A [ X , X − 1 ] R = A[X,{X^{ - 1}}] is a Laurent polynomial ring. We prove that for projective R R -modules P P and Q Q with rank Q > Q > rank P P , if Q f {Q_f} is a direct summand of P f {P_f} for a doubly monic polynomial f f then Q Q is also a direct summand of P P . We also prove the analogue of the Horrock’s theorem for Laurent polynomials rings.