About direct summands of projective modules over Laurent polynomial rings

Abstract:

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