We propose to apply the recently introduced local projection stabilization to the numerical computation of the Oseen equation at high Reynolds number. The discretization is done by nested finite element spaces. Using a priori error estimation techniques, we prove the convergence of the method. The a priori estimates are independent of the local Peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth. Moreover, we show how this method may be cast in the framework of variational multiscale methods. We indicate what modeling assumptions must be made to use the method for large eddy simulations.