We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a K[X1,…,Xr]-module M of finite dimension D as a K-vector space, and given elements f1,…,fm in M, the problem is to compute syzygies between the fi’s, that is, polynomials (p1,…,pm) in K[X1,…,Xr]m such that p1f1+⋯+pmfm=0 in M. Assuming that the multiplication matrices of the r variables with respect to some basis of M are known, we give an algorithm which computes the reduced Gröbner basis of the module of these syzygies, for any monomial order, using O(mDω−1+rDωlog(D)) operations in the base field K, where ω is the exponent of matrix multiplication. Furthermore, assuming that M is itself given as M=K[X1,…,Xr]n∕N, under some assumptions on N we show that these multiplication matrices can be computed from a Gröbner basis of N within the same complexity bound. In particular, taking n=1, m=1 and f1=1 in M, this yields a change of monomial order algorithm along the lines of the FGLM algorithm with a complexity bound which is sub-cubic in D.