Effective normality criteria for algebras of linear type

The algebras studied here are subalgebras of rings of polynomials generated by 1-forms (so-called Rees algebras), with coefficients in a Noetherian ring. Given a normal domain R and a torsionfree module E with a free resolution, , we study the role of the matrices of syzygies in the normality of the Rees algebra of E. When the Rees algebra and the symmetric algebra S(E) coincide, the main results characterize normality in terms of the ideal Ic(ψ)S(E) and of the completeness of the first s symmetric powers of E, where c=rankψ, and s=rankF0−rankE. It requires that R be a regular domain. Special results, under broader conditions on R, are still more effective.