Bounds for multiplicities

Let R = K[x(1), x(2), ..., x(n)] and S = R/I be a homogeneous K-algebra. We establish bounds for the multiplicity of certain homogeneous K-algebras S in terms of the shifts in a free resolution of S over R. Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasipure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals I. As a consequence, we get a bound for the regularity of S. Further, when S is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a p-linear resolution.