The Hilbert Function of a Maximal Cohen-Macaulay Module

الأصل · EN

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if A is a hypersurface ring with dimension d > 0 then the Hilbert function of M is non-decreasing. If A = Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M). We analyze the case when equality holds and prove that in this case G(M) is CM. Furthermore in this case we also determine the Hilbert function of M. When A is Gorenstein then M is the first syzygy of Sᵃ(M) = (ᵃ₁(M*))*. A relation between the second Hilbert coefficient of M, A and Sᵃ(M) is found when G(M) is and G(A) ≥ d-1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyse when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.