The Bernstein Inequality for Formal Power Series

Abstract

n this thesis, we give a short proof of the Bernstein inequality for the ring of formal power series. Let R = k[[x1,...,xn]] be the ring of formal power series over a field of characteristic zero k and let D(R, k) stand for the ring of k−linear differential operators in R. If M is a finitely generated D(R,k)−module then d(M) ≥ n where d(M) is defined to be the Krull dimension of the graded ring grΣ(D(R,k))/Ann(grΓ(M)) in which Σ = Σ0 ⊆ Σ1 ⊆ Σ2 ⊆ ··· is the standard filtration on D(R,k) and Γ = Γ0 ⊆ Γ1 ⊆ Γ2 ⊆ ··· is any good filtration on M. This is a celebrated inequality called the Bernstein inequality. The Bernstein inequality was originally proved by I. N. Bernstein [2] for the ring of poly- nomials by a beautiful short argument. It was extended to formal power series by J, E, Bjo ̈rk [3]. His proof is far from simple. In chapter 1 we reproduce Bernstein’s simple proof for the ring of polynomials and we are going to see that the proof is short and simple. Unfortunately this simple proof does not extend to formal power series since the Bernstein filtration in the ring of formal power series has quite more difficult structure compared to that of ring of polynomials. To be more precise the Bernstein filtration in the ring of polynomials T = k[x1,...,xn] is Ξ = Ξ0 ⊆ Ξ1 ⊆ Ξ2 ⊆ ··· in whichΞ isthek−vectorspacegeneratedbymonomialsoftheformxα1 ···xα1∂β1 ···∂βn oftotal j 111n degree j where ∂i = ∂ /∂ xi : T → T whereas the Bernstein filtration in the ring of formal power series is Σ = Σ0 ⊆ Σ1 ⊆ Σ2 ⊆ ··· in which Σj is the left R−module generated by monomials of theformdα1 ···dαn oftotaldegree jinwhichd =∂/∂x :R→R. ClearlytheBernsteinfiltration 1nii for the ring of polynomials is of simpler structure. In chapter 2 we give Bjo ̈rk’s proof of the Bernstein inequality and as we will see, the proof is utterly difficult as it is comprised of many elegant homological algebra tools. In the third chapter a new and short proof for the Bernstein inequality for formal power series is given which is inspired by [6] and is of elementary nature.