On completion of graded D-modules

2018/08/27 by Switala, Nicholas, Zhang, Wenliang
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.09035

Abstract

Let R = k[x₁, …, x_n] be a polynomial ring over a field k of characteristic zero and \cR be the formal power series ring k[[x₁, …, x_n]]. If M is a \D-module over R, then \cR ⊗_R M is naturally a \D-module over \cR. Hartshorne and Polini asked whether the natural maps Hⁱ_\dR(M)→ Hⁱ_\dR(\cR ⊗_R M) (induced by M→ \cR ⊗_R M) are isomorphisms whenever M is graded and holonomic. We give a positive answer to their question, as a corollary of the following stronger result. Let M be a finitely generated graded \D-module: for each integer i such that dim_kHⁱ_\dR(M)

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