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Connections on modules over quasi-homogeneous plane curves

Eivind Eriksen
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Abstract

Let k be an algebraically closed field of characteristic 0, and let $A = k[x,y]/(f)$ be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e. a graded A-linear homomorphism $\nabla: \operatorname{Der}_k(A) \to \operatorname{End}_k(M)$ that satisfy the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring $B = \hat A$ admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module.

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Eivind Eriksen. 2007-08-01. Connections on modules over quasi-homogeneous plane curves. https://doi.org/10.1080/00927870802110797

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