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Monadicity theorem and weighted projective lines of tubular type

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arxiv 1408.0028 v1 pith:X6H2UZQW submitted 2014-07-31 math.RA math.AGmath.RT

Monadicity theorem and weighted projective lines of tubular type

classification math.RA math.AGmath.RT
keywords abeliancategoriescategoryequivariantizationgradedequivalencesmodulemonadicity
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We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve, where the acting groups are cyclic and the two equivalences are somehow adjoint to each other.

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    Indecomposable rigid objects in the derived category of the (2,2,2,2)-weighted projective line are shown to correspond to graded simple arcs on a sphere with four binaries, with Hom-dimensions given by oriented inters...