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Free extensions and Lefschetz properties, with an application to rings of relative coinvariants. (arXiv:1807.05869v1 [math.AC])
来源于:arXiv
Graded Artinian rings can be regarded as algebraic analogues of cohomology
rings (in even degrees) of compact topological manifolds. In this analogy, a
free extension of rings corresponds to a fiber bundle of manifolds. For rings,
as with manifolds, it is a natural question to ask: to what extent do
properties of the base and the fiber carry over to the extension ring? For
example if the base and fiber both satisfy a strong Lefschetz property, can we
conclude the same for the extension? We address these questions using certain
relative coinvariant rings as a prototypical model. We show in particular that
if the subgroup $W$ of the general linear group $Gl (V)$, $V$ a vector space,
is a non-modular finite reflection group and $K\subset W$ is a non parabolic
reflection subgroup, then the relative coinvariant ring $R^K_W$ does not
satisfy the strong Lefschetz property. We give many examples, including those
of relative coinvariant rings with non-unimodal Hilbert functions, and pose
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