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arXiv · 1603.06215

Which series are Hilbert series of graded modules over polynomial rings?

Lukas Katthän·Julio José Moyano-Fernández·Jan Uliczka
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Abstract

Let $S$ be a multigraded polynomial ring such that the degree of each variable is a unit vector; so $S$ is the homogeneous coordinate ring of a product of projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated $S$-modules. Also we provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive depth.

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Lukas Katthän, Julio José Moyano-Fernández, Jan Uliczka. 2016-03-20. Which series are Hilbert series of graded modules over polynomial rings?. https://doi.org/10.1002/mana.201800436

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