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

Lukas Katthän, Julio José Moyano‐Fernández and Jan Uliczka

Mathematische Nachrichten, 2020, vol. 293, issue 1, 129-146

Abstract: Consider a polynomial ring R with the Zn‐grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of n projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated R‐modules. We also 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.

Date: 2020
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https://doi.org/10.1002/mana.201800436

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