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Which series are Hilbert series of graded modules over standard multigraded polynomial rings?
Author(s) -
Katthän Lukas,
MoyanoFernández Julio José,
Uliczka Jan
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800436
Subject(s) - mathematics , laurent series , hilbert–poincaré series , pure mathematics , hilbert series and hilbert polynomial , polynomial ring , series (stratigraphy) , laurent polynomial , hilbert scheme , formal power series , polynomial , algebra over a field , power series , hilbert space , mathematical analysis , paleontology , biology
Consider a polynomial ring R with the Z n ‐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.