ON THE RATIONALITY OF POINCARÉ SERIES OF GORENSTEIN ALGEBRAS VIA MACAULAY'S CORRESPONDENCE

Let A be a local Artinian Gorenstein algebra with maximal ideal M, P A ( z ) : = ∑ p = 0 ∞ ( T o r p A ( k , k ) ) z p its Poicaré series. We prove that PA(z) is rational if either dimk(M2/M3) ≤ 4 and dimk(A) ≤ 16, or there exist m ≤ 4 and c such that the Hilbert function HA(n) of A is equal to m fo...

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Bibliographic Details
Published in:The Rocky Mountain journal of mathematics 2016-01, Vol.46 (2), p.413-433
Main Authors: CASNATI, GIANFRANCO, JELISIEJEW, JOACHIM, NOTARI, ROBERTO
Format: Article
Language:English
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Summary:Let A be a local Artinian Gorenstein algebra with maximal ideal M, P A ( z ) : = ∑ p = 0 ∞ ( T o r p A ( k , k ) ) z p its Poicaré series. We prove that PA(z) is rational if either dimk(M2/M3) ≤ 4 and dimk(A) ≤ 16, or there exist m ≤ 4 and c such that the Hilbert function HA(n) of A is equal to m for n ∈ [2, c] and equal to 1 for n = c + 1. The results are obtained due to a decomposition of the apolar ideal Ann(F) when F = G + H and G and H belong to polynomial rings in different variables.
ISSN:0035-7596
1945-3795
DOI:10.1216/RMJ-2016-46-2-413