scholarly journals An inequality for Hilbert series of graded algebras.

1985 ◽  
Vol 56 ◽  
pp. 117 ◽  
Author(s):  
Ralf Fröberg
Keyword(s):  
Hilbert Series ◽  
Graded Algebras

scholarly journals Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 881 ◽  
Author(s):  
Oswaldo Lezama ◽  
Jaime Gomez
Keyword(s):  
Mirror Symmetry ◽  
Hilbert Series ◽  
Poincaré Series ◽  
Graded Rings ◽  
Yoneda Algebra ◽  
Graded Algebras ◽  
Poincare Series ◽  
Rings And Algebras ◽  
Lattice Of Ideals ◽  
Point Modules

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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