Laurent coefficients and Ext of finite graded modules

1994/09/23 by Luchezar L. Avramov, Luchézar L. Avramov, Ragnar-Olaf Buchweitz +4
Chemistry · Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Chemistry #Combinatorics #Commutative Algebra and Its Applications #Commutative property #Commutative ring #Discrete mathematics #Field (mathematics) #Finite field #Graded ring #Hilbert series and Hilbert polynomial #Hilbert space #Homogeneous #Laurent series #Mathematics #Pure mathematics #Rank (graph theory) #Ring (chemistry) #math.AC

paper · pdf · doi:10.48550/arxiv.math/9409208

arxiv created 1994/09/23 · openalex publication_date 1994/09/23 · arxiv updated 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let R=\bigoplus_n\ges0R_n be a graded commutative ring generated over a field K=R₀ by homogeneous elements x₁,…,x_e of positive degrees d₁,…,d_e. The Hilbert-Serre Theorem shows that for each finite graded R--module M=\bigoplus_n∈\BZM_n the \it Hilbert series\/ ∑_n∈\BZ(\rank_K M_n)tⁿ is the Laurent expansion around 0 of a rational function H_M(t)=\fracq_M(t)∏_i=1^e(1-t^d_i) with q_M(t)∈\BZ[t,\ti]. We demonstrate that Laurent expansions [M]_z of H_M(t) around other points z of the extended complex plane \BC also carry important structural information.

Related

• The Hilbert Series of a Linear Symplectic Circle Quotient
• An Upper Bound for the First Hilbert Coefficient of Gorenstein Algebras and Modules
• Bounds for Hilbert coefficients
• The equivariant Hilbert series of the canonical ring of Fermat curves
• Hilbert regularity of ZZ-graded modules over polynomial rings