2008/12/31 by Lars Winther Christensen, Janet Striuli, Oana Veliche · 1 citation
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics #Commutative Algebra and Its Applications #Commutative property #Discrete mathematics #Field (mathematics) #Injective function #Local ring #Mathematics #Noetherian #Noetherian ring #Pure mathematics #Residue field #Resolution (logic) #Ring (chemistry) #math.AC #msc:13D02 #msc:13D07 #msc:13H10
paper · pdf · doi:10.1112/jlms/jdp058
Final version, to appear in J. London Math. Soc.; 21 pp
arxiv created 2009/07/06 · openalex publication_date 2009/11/25 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module ExtiR(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero.