Free resolutions over short local rings
Luchezar L. Avramov, Srikanth B. Iyengar, Liana M. Sega
TL;DR
This paper investigates the structure of minimal free resolutions over certain local rings, identifying conditions for Koszul modules, classifying non-Koszul modules in Gorenstein cases, and describing Ext algebra structures.
Contribution
It provides new classifications of Koszul and non-Koszul modules over local rings with m^3=0, and describes the Ext algebra structures in these contexts.
Findings
Over generic rings, every module has a Koszul syzygy.
Explicit families of Koszul modules are identified.
Non-Koszul modules are classified when the ring is Gorenstein.
Abstract
The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k).
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Algebraic structures and combinatorial models