2004/09/10 by Zhaoyong Huang, Huang, Zhaoyong · 1 citation
Mathematics · #16E30 #16E65 #16P40 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16E30 #msc:16E65 #msc:16P40
paper · pdf · doi:10.48550/arxiv.math/0409174
13 pages
arxiv created 2004/09/10 · arxiv updated 2009/12/01
Let Λ be a quasi k-Gorenstein ring. For each dth syzygy module M in mod Λ (where 0 ≤ d ≤ k-1), we obtain an exact sequence 0 → B → M \bigoplus P → C → 0 in mod Λ with the properties that it is dual exact, P is projective, C is a (d+1)st syzygy module, B is a dth syzygy of ExtΛd+1(D(M), Λ) and the right projective dimension of B^* is less than or equal to d-1. We then give some applications of such an exact sequence as follows. (1) We obtain a chain of epimorphisms concerning M, and by dualizing it we then get the spherical filtration of Auslander and Bridger for M^*. (2) We get Auslander and Bridger's Approximation Theorem for each reflexive module in mod Λop. (3) We show that for any 0 ≤ d ≤ k-1 each dth syzygy module in mod Λ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if Λ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer d, each dth syzygy module in mod Λ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting.