2007/03/29 by Anne-Marie Simon, Simon, Anne-Marie, Jan R. Strooker +1
Mathematics · #13E10 #13H10 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:13E10 #msc:13H10
paper · pdf · doi:10.48550/arxiv.math/0703880
37 pages
arxiv created 2007/03/29 · openalex publication_date 2007/03/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Wiebe's criterion, which recognizes complete intersections of dimension zero among the class of noetherian local rings, is revisited and exploited in order to provide information on what we call C.I.0-ideals (those such that the corresponding quotient is a complete intersection of dimension zero) and also on chains of C.I.0-ideals. A correspondence is established between C.I.0-ideals and a certain kind of matrices which we call x-nice, and a chain of C.I.0-ideals corresponds to a factorization of some x-nice matrix. When the local ring A itself is a complete intersection of dimension zero, a C.I.0-ideal is necessarily of the form (0:bA) for some b∈ A. Some criteria are provided to recognize whether an ideal (0:bA) is C.I.0 or not. When y is a minimal generator of the maximal ideal of A, it is also proved that the ideals yA and (0:yA) are C.I.0 simultaneously and that this is the case exactly when the ideal (0:yA) is principal. These C.I.0-ideals of the form (0:yA), y being a minimal generator of the maximal ideal, are investigated. They are of interest because the smallest nonnull C.I.0-ideal in a strict chain of C.I.0-ideals of the maximal length is necessarily of that form, and their existence has some implications for a realization of the ring, i.e. for the way the ring can can be viewed as a quotient of a regular local ring.