2015/04/01 by Helmut Zöschinger, Zöschinger, Helmut
Mathematics · #13B35 #13C11 #13J10 #16D40 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.AC #math.RA #msc:13B35 #msc:13C11 #msc:13J10 #msc:16D40
paper · pdf · doi:10.48550/arxiv.1504.00168
12 pages, in German
arxiv created 2015/04/01 · openalex publication_date 2015/04/01 · arxiv updated 2015/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (R, \mathfrakm) be a noetherian local ring, M a separated R-module (i.e. \bigcapn≥ 1\mathfrakmn M = 0) and \widehatM = lim← M/\mathfrakmn M its completion. Generally, M is not pure in \widehatM and \widehatM is not pure-injective. But if M is totally separated, i.e. X\undersetR⊗ M is separated for all finitely generated R-modules X, the situation improves: In this case, M is pure in \widehatM and, under additional conditions, \widehatM is even pure-injective, e.g. if M≅ X(I) holds with X finitely generated or M ≅\coprodi=1∞ R/\mathfrakmi. In section 2, we investigate the question under which conditions both M and \widehatM are totally separated and establish a close connection to the class of strictly pure-essential extensions. In section 3, we replace the completion \widehatM in the case M = \coprodi∈ IMi with the \mathfrakm-adic closure A of M in P = ∏i∈ I Mi, i.e. with A = \bigcapn ≥ 1(M + \mathfrakmn P). We give criteria so that A/M is radical and show that this always holds in the countable case M = \coprodi=1∞ Mi. Finally, we deal with the case that A is even totally separated and additionally determine the coassociated prime ideals of A/M.