2019/07/30 by Monica Lewis, Lewis, Monica Ann · 1 citation
Mathematics · #13D45 (Primary) 13H05 #13H10 (Secondary) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1907.12873
openalex publication_date 2019/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a regular ring, let J be an ideal generated by a regular\nsequence of codimension at least 2, and let I be an ideal containing J.\nWe give an example of a module H3I(J) with infinitely many associated\nprimes, answering a question of Hochster and N 'u ~nez-Betancourt in the\nnegative. In fact, for i\≤ 4, we show that under suitable hypotheses on\nR/J, \Ass ,HiI(J) is finite if and only if\n\Ass ,Hi-1I(R/J) is finite. Our proof of this statement involves a\nnovel generalization of an isomorphism of Hellus, which may be of some\nindependent interest. The finiteness comparison between \Ass , HiI(J)\nand \Ass , Hi-1I(R/J) tends to improve as our hypotheses on R/J\nbecome more restrictive. To illustrate the extreme end of this phenomenon, at\nleast in the prime characteristic p>0 setting, we show that if R/J is\nregular, then \Ass , HiI(J) is finite for all i\≥ 0.\n