2016/06/06 by Yekutieli, Amnon · 3 citations
#13B35 #13C10 #13C11 #13D07 (Secondary) #13E05 #13J10 (Primary) #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1606.01832
We continue investigating the interaction between flatness and \mathfraka-adic completion for infinitely generated modules over a commutative ring A. We introduce the concept of \mathfraka-adic flatness, which is weaker than flatness. We prove that \mathfraka-adic flatness is preserved under completion when the ideal \mathfraka is weakly proregular. We also prove that when A is noetherian, \mathfraka-adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring A, with a weakly proregular ideal \mathfraka, for which the completion A is not flat. We also study \mathfraka-adic systems, and prove that if the ideal \mathfraka is finitely generated, then the limit of any \mathfraka-adic system is a complete module.