Lazard's Theorem for S -posets

✍ Scribed by Sydney Bulman-Fleming; Valdis Laan

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 191 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310338

✦ Synopsis

Abstract

In 1971, inspired by the work of Lazard and Govorov for modules over a ring, Stenström proved that the strongly flat right acts A ~S~ over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor A ~S~ ⊗ (from the category of left S ‐acts into the category of sets) preserves pullbacks and equalizers. He also provided interpolation‐type conditions (now referred to in the literature as Property (P) and Property (E)) characterizing strong flatness. Unlike the situation for modules over a ring, strong flatness is strictly stronger than (mono‐) flatness (wherein the functor A ~S~ ⊗ is required only to preserve monomorphisms). The study of flatness properties of partially ordered monoids acting on partially ordered sets was initiated by S. M. Fakhruddin in the 1980s, and has been continued recently in the paper “Indecomposable, projective, and flat S ‐posets” by Shi, Liu, Wang, and Bulman–Fleming, Comm. Algebra 33, 235–251 (2005). In that paper, a criterion for the equality of elements in a tensor product of S ‐posets is given and a version of Property (P) is presented that, as in the unordered case, implies flatness and is implied by projectivity. The present paper introduces a corresponding Property (E) and establishes an analogue of the Lazard–Govorov–Stenström theorem in the context of S ‐posets. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)