A bounded below, noncontractible, acyclic complex of projective modules

0585949 - MÚ 2025 RIV DE eng J - Journal Article
Positselski, Leonid
A bounded below, noncontractible, acyclic complex of projective modules.
Acta Mathematica Hungarica. Roč. 172, č. 2 (2024), s. 324-345. ISSN 0236-5294. E-ISSN 1588-2632
R&D Projects: GA ČR(CZ) GA23-05148S
Institutional support: RVO:67985840
Keywords : acyclic complex * coalgebra over a field * comodule
OECD category: Pure mathematics
Impact factor: 0.9, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s10474-024-01414-1

We construct examples of bounded below, noncontractible, acycliccomplexes of finitely generated projective modules over some rings S,as well as bounded above, noncontractible, acyclic complexes ofinjective modules. The rings S are certain rings of infinite matrices with entries inthe rings of commutative polynomials or formal power series ininfinitely many variables. In the world of comodules or contramodules over coalgebras overfields, similar examples exist over the cocommutative symmetriccoalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible,acyclic complex of free modules with one generator, communicated tothe author by Canonaco, is included at the end of the paper.
Permanent Link: https://hdl.handle.net/11104/0353583