TY - JOUR
T1 - A Bounded Below, Noncontractible, Acyclic Complex Of Projective Modules
AU - Positselski, L.
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - 16D40
KW - 16D50
KW - 16E05
KW - 16T15
KW - acyclic complex
KW - coalgebra over a field
KW - comodule
KW - contramodule
KW - flat module
KW - injective module
KW - projective module
UR - http://www.scopus.com/inward/record.url?scp=85187716271&partnerID=8YFLogxK
U2 - 10.1007/s10474-024-01414-1
DO - 10.1007/s10474-024-01414-1
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AN - SCOPUS:85187716271
SN - 0236-5294
JO - Acta Mathematica Hungarica
JF - Acta Mathematica Hungarica
ER -