Abstract
We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper.
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References
L. Angeleri Hügel, Covers and envelopes via endoproperties of modules, Proc. London Math. Soc., 86 (2003), 649–665.
H. Cartan and S. Eilenberg, Homological Algebra, Princeton Landmarks in Mathematics and Physics, Princeton University Press (Princeton, NJ, 1999).
A. Dress, On the decomposition of modules, Bull. Amer. Math. Soc., 75 (1969), 984–986.
C. U. Jensen, Les foncteurs dérivés de \(\underleftarrow {\rm lim}\) et leurs applications en théorie des modules,Lecture Notes in Math., vol. 254, Springer (1972).
A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Inventiones Math., 174 (2008), 225–308.
L. Positselski, Homological Algebra of Semimodules and Semicontramodules: Semiinfinite Homological Algebra of Associative Algebraic Structures, Appendix C in collaboration with D. Rumynin, Appendix D in collaboration withS. Arkhipov, IMPAN Monogr. Mat. (N.S.), vol. 70, Birkhäuser/Springer Basel AG (Basel, 2010).
L. Positselski, Contramodules, Confluentes Math., 13 (2021), 93–182.
L. Positselski, Contramodules over pro-perfect topological rings, Forum Math., 34 (2022), 1–39.
L. Positselski, Relative Nonhomogeneous Koszul Duality, Frontiers in Mathematics, Birkhäuser/Springer Nature (Cham, Switzerland, 2021).
L. Positselski, Differential graded Koszul duality: An introductory survey, Bull. London Math. Soc., 55 (2023), 1551–1640.
L. Positselski and J. Št’ovíček, The tilting-cotilting correspondence, Int. Math. Res. Not. IMRN, 2021 (2021), 189–274.
L. Positselski and J. Št’ovíček , Topologically semisimple and topologically perfect topological rings, Publ. Mat., 66 (2022), 457–540.
F. Prosmans, Derived limits in quasi-abelian categories, Bull. Soc. Roy. Sci. Liège, 68 (1999), 335–401.
J. Rickard, Unbounded derived categories and the finitistic dimension conjecture, Adv. in Math., 354 (2019), Paper No. 106735, 21 pp.
L. Shaul, The finitistic dimension conjecture via DG-rings, arXiv:2209.02068 (2022).
L. Shaul, Acyclic complexes of injectives and finitistic dimensions, With an appendix by T. Nakamura and P. Thompson, arXiv:2303.08756 (2023).
M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin,Inc. (New York, 1969).
Acknowledgements
This paper was inspired by Liran Shaul’s talk at the Algebra seminar in Prague, organized by Jan Trlifaj. I want to thank both the speaker and the organizer of the seminar. I also wish to thank Alberto Canonaco for communicating his example to me and giving a kind permission to reproduce it here (see Example 8.4).
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The author is supported by the GAČR project 23-05148S and the Czech Academy of Sciences (RVO 67985840).
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Positselski, L. A Bounded Below, Noncontractible, Acyclic Complex Of Projective Modules. Acta Math. Hungar. 172, 324–345 (2024). https://doi.org/10.1007/s10474-024-01414-1
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DOI: https://doi.org/10.1007/s10474-024-01414-1
Key words and phrases
- acyclic complex
- projective module
- injective module
- flat module
- coalgebra over a field
- comodule
- contramodule