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Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors

Part of: Commutative algebra: Homological methods Homological algebra Homological methods Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  21 May 2025

Leonid Positselski Open the ORCID record for Leonid Positselski [Opens in a new window]
Leonid Positselski*
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
*
Email: positselski@math.cas.cz
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Abstract

In this paper, we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon :\ C{{-\mathsf{Comod}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta :\ C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\textrm {Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.

Keywords

conilpotent coalgebras over a fieldcomodulescontramodulesfaithful exact functorsExt spaces

MSC classification

Primary: 16T15: Coalgebras and comodules; corings 16E30: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D09: Derived categories 18G15: Ext and Tor, generalizations, Künneth formula
Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 34
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Anick, D. J., Generic algebras and CW complexes, in Algebraic topology and algebraic K-theory (Princeton, 1983) (Princeton Univ. Press, 1987), 247–321.Google Scholar
Beilinson, A., Bernstein, J., Deligne, P. and Gabber, O., Faisceaux pervers, Astérisque 100 (2018), vi+180.10.24033/ast.1042CrossRefGoogle Scholar
Cuadra, J., Extensions of rational modules, Int. J. Math. Math. Sci. 2003(69) (2003), 4363–4371.10.1155/S0161171203203471CrossRefGoogle Scholar
Cuadra, J., Năstăsescu, C. and van Oystaeyen, F., Graded almost noetherian rings and applications to coalgebras, J. Algebra 256(1) (2002), 97–110.10.1016/S0021-8693(02)00099-6CrossRefGoogle Scholar
Eilenberg, S. and Moore, J. C., Foundations of relative homological algebra, Memoirs Am. Math. Soc. 55(1965).10.1090/memo/0055CrossRefGoogle Scholar
Eklof, P. C. and Trlifaj, J., How to make Ext vanish, Bull. Lond. Math. Soc 33(1) (2001), 41–51.10.1112/blms/33.1.41CrossRefGoogle Scholar
Fröberg, R., Gulliksen, T. and Löfwall, C., Flat families of local, Artinian algebras with an infinite number of Poincaré series, in Algebra, algebraic topology and their interactions (Stockholm, 1983) (Springer, Berlin, 1986), 170–191,CrossRefGoogle Scholar
Geigle, W. and Lenzing, H., Perpendicular categories with applications to representations and sheaves, J, Algebra 144(2) (1991), 273–343.10.1016/0021-8693(91)90107-JCrossRefGoogle Scholar
Gómez-Torrecillas, J., Năstăsescu, C. and Torrecillas, B., Localization in coalgebras. Applications to finiteness conditions, J. Algebra Appl. 6(02) (2007), 233–243,10.1142/S0219498807002156CrossRefGoogle Scholar
Iovanov, M. C., On extensions of rational modules, Israel J. Math. 199(2) (2014), 585–622.10.1007/s11856-013-0070-3CrossRefGoogle Scholar
Jacobson, N., Lectures in abstract algebra. II. Linear algebra, Graduate Texts in Mathematics. vol. 31, (Springer, New York, 1951-1975), xii+280.Google Scholar
Kashiwara, M. and Schapira, P., Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften, vol. 292 (Springer, 1994), x+512.Google Scholar
Kashiwara, M. and Schapira, P., Categories and sheaves, Grundlehren der mathematischen Wissenschaften, vol. 332 (Springer, 2006), x+497.Google Scholar
Keller, B., Derived categories and their uses, in Handbook of Algebra, (Hazewinkel, M.editor), vol. 1 (Elsevier, 1996), 671–701.10.1016/S1570-7954(96)80023-4CrossRefGoogle Scholar
Lin, B. I. P., Semiperfect coalgebras, J. Algebra 49(2) (1977), 357–373.CrossRefGoogle Scholar
Montgomery, S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, (American Math. Society, Providence, 1993), xiv+238.Google Scholar
Polishchuk, A. and Positselski, L., Quadratic algebras, University Lecture Series, vol. 37 (American Math. Society, Providence, RI, 2005), xii+159.Google Scholar
Porta, M., Shaul, L. and Yekutieli, A., On the homology of completion and torsion, Algebr. Represent. Theory 17(1) (2014), 31–67.10.1007/s10468-012-9385-8CrossRefGoogle Scholar
Positselski, L., Koszul property and Bogomolov’s conjecture, Internat. Math. Res. Not. 2005(31) (2005), 1901–1936.10.1155/IMRN.2005.1901CrossRefGoogle Scholar
Positselski, L., Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov, Monografie Matematyczne 70 (2010), xxiv+349,Google Scholar
Positselski, L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Am. Math. Soc. 212(996) (2011), vi+133.Google Scholar
Positselski, L., Weakly curved A ${}_\infty$ -algebras over a topological local ring, Mém. Soc. Math. France 159 (2018), vi+206.Google Scholar
Positselski, L., Contraherent cosheaves on schemes, Electronic preprint arXiv: 1209.2995v23 [math.CT], (2025), 1–470.Google Scholar
Positselski, L., Contramodules, Confluentes Math. 13(2) (2021), 93–182.10.5802/cml.78CrossRefGoogle Scholar
Positselski, L., Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220(12) (2016), 3866–3909.10.1016/j.jpaa.2016.05.019CrossRefGoogle Scholar
Positselski, L., Koszulity of cohomology $=$ $K(\pi, 1)$ -ness $+$ quasi-formality, J. Algebra 483 (2017), 188–229.10.1016/j.jalgebra.2017.03.022CrossRefGoogle Scholar
Positselski, L., Contraadjusted modules, contramodules, and reduced cotorsion modules, Moscow Math. J. 17(3) (2017), 385–455.10.17323/1609-4514-2017-17-3-385-455CrossRefGoogle Scholar
Positselski, L., Triangulated Matlis equivalence, J. Algebra Appl. 17(04) (2018), 1850067.CrossRefGoogle Scholar
Positselski, L., Dedualizing complexes of bicomodules and MGM duality over coalgebras, Algebr. Represent. Theory 21(4) (2018), 737–767.10.1007/s10468-017-9736-6CrossRefGoogle Scholar
Positselski, L., Smooth duality and co-contra correspondence, J. Lie Theory 30 (2020), 85–144.Google Scholar
Positselski, L., Abelian right perpendicular subcategories in module categories, Electronic preprint arXiv: 1705.04960 [math.CT], (2022), 1–57.Google Scholar
Positselski, L., Contramodules over pro-perfect topological rings, Forum Math. 34(1) (2022), 1–39.10.1515/forum-2021-0010CrossRefGoogle Scholar
Positselski, L., Pseudo-dualizing complexes of bicomodules and pairs of t-structures, Appl. Categ. Struct. 30(2) (2022), 379–416.10.1007/s10485-021-09660-yCrossRefGoogle Scholar
Positselski, L., Remarks on derived complete modules and complexes, Math. Nachrichten 296(2) (2023), 811–839.10.1002/mana.202000140CrossRefGoogle Scholar
Positselski, L., Relative nonhomogeneous Koszul duality, Frontiers in Mathematics (Birkhäuser/Springer Nature, Cham, Switzerland, 2021), xxix+278.Google Scholar
Positselski, L., Differential graded Koszul duality: An introductory survey, Bull. Lond. Math. Soc. 55(4) (2023), 1551–1640.10.1112/blms.12797CrossRefGoogle Scholar
Positselski, L. and Schnürer, O. M., Unbounded derived categories of small and big modules: Is the natural functor fully faithful?, J. Pure Appl. Algebra 225(11) (2021), 23.10.1016/j.jpaa.2021.106722CrossRefGoogle Scholar
Positselski, L. and Št’ovíček, J., $\infty$ -tilting theory, Pacific J. Math 301 (2019), 297–334.10.2140/pjm.2019.301.297CrossRefGoogle Scholar
Positselski, L. and Vishik, A., Koszul duality and Galois cohomology, Math. Res. Lett. 2 (1995), 771–781.10.4310/MRL.1995.v2.n6.a8CrossRefGoogle Scholar
Priddy, S., Koszul resolutions, Trans. Am. Math. Soc. 152 (1970), 39–60.10.1090/S0002-9947-1970-0265437-8CrossRefGoogle Scholar
Radford, D. E., Coreflexive coalgebras, J. Algebra 26 (1973), 512–535.10.1016/0021-8693(73)90012-4CrossRefGoogle Scholar
Simon, A.-M., Approximations of complete modules by complete big Cohen–Macaulay modules over a Cohen–Macaulay local ring, Algebr. Rep. Theory 12 (2009), 385–400.CrossRefGoogle Scholar
Shudo, T., A note on coalgebras and rational modules, Hiroshima Math. J. 6 (1976), 297–304.CrossRefGoogle Scholar
Sweedler, M. E., Hopf algebras, Mathematics Lecture Note Series (W. A. Benjamin, Inc, New York, 1969), vii+336.Google Scholar
Takeuchi, M., Morita theorems for categories of comodules, J. Faculty Sci. Univ. Tokyo 24 (1977), 629–644.Google Scholar
Teply, M. L. and Torrecillas, B., Coalgebras with a radical rational functor, J. Algebra 290(2) (2005), 491–502.10.1016/j.jalgebra.2005.05.016CrossRefGoogle Scholar
Wang, M. and Wu, Z., Conoetherian coalgebras, Algebra Colloq. 5 (1998), 117–120.Google Scholar
Yekutieli, A., On flatness and completion for infinitely generated modules over noetherian rings, Commun. Algebra 39(11) (2011), 4221–4245.CrossRefGoogle Scholar