×
Zhao, Tiwei; Huang, Zhaoyong

Phantom ideals and cotorsion pairs in extriangulated categories. (English) Zbl 1422.18013

Taiwanese J. Math. 23, No. 1, 29-61 (2019).
The approximation theory of an exact category was developed by X. H. Fu et al. [Adv. Math. 244, 750–790 (2013; Zbl 1408.18022)] as a generalization of the classical approximation theory for subcategories. In the paper under review, the authors establish the approximation theory in an additive category equipped with an additive bifunctor, and consider it in extriangulated categories. It is proved in the paper that if \((\mathcal{C}, \mathbb{E}, \mathfrak{s})\) is an extriangulated category with enough injective objects and projective objects then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of \(\mathcal{C}\); (2) special preenveloping ideals of \(\mathcal{C}\); (3) additive subfunctors of \(\mathbb{E}\) having enough special injective morphisms; and (4) additive subfunctors of \(\mathbb{E}\) having enough special projective morphisms. Moreover, it is proved that if \((\mathcal{C}, \mathbb{E}, \mathfrak{s})\) is an extriangulated category with enough injective objects and projective morphisms then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of \(\mathcal{C}\) ; (2) all additive subfunctors of \(\mathbb{E}\) having enough special injective objects.
Reviewer: Li Liang (Lanzhou)

Cited in 11 Documents

MSC:

18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
18E40 Torsion theories, radicals

Keywords:

phantom ideals; cotorsion pairs; extriangulated categories; (co)phantom morphisms; special precovering ideals; special preenveloping ideals

Citations:

Zbl 1408.18022
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

[1] N. Abe and H. Nakaoka, General heart construction on a triangulated category (II): Associated homological functor, Appl. Categ. Structures 20 (2012), no. 2, 161-174. · Zbl 1264.18015 · doi:10.1007/s10485-010-9226-z
[2] J. F. Adams and G. Walker, An example in homotopy theory, Proc. Cambridge Philos. Soc. 60 (1964), 699-700. · Zbl 0126.18603 · doi:10.1017/S0305004100077422
[3] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisgue 100, Soc. Math. France, Paris, 1982. · Zbl 0536.14011
[4] D. J. Benson, Phantom maps and purity in modular representation theory III, J. Algebra 248 (2002), no. 2, 747-754. · Zbl 1002.20008 · doi:10.1006/jabr.2001.9043
[5] D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory I, Fund. Math. 161 (1999), no. 1-2, 37-91. · Zbl 0944.20004
[6] ——–, Phantom maps and purity in modular representation theory II, Algebr. Represent. Theory 4 (2001), no. 4, 395-404. · Zbl 0998.20004
[7] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Second editor, Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI, 2000. · Zbl 0980.22015
[8] S. Breaz and G.-C. Modoi, Ideal cotorsion theories in triangulated categories, arXiv:1501.06810v2. · Zbl 1461.18007
[9] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2000. · Zbl 0952.13001
[10] X. H. Fu, P. A. Guil Asensio, I. Herzog and B. Torrecillas, Ideal approximation theory, Adv. Math. 244 (2013), 750-790. · Zbl 1408.18022
[11] A. Grothendieck, The cohomology theory of abstract algebraic varieties, in: 1960 Proc. Internat. Congress Math. (Edinburgh, 1958), 103-118, Cambridge Univ. Press, New York. · Zbl 0119.36902
[12] D. Happel, Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988. · Zbl 0635.16017
[13] R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin, Heidelberg, New York, 1966. · Zbl 0212.26101
[14] M. Herschend, Y. Liu and H. Nakaoka, \(n\)-exangulated categories, arXiv:1709.06689v2.
[15] I. Herzog, The phantom cover of a module, Adv. Math. 215 (2007), no. 1, 220-249. · Zbl 1128.16005 · doi:10.1016/j.aim.2007.03.010
[16] N. Hoffmann and M. Spitzweck, Homological algebra with locally compact abelian groups, Adv. Math. 212 (2007), no. 2, 504-524. · Zbl 1123.22002 · doi:10.1016/j.aim.2006.09.019
[17] B. E. Johnson, Introduction to cohomology in Banach algebras, in: Algebras in Analysis, (Proceedings of the Instructional Conference and NATO Advanced Study Institute, Birmingham, 1973), 84-100, Academic Press, London, 1975. · Zbl 0306.46065
[18] R. Kiełpiński and D. Simson, On pure homological dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 1-6. · Zbl 0303.16016
[19] S. Koenig and B. Zhu, From triangulated categories to abelian categories: cluster tilting in a general framework, Math. Z. 258 (2008), no. 1, 143-160. · Zbl 1133.18005 · doi:10.1007/s00209-007-0165-9
[20] Y. Liu, Hearts of twin cotorsion pairs on exact categories, J. Algebra 394 (2013), 245-284. · Zbl 1319.18002 · doi:10.1016/j.jalgebra.2013.07.028
[21] C. A. McGibbon, Phantom maps, in: Handbook of Algebraic Topology, 1209-1257, North-Holland, Amsterdam, 1995. · Zbl 0867.55013
[22] D. Miličić, Lectures on Derived Categories, Preprint available at https://www.math.utah.edu/ milicic/Eprints/dercat.pdf.
[23] H. Nakaoka, General heart construction on a triangulated category (I): Unifying \(t\)-structures and cluster tilting subcategories, Appl. Categ. Structures 19 (2011), no. 6, 879-899. · Zbl 1254.18011 · doi:10.1007/s10485-010-9223-2
[24] H. Nakaoka and Y. Palu, Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories, arXiv:1605.05607v2. · Zbl 1451.18021
[25] A. Neeman, The Brown representability theorem and phantomless triangulated categories, J. Algebra 151 (1992), no. 1, 118-155. · Zbl 0799.18006 · doi:10.1016/0021-8693(92)90135-9
[26] D. Quillen, Higher algebraic \(K\)-theory I, in: Algebraic \(K\)-theory I: Higher \(K\)-theories, (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington, 1972), 85-147, Lecture Notes in Math. 341, Springer, Berlin, 1973. · Zbl 0292.18004
[27] L. Salce, Cotorsion theories for abelian groups, in: Symposia Mathematica XXIII, 11-32, Academic Press, New York, 1979. · Zbl 0426.20044
[28] M. Schlichting, Hermitian \(K\)-theory of exact categories, J. K-Theory 5 (2010), no. 1, 105-165. · Zbl 1328.19009
[29] D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), no. 2, 91-116. · Zbl 0361.18010 · doi:10.4064/fm-96-2-91-116
[30] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), 253 pp. · Zbl 0882.18010
[31] P. Zhou and B. Zhu, Triangulated quotient categories revisited, J. Algebra, 502 (2018), 196-232. · Zbl 1388.18014 · doi:10.1016/j.jalgebra.2018.01.031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.