## Taiwanese Journal of Mathematics

### Phantom Ideals and Cotorsion Pairs in Extriangulated Categories

#### Abstract

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ 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 $\mathscr{C}$; (2) special preenveloping ideals of $\mathscr{C}$; (3) additive subfunctors of $\mathbb{E}$ having enough special injective morphisms; and (4) additive subfunctors of $\mathbb{E}$ having enough special projective morphisms. Moreover, we show that if $(\mathscr{C},\mathbb{E}, \mathfrak{s})$ 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 $\mathscr{C}$; (2) all additive subfunctors of $\mathbb{E}$ having enough special injective objects.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 1 (2019), 29-61.

Dates
Revised: 28 April 2018
Accepted: 20 May 2018
First available in Project Euclid: 9 June 2018

https://projecteuclid.org/euclid.twjm/1528509854

Digital Object Identifier
doi:10.11650/tjm/180504

Mathematical Reviews number (MathSciNet)
MR3909989

Zentralblatt MATH identifier
07021717

#### Citation

Zhao, Tiwei; Huang, Zhaoyong. Phantom Ideals and Cotorsion Pairs in Extriangulated Categories. Taiwanese J. Math. 23 (2019), no. 1, 29--61. doi:10.11650/tjm/180504. https://projecteuclid.org/euclid.twjm/1528509854

#### References

• 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.
• J. F. Adams and G. Walker, An example in homotopy theory, Proc. Cambridge Philos. Soc. 60 (1964), 699–700.
• A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisgue 100, Soc. Math. France, Paris, 1982.
• D. J. Benson, Phantom maps and purity in modular representation theory III, J. Algebra 248 (2002), no. 2, 747–754.
• 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.
• ––––, Phantom maps and purity in modular representation theory II, Algebr. Represent. Theory 4 (2001), no. 4, 395–404.
• 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.
• S. Breaz and G.-C. Modoi, Ideal cotorsion theories in triangulated categories, arXiv:1501.06810v2.
• E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2000.
• X. H. Fu, P. A. Guil Asensio, I. Herzog and B. Torrecillas, Ideal approximation theory, Adv. Math. 244 (2013), 750–790.
• A. Grothendieck, The cohomology theory of abstract algebraic varieties, in: 1960 Proc. Internat. Congress Math. (Edinburgh, 1958), 103–118, Cambridge Univ. Press, New York.
• D. Happel, Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988.
• R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
• M. Herschend, Y. Liu and H. Nakaoka, $n$-exangulated categories, arXiv:1709.06689v2.
• I. Herzog, The phantom cover of a module, Adv. Math. 215 (2007), no. 1, 220–249.
• N. Hoffmann and M. Spitzweck, Homological algebra with locally compact abelian groups, Adv. Math. 212 (2007), no. 2, 504–524.
• 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.
• R. Kiełpiński and D. Simson, On pure homological dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 1–6.
• 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.
• Y. Liu, Hearts of twin cotorsion pairs on exact categories, J. Algebra 394 (2013), 245–284.
• C. A. McGibbon, Phantom maps, in: Handbook of Algebraic Topology, 1209–1257, North-Holland, Amsterdam, 1995.
• D. Miličić, Lectures on Derived Categories, Preprint available at https://www.math.utah.edu/~milicic/Eprints/dercat.pdf.
• 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.
• H. Nakaoka and Y. Palu, Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories, arXiv:1605.05607v2.
• A. Neeman, The Brown representability theorem and phantomless triangulated categories, J. Algebra 151 (1992), no. 1, 118–155.
• 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.
• L. Salce, Cotorsion theories for abelian groups, in: Symposia Mathematica XXIII, 11–32, Academic Press, New York, 1979.
• M. Schlichting, Hermitian $K$-theory of exact categories, J. K-Theory 5 (2010), no. 1, 105–165.
• D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), no. 2, 91–116.
• J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), 253 pp.
• P. Zhou and B. Zhu, Triangulated quotient categories revisited, J. Algebra, 502 (2018), 196–232.