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Duality pairs, phantom maps, and definability in triangulated categories

Part of: Commutative algebra: Homological methods Applied homological algebra and category theory

Published online by Cambridge University Press:  13 September 2024

Isaac Bird Open the ORCID record for Isaac Bird [Opens in a new window]  and
Jordan Williamson Open the ORCID record for Jordan Williamson [Opens in a new window]
Isaac Bird*
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha, Czech Republic (bird@karlin.mff.cuni.cz; williamson@karlin.mff.cuni.cz)
Jordan Williamson
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha, Czech Republic (bird@karlin.mff.cuni.cz; williamson@karlin.mff.cuni.cz)
*
*Corresponding author.
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Abstract

We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure quotients and pure extensions, as well as providing a way to show the existence of approximations. One key ingredient is a new characterization of phantom maps. We then introduce an axiomatic form of Auslander–Gruson–Jensen duality, from which we define dual definable categories, and show that these coincide with symmetric coproduct closed duality pairs. This framework is ubiquitous, encompassing both algebraic triangulated categories and stable homotopy theories. Accordingly, we provide many applications in both settings, with a particular emphasis on silting theory and stratified tensor-triangulated categories.

Keywords

triangulated categoriespuritysiltingdefinableduality

MSC classification

Secondary: 55U30: Duality 13D09: Derived categories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 46
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Adámek, J. and Rosický, J., Locally presentable and accessible categories, volume 189 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1994).Google Scholar
Angeleri Hügel, L.. Silting objects. Bull. Lond. Math. Soc. 51 (2019), 658–690.CrossRefGoogle Scholar
Angeleri Hügel, L. and Hrbek, M.. Parametrizing torsion pairs in derived categories. Represent. Theory 25 (2021), 679–731.CrossRefGoogle Scholar
Angeleri Hügel, L., Marks, F. and Vitória, J.. Silting modules. Int. Math. Res. Not. IMRN 4 (2016), 1251–1284.CrossRefGoogle Scholar
Auslander, M., Isolated singularities and existence of almost split sequences. In Representation theory, II (Ottawa, Ont., 1984), volume 1178 of Lecture Notes in Math. (Springer, Berlin, 1986), pp. 194–242.CrossRefGoogle Scholar
Avramov, L. L. and Foxby, H.-B.. Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71 (1991), 129–155.CrossRefGoogle Scholar
Balmer, P.. The spectrum of prime ideals in tensor triangulated categories. J. Reine. Angew. Math. 588 (2005), 149–168.CrossRefGoogle Scholar
Balmer, P., Krause, H. and Stevenson, G.. Tensor-triangular fields: ruminations. Selecta Math. (N.S.) 25 (2019), 36.CrossRefGoogle Scholar
Barthel, T., Castellana, N., Heard, D. and Sanders, B., Cosupport in tensor triangular geometry. eprint arXiv:2303.13480 (2023).CrossRefGoogle Scholar
Barthel, T., Heard, D. and Sanders, B.. Stratification in tensor triangular geometry with applications to spectral Mackey functors. Camb. J. Math. 11 (2023), 829–915.CrossRefGoogle Scholar
Barthel, T., Heard, D. and Valenzuela, G.. Local duality in algebra and topology. Adv. Math. 335 (2018), 563–663.CrossRefGoogle Scholar
Benson, D., Iyengar, S. and Krause, H.. Local cohomology and support for triangulated categories. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 573–619.Google Scholar
Benson, D., Iyengar, S. and Krause, H.. Stratifying triangulated categories. J. Topol. 4 (2011), 641–666.CrossRefGoogle Scholar
Benson, D. J., Iyengar, S. and Krause, H.. Stratifying modular representations of finite groups. Ann. of Math. (2) 174 (2011), 1643–1684.CrossRefGoogle Scholar
Beĭlinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque (Soc. Math. France, Paris, 1982), pp. 5–171.Google Scholar
Bird, I. and Williamson, J., The homological spectrum via definable subcategories. eprint arXiv:2304.05179 (2023).Google Scholar
Burklund, R., Hahn, J., Levy, I. and Schlank, T., $K$-theoretic counterexamples to Ravenel's telescope conjecture. eprint arXiv:2310.17459 (2023).Google Scholar
Christensen, J. D. and Strickland, N. P.. Phantom maps and homology theories. Topology 37 (1998), 339–364.CrossRefGoogle Scholar
Christensen, L. W., Foxby, H.-B. and Holm, H., Derived category methods in commutative algebra. Draft version of 11/21. Obtained at https://www.math.ttu.edu/lchriste/download/dcmca.pdf.Google Scholar
Crivei, S., Prest, M. and Torrecillas, B.. Covers in finitely accessible categories. Proc. Amer. Math. Soc. 138 (2010), 1213–1221.CrossRefGoogle Scholar
Enochs, E. E. and Iacob, A.. Gorenstein injective covers and envelopes over Noetherian rings. Proc. Amer. Math. Soc. 143 (2015), 5–12.CrossRefGoogle Scholar
Frankild, A. and Jørgensen, P.. Foxby equivalence, complete modules, and torsion modules. J. Pure Appl. Algebra 174 (2002), 135–147.CrossRefGoogle Scholar
Gillespie, J.. Duality pairs and stable module categories. J. Pure Appl. Algebra 223 (2019), 3425–3435.CrossRefGoogle Scholar
Gillespie, J. and Iacob, A.. Duality pairs, generalized Gorenstein modules, and Ding injective envelopes. C. R. Math. Acad. Sci. Paris 360 (2022), 381–398.CrossRefGoogle Scholar
Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules. Volume 1, volume 41 of De Gruyter Expositions in Mathematics, extended edition (Walter de Gruyter GmbH & Co. KG, Berlin, 2012), Approximations.CrossRefGoogle Scholar
Gruson, L. and Jensen, C. U., Dimensions cohomologiques reliées aux foncteurs $\varprojlim^{(i)}$. In Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980), volume 867 of Lecture Notes in Math. (Springer, Berlin, 1981), pp. 234–294.CrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1988).Google Scholar
Holm, H.. Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
Holm, H. and Jørgensen, P.. Compactly generated homotopy categories. Homology Homotopy Appl. 9 (2007), 257–274.CrossRefGoogle Scholar
Holm, H. and Jørgensen, P.. Cotorsion pairs induced by duality pairs. J. Commut. Algebra 1 (2009), 621–633.CrossRefGoogle Scholar
Hovey, M., Palmieri, J. H. and Strickland, N. P.. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128 (1997), x+114.Google Scholar
Hügel, L. A., Laking, R., Šťovíček, J. and Vitória, J., Mutation and torsion pairs. eprint arXiv:2201.02147 (2022).Google Scholar
Jensen, C. U. and Lenzing, H., Model-theoretic algebra with particular emphasis on fields, rings, modules, volume 2 of Algebra, Logic and Applications (Gordon and Breach Science Publishers, New York, 1989).Google Scholar
Jørgensen, P.. The homotopy category of complexes of projective modules. Adv. Math. 193 (2005), 223–232.CrossRefGoogle Scholar
Keller, B.. Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27 (1994), 63–102.CrossRefGoogle Scholar
Keller, B., On differential graded categories. In International Congress of Mathematicians. Vol. II, (Eur. Math. Soc., Zürich, 2006), pp. 151–190.CrossRefGoogle Scholar
Krause, H.. Smashing subcategories and the telescope conjecture—an algebraic approach. Invent. Math. 139 (2000), 99–133.CrossRefGoogle Scholar
Krause, H.. Coherent functors in stable homotopy theory. Fund. Math. 173 (2002), 33–56.CrossRefGoogle Scholar
Krause, H.. The stable derived category of a Noetherian scheme. Compos. Math. 141 (2005), 1128–1162.CrossRefGoogle Scholar
Krause, H.. Approximations and adjoints in homotopy categories. Math. Ann. 353 (2012), 765–781.CrossRefGoogle Scholar
Laking, R.. Purity in compactly generated derivators and t-structures with Grothendieck hearts. Math. Z. 295 (2020), 1615–1641.CrossRefGoogle Scholar
Laking, R. and Vitória, J.. Definability and approximations in triangulated categories. Pacific J. Math. 306 (2020), 557–586.CrossRefGoogle Scholar
Marks, F. and Vitória, J.. Silting and cosilting classes in derived categories. J. Algebra 501 (2018), 526–544.CrossRefGoogle Scholar
Mehdi, A. R. and Prest, M.. Almost dual pairs and definable classes of modules. Comm. Algebra 43 (2015), 1387–1397.CrossRefGoogle Scholar
Neeman, A.. The chromatic tower for $D(R)$. Topology 31 (1992), 519–532. With an appendix by Marcel Bökstedt.CrossRefGoogle Scholar
Neeman, A.. The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25 (1992), 547–566.CrossRefGoogle Scholar
Neeman, A., Triangulated categories, volume 148 of Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 2001).CrossRefGoogle Scholar
Neeman, A.. The homotopy category of flat modules, and Grothendieck duality. Invent. Math. 174 (2008), 255–308.CrossRefGoogle Scholar
Neeman, A.. Some adjoints in homotopy categories. Ann. of Math. (2) 171 (2010), 2143–2155.CrossRefGoogle Scholar
nLab authors, Nerve and realization. https://ncatlab.org/nlab/show/nerve 2022.Google Scholar
Prest, M., Purity, spectra and localisation, volume 121 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2009).CrossRefGoogle Scholar
Prest, M.. Definable additive categories: purity and model theory. Mem. Amer. Math. Soc. 210 (2011), vi+109.Google Scholar
Prest, M. and Slávik, A.. Purity in categories of sheaves. Math. Z. 297 (2021), 429–451.CrossRefGoogle Scholar
Rickard, J.. Morita theory for derived categories. J. London Math. Soc. (2) 39 (1989), 436–456.CrossRefGoogle Scholar
Rothmaler, P.. When are pure-injective envelopes of flat modules flat?. Comm. Algebra 30 (2002), 3077–3085.CrossRefGoogle Scholar
Shaul, L. and Williamson, J., Lifting (co)stratifications between tensor-triangulated categories. arXiv:2012.05190v4, to appear in Israel J. Math., https://doi.org/10.1007/s11856-023-2578-5, 2023.CrossRefGoogle Scholar
Stenström, B.. Purity in functor categories. J. Algebra 8 (1968), 352–361.CrossRefGoogle Scholar
Šťovíček, J.. Locally well generated homotopy categories of complexes. Doc. Math. 15 (2010), 507–525.CrossRefGoogle Scholar
Šťovíček, J. and Pospíšil, D.. On compactly generated torsion pairs and the classification of co-$t$-structures for commutative noetherian rings. Trans. Amer. Math. Soc. 368 (2016), 6325–6361.CrossRefGoogle Scholar
Wagstaffe, R., Definability in monoidal additive and tensor triangulated categories. PhD thesis, The University of Manchester (2021).Google Scholar