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Local duality in algebra and topology. (English) Zbl 1403.55008
Local duality is a relation between local cohomology and local homology. Examples include Grothendieck’s local duality, and its generalisation by Greenlees and May. This paper constructs a simple abstract framework for local duality statements in general. The named examples are then deducible from this framework and the authors consider a number of further examples of local duality that fit into their theory. These examples include categories of comodules over Hopf algebroids, chromatic homotopy theory, equivariant homotopy theory, motivic homotopy theory and local duality for schemes.
The general results are phrased in the context of a symmetric monoidal triangulated category \(\mathcal{C}\) with some underlying homotopy theory, such as a pre-triangulated dg-category, stable model category or stable \(\infty\)-category. These categories are called stable categories.
Let \(\mathcal{K}\) denote a set of compact objects of \(\mathcal{C}\). From this set, three more stable categories are defined. Let \(\mathcal{C}^{\mathcal{K}\text{-tors}}\) be the localising ideal generated by \(\mathcal{K}\): the full triangulated subcategory of \(\mathcal{C}\) that contains \(\mathcal{K}\), is closed under retracts, desuspensions and filtered colimits and tensor products with objects of \(\mathcal{C}\). Let \(\mathcal{C}^{\mathcal{K}\text{-loc}}\) be those objects of \(\mathcal{C}\) which admit no maps from objects of \(\mathcal{C}^{\mathcal{K}\text{-tors}}\). Let \(\mathcal{C}^{\mathcal{K}\text{-comp}}\) be those objects of \(\mathcal{C}\) which admit no maps from objects of \(\mathcal{C}^{\mathcal{K}\text{-loc}}\).
The inclusion \(i_{\text{tors}}\) of \(\mathcal{C}^{\mathcal{K}\text{-tors}}\) into \(\mathcal{C}\) has a right adjoint \(\Gamma\), called the torsion functor, and may be thought of as a local cohomology functor. Similarly, the inclusion \(i_{\text{loc}}\) of \(\mathcal{C}^{\mathcal{K}\text{-loc}}\) into \(\mathcal{C}\) has a left adjoint \(L\) and the inclusion \(i_{\text{comp}}\) of \(\mathcal{C}^{\mathcal{K}\text{-comp}}\) into \(\mathcal{C}\) has a left adjoint \(\Lambda\). The functor \(\Lambda\) is called the completion functor and may be thought of as local homology. The functors \(\Gamma\) and \(L\) are smashing and the inclusions and their adjoints induce an equivalence of stable categories \[ \mathcal{C}^{\mathcal{K}\text{-loc}} \simeq \mathcal{C}^{\mathcal{K}\text{-comp}}. \] The duality statement is the adjunction \[ \underline{\text{Hom}}( i_{\text{tors}} \Gamma X, Y ) \simeq \underline{\text{Hom}}( X, i_{\text{comp}} \Lambda Y ) \] where \(\underline{\text{Hom}}\) denotes the internal function object of \(\mathcal C\). The pullback of the functors \[ L \longrightarrow L \Lambda \longleftarrow \Lambda \] is the identity, giving a fracture square analogous to the Hasse squares in stable homotopy theory.
The rest of the paper then examines various examples of where this formalism applies as above. Of particular note is the new duality result on comodules over a Hopf algebroid, which is then extended to quasi-coherent sheaves on a presentable algebraic stack. There is a substantial section on chromatic homotopy theory, considering three settings: \(p\)-local spectra, \(E(n)\)-local spectra and \(E(n)\)-modules. The paper demonstrates how these topological duality statements are related to algebraic duality statements of comodules: taking \(R\) to be \(BP\), \(E(n)\) or the sphere spectrum (respectively) the homology functor \(R_*\) will send spectra to \(R_*R\)-comodules in a manner compatible with the duality constructions.

MSC:
55P60 Localization and completion in homotopy theory
13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
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[1] Abrams, G.; Menini, C., Categorical equivalences and realization theorems, J. Pure Appl. Algebra, 113, 2, 107-120, (1996) · Zbl 0862.16002
[2] Antolin-Camarena, O.; Barthel, T., Chromatic fracture cubes, (2014), preprint
[3] Baker, A.; Richter, B., Invertible modules for commutative \(\mathbb{S}\)-algebras with residue fields, Manuscripta Math., 118, 1, 99-119, (2005) · Zbl 1092.55007
[4] Barnes, D.; Roitzheim, C., Monoidality of Franke’s exotic model, Adv. Math., 228, 6, 3223-3248, (2011) · Zbl 1246.55009
[5] Barthel, T.; Heard, D.; Valenzuela, G., Local duality for structured ring spectra, J. Pure Appl. Algebra, 222, 2, 433-463, (2018) · Zbl 1384.55008
[6] Barthel, T.; Stapleton, N., Centralizers in good groups are good, Algebr. Geom. Topol., 16, 3, 1453-1472, (2016) · Zbl 1365.55001
[7] Basterra, M.; Mandell, M. A., The multiplication on BP, J. Topol., 6, 2, 285-310, (2013) · Zbl 1317.55005
[8] Benson, D. J.; Krause, H., Complexes of injective kg-modules, Algebra Number Theory, 2, 1, 1-30, (2008) · Zbl 1167.20006
[9] Bondal, A.; van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., 3, 1, 1-36, (2003), 258 · Zbl 1135.18302
[10] Borghesi, S., Algebraic Morava K-theories, Invent. Math., 151, 2, 381-413, (2003) · Zbl 1030.55003
[11] Brodmann, M. P.; Sharp, R. Y., Local cohomology, (An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math., vol. 136, (2013), Cambridge University Press Cambridge)
[12] Bruns, W.; Herzog, J., Cohen-Macaulay rings, Cambridge Stud. Adv. Math., vol. 39, (1993), Cambridge University Press Cambridge · Zbl 0788.13005
[13] Devinatz, E. S.; Hopkins, M. J.; Smith, J. H., Nilpotence and stable homotopy theory. I, Ann. of Math. (2), 128, 2, 207-241, (1988) · Zbl 0673.55008
[14] Dugger, D.; Isaksen, D. C., Motivic cell structures, Algebr. Geom. Topol., 5, 615-652, (2005) · Zbl 1086.55013
[15] Dwyer, W. G.; Greenlees, J. P.C., Complete modules and torsion modules, Amer. J. Math., 124, 1, 199-220, (2002) · Zbl 1017.18008
[16] Dwyer, W. G.; Greenlees, J. P.C.; Iyengar, S., Duality in algebra and topology, Adv. Math., 200, 2, 357-402, (2006) · Zbl 1155.55302
[17] Goerss, P. G., Quasi-coherent sheaves on the moduli stack of formal groups, (2008), preprint
[18] Greenlees, J. P.C., Tate cohomology in axiomatic stable homotopy theory, (Cohomological Methods in Homotopy Theory, Bellaterra, 1998, Progr. Math., vol. 196, (2001), Birkhäuser Basel), 149-176 · Zbl 1002.55005
[19] Greenlees, J. P.C.; May, J. P., Derived functors of I-adic completion and local homology, J. Algebra, 149, 2, 438-453, (1992) · Zbl 0774.18007
[20] Greenlees, J. P.C.; May, J. P., Completions in algebra and topology, (Handbook of Algebraic Topology, (1995), North-Holland Amsterdam), 255-276 · Zbl 0869.55007
[21] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux \((S G A 2)\), (Masson; Cie, Augmenté d’un exposé par Michèle Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Adv. Stud. Pure Math., vol. 2, (1968), North-Holland Publishing Co. Amsterdam) · Zbl 0159.50402
[22] Hartshorne, R., Local cohomology, (A Seminar Given by A. Grothendieck, Harvard University, Fall, vol. 1961, (1967), Springer-Verlag Berlin-New York)
[23] Hartshorne, R., Affine duality and cofiniteness, Invent. Math., 9, 145-164, (1969) · Zbl 0196.24301
[24] Hashimoto, M., Equivariant twisted inverses, (Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Math., vol. 1960, (2009), Springer Berlin), 261-478
[25] Haynes, M., Finite localizations, Bol. Soc. Mat. Mexicana (2), 37, 1-2, 383-389, (1992), Papers in honor of José Adem (Spanish) · Zbl 0852.55015
[26] Hopkins, M. J.; Smith, J. H., Nilpotence and stable homotopy theory. II, Ann. of Math. (2), 148, 1, 1-49, (1998) · Zbl 0927.55015
[27] Hovey, M., Bousfield localization functors and hopkins’ chromatic splitting conjecture, (The Čech Centennial, Boston, MA, 1993, Contemp. Math., vol. 181, (1995), Amer. Math. Soc. Providence, RI), 225-250 · Zbl 0830.55004
[28] Hovey, M., Morita theory for Hopf algebroids and presheaves of groupoids, Amer. J. Math., 124, 6, 1289-1318, (2002) · Zbl 1033.55002
[29] Hovey, M., Homotopy theory of comodules over a Hopf algebroid, (Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemp. Math., vol. 346, (2004), Amer. Math. Soc. Providence, RI), 261-304 · Zbl 1067.18012
[30] Hovey, M., Chromatic phenomena in the algebra of \(\operatorname{BP}_\ast \operatorname{BP}\)-comodules, (Elliptic Cohomology, London Math. Soc. Lecture Note Ser., vol. 342, (2007), Cambridge Univ. Press Cambridge), 170-203 · Zbl 1236.55019
[31] Hovey, M.; Palmieri, J. H.; Strickland, N. P., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., 128, 610, (1997), x+114 · Zbl 0881.55001
[32] Hovey, M.; Sadofsky, H., Invertible spectra in the \(E(n)\)-local stable homotopy category, J. Lond. Math. Soc. (2), 60, 1, 284-302, (1999) · Zbl 0947.55013
[33] Hovey, M. A.; Strickland, N. P., Morava K-theories and localisation, Mem. Amer. Math. Soc., 139, 666, (1999), viii+100-100 · Zbl 0929.55010
[34] Hovey, M.; Strickland, N., Comodules and Landweber exact homology theories, Adv. Math., 192, 2, 427-456, (2005) · Zbl 1075.55001
[35] Hovey, M.; Strickland, N., Local cohomology of \(\operatorname{BP}_\ast \operatorname{BP}\)-comodules, Proc. Lond. Math. Soc. (3), 90, 2, 521-544, (2005) · Zbl 1135.55001
[36] Isaksen, D. C.; Shkembi, A., Motivic connective K-theories and the cohomology of A(1), J. K-Theory, 7, 3, 619-661, (2011) · Zbl 1266.14015
[37] Joachimi, R., Thick ideals in equivariant and motivic stable homotopy categories, (March 2015), preprint
[38] Joyal, A., Quasi-categories and kan complexes, Special Volume Celebrating the 70th Birthday of Professor Max Kelly, J. Pure Appl. Algebra, 175, 1-3, 207-222, (2002) · Zbl 1015.18008
[39] Krause, H., The stable derived category of a Noetherian scheme, Compos. Math., 141, 5, 1128-1162, (2005) · Zbl 1090.18006
[40] Krause, H., Deriving Auslander’s formula, Doc. Math., 20, 669-688, (2015) · Zbl 1348.18018
[41] Landweber, P. S., Associated prime ideals and Hopf algebras, J. Pure Appl. Algebra, 3, 43-58, (1973) · Zbl 0257.55005
[42] Laumon, G.; Moret-Bailly, L., Champs algébriques, Ergeb. Math. Grenzgeb. (3), vol. 39, (2000), Springer-Verlag Berlin · Zbl 0945.14005
[43] T. Lawson, Secondary power operations and the Brown-Peterson spectrum at the prime 2, March 2017. To appear in Annals of Mathematics. · Zbl 1431.55011
[44] Lewis, L. G.; May, J. P.; Steinberger, M.; McClure, J. E., Equivariant stable homotopy theory, Lecture Notes in Math., vol. 1213, (1986), Springer-Verlag Berlin, With contributions by J.E. McClure · Zbl 0611.55001
[45] Lurie, J., Higher topos theory, Ann. of Math. Stud., vol. 170, (2009), Princeton University Press Princeton, NJ · Zbl 1175.18001
[46] J. Lurie, Proper morphisms, completions, and the Grothendieck existence theorem. Draft available from author’s website as http://www.math.harvard.edu/ lurie/papers/DAG-XII.pdf, 2011.
[47] J. Lurie, Higher Algebra, 2014, Draft available from author’s website as http://www.math.harvard.edu/lurie/papers/higheralgebra.pdf.
[48] Mahowald, M.; Sadofsky, H., \(v_n\) telescopes and the Adams spectral sequence, Duke Math. J., 78, 1, 101-129, (1995) · Zbl 0984.55008
[49] Mandell, M. A.; May, J. P., Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc., 159, 755, (2002), x+108 · Zbl 1025.55002
[50] Mathew, A.; Naumann, N.; Noel, J., Nilpotence and descent in equivariant stable homotopy theory, Adv. Math., 305, 994-1084, (2017) · Zbl 1420.55024
[51] May, J. P., Equivariant homotopy and cohomology theory, CBMS Reg. Conf. Ser. Math., vol. 91, (1996), Published for the Conference Board of the Mathematical Sciences Washington, DC, With contributions by M. Cole, G. Comezaña, S. Costenoble, A.D. Elmendorf, J.P.C. Greenlees, L.G. Lewis, Jr., R.J. Piacenza, G. Triantafillou, and S. Waner · Zbl 0890.55001
[52] Miller, H. R.; Ravenel, D. C., Morava stabilizer algebras and the localization of Novikov’s \(E_2\)-term, Duke Math. J., 44, 2, 433-447, (1977) · Zbl 0358.55019
[53] Morel, F.; Voevodsky, V., \(\mathbf{A}^1\)-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci., 90, 45-143, (2001), 1999
[54] Naumann, N., The stack of formal groups in stable homotopy theory, Adv. Math., 215, 2, 569-600, (2007) · Zbl 1157.55006
[55] Naumann, N.; Spitzweck, M.; Østvær, P. A., Motivic Landweber exactness, Doc. Math., 14, 551-593, (2009) · Zbl 1230.55005
[56] Porta, M.; Shaul, L.; Yekutieli, A., On the homology of completion and torsion, Algebr. Represent. Theory, 17, 1, 31-67, (2014) · Zbl 1316.13020
[57] Ravenel, D. C., Localization with respect to certain periodic homology theories, Amer. J. Math., 106, 2, 351-414, (1984) · Zbl 0586.55003
[58] Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres, Pure Appl. Math., vol. 121, (1986), Academic Press, Inc. Orlando, FL · Zbl 0608.55001
[59] Ravenel, D. C., Nilpotence and periodicity in stable homotopy theory, Ann. of Math. Stud., vol. 128, (1992), Princeton University Press Princeton, NJ, Appendix C by Jeff Smith · Zbl 0774.55001
[60] Ravenel, D. C., Progress report on the telescope conjecture, (Adams Memorial Symposium on Algebraic Topology, 2, Manchester, 1990, London Math. Soc. Lecture Note Ser., vol. 176, (1992), Cambridge Univ. Press Cambridge), 1-21 · Zbl 0751.55006
[61] Ravenel, D. C., Life after the telescope conjecture, (Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991, NATO Adv. Stud. Inst. Ser., Ser. C, Math. Phys. Sci., vol. 407, (1993), Kluwer Acad. Publ. Dordrecht), 205-222 · Zbl 0899.55009
[62] Robalo, M., K-theory and the bridge from motives to noncommutative motives, Adv. Math., 269, 399-550, (2015) · Zbl 1315.14030
[63] Schäppi, D., Ind-abelian categories and quasi-coherent sheaves, Math. Proc. Cambridge Philos. Soc., 157, 3, 391-423, (2014) · Zbl 1305.18043
[64] D. Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks, June 2012.
[65] Schenzel, P., Proregular sequences, local cohomology, and completion, Math. Scand., 92, 2, 161-180, (2003) · Zbl 1023.13011
[66] Schwede, S.; Shipley, B., Stable model categories are categories of modules, Topology, 42, 1, 103-153, (2003) · Zbl 1013.55005
[67] Shipley, B., \(H \mathbb{Z}\)-algebra spectra are differential graded algebras, Amer. J. Math., 129, 2, 351-379, (2007) · Zbl 1120.55007
[68] Sitte, T., Local cohomology sheaves on algebraic stacks, (2014), PhD thesis
[69] Tarrío, L. A.; López, A. J.; Lipman, J., Local homology and cohomology on schemes, Ann. Sci. Éc. Norm. Supér. (4), 30, 1, 1-39, (1997) · Zbl 0894.14002
[70] Tarrío, L. A.; López, A. J.; Lipman, J., Studies in duality on noetherian formal schemes and non-Noetherian ordinary schemes, Contemp. Math., vol. 244, (1999), American Mathematical Society Providence, RI · Zbl 0927.00024
[71] Tarrío, L. A.; López, A. J.; Pérez Rodríguez, M.; Vale Gonsalves, M. J., The derived category of quasi-coherent sheaves and axiomatic stable homotopy, Adv. Math., 218, 4, 1224-1252, (2008) · Zbl 1149.14016
[72] Tarrío, L. A.; López, A. J.; Pérez Rodríguez, M.; Vale Gonsalves, M. J., A functorial formalism for quasi-coherent sheaves on a geometric stack, Expo. Math., (2014)
[73] The Stacks Project Authors, Stacks project, (2015)
[74] Thomason, R. W., Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., 65, 1, 16-34, (1987) · Zbl 0624.14025
[75] Tibor, B., Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc., 129, 3, 447-475, (2000) · Zbl 0964.55018
[76] Voevodsky, V., \(\mathbf{A}^1\)-homotopy theory, (Proceedings of the International Congress of Mathematicians, vol. I, Berlin, 1998, (1998)), 579-604, (electronic) · Zbl 0907.19002
[77] Weibel, C. A., An introduction to homological algebra, Cambridge Stud. Adv. Math., vol. 38, (1994), Cambridge University Press Cambridge · Zbl 0797.18001
[78] White, D., Monoidal bousfield localizations and algebras over operads, (2014), preprint
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