# zbMATH — the first resource for mathematics

Chromatic homotopy theory is asymptotically algebraic. (English) Zbl 1442.55002
Localizing at the Johnson-Wilson theories $$E(n)$$ provides a filtration of ($$p$$-local) spectra $L_0\mathrm{Sp}_{(p)} \subset L_1\mathrm{Sp}_{(p)} \subset \cdots \subset L_n\mathrm{Sp}_{(p)}\subset \cdots \subset \mathrm{Sp}_{(p)}$ in which the bottom layer $$L_0\mathrm{Sp}_{(p)}$$ is the category of rational spectra. Serre’s work shows that the category of rational spectra is equivalent to the derived category of $$\mathbb{Q}$$, and one can ask if algebraic models can be found at higher chromatic heights. In fact, for all primes $$p$$ and $$n>0$$, there is no algebraic model for $$L_n\mathrm{Sp}_{(p)}$$. Instead of fixing the prime $$p$$ and varying the height $$n$$, one can fix $$n$$ and vary the prime. The main result of this paper states that at a fixed height $$n$$, as $$p \to \infty$$ chromatic homotopy theory admits a symmetric monoidal algebraic model. This algebraic model is built from the categories of twisted complexes of quasi-coherent sheaves on the moduli stack of formal groups introduced by Franke and further studied by M. Hovey [Contemp. Math. 346, 261–304 (2004; Zbl 1067.18012)] and D. Barnes and C. Roitzheim [Adv. Math. 228, No. 6, 3223–3248 (2011; Zbl 1246.55009)].
In order to make sense of such a statement, the authors develop a notion of ultraproduct for $$\infty$$-categories, motivated by the notion of an ultraproduct in model theory. This definition captures the asympotic behaviour of a collection of objects, and is likely to be of independent interest. The authors suggest many interesting applications of their main result to be returned to in future work, and give details of one particular application: the existence of multiplicative structures on local generalized Moore spectra.

##### MSC:
 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55P42 Stable homotopy theory, spectra 03C20 Ultraproducts and related constructions
##### Keywords:
ultraproduct; chromatic homotopy theory; algebraic model
Full Text:
##### References:
 [1] Ayala, D.; Francis, J.; Tanaka, HL, Factorization homology of stratified spaces, Sel. Math. (N.S.), 23, 1, 293-362 (2017) · Zbl 1365.57037 [2] Ax, J.; Kochen, S., Diophantine problems over local fields. I, Am. J. Math., 87, 605-630 (1965) · Zbl 0136.32805 [3] Ax, J.; Kochen, S., Diophantine problems over local fields. II. A complete set of axioms for $$p$$-adic number theory, Am. J. Math., 87, 631-648 (1965) · Zbl 0136.32805 [4] Ax, J.; Kochen, S., Diophantine problems over local fields. III. Decidable fields, Ann. Math. (2), 83, 437-456 (1966) · Zbl 0223.02050 [5] Adámek, J.; Lawvere, FW; Rosický, J., Continuous categories revisited, Theory Appl. Categ., 11, 11, 252-282 (2003) · Zbl 1018.18003 [6] Ax, J., The elementary theory of finite fields, Ann. Math. (2), 88, 239-271 (1968) · Zbl 0195.05701 [7] Beaudry, A., The chromatic splitting conjecture at $$n = p = 2$$, Geom. Topol., 21, 6, 3213-3230 (2017) · Zbl 1421.55010 [8] Beaudry, A., Goerss, P. G., Henn, H.-W.: Chromatic splitting for the $$K(2)$$-local sphere at $$p=2$$, ArXiv e-prints (2017) [9] Barthel, Tobias, Heard, Drew: Algebraic chromatic homotopy theory for $$BP_*BP$$-comodules, preprint. https://arxiv.org/abs/1708.09261 · Zbl 1412.55005 [10] Blass, A., A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 25, 4, 329-331 (1977) · Zbl 0365.02054 [11] Bousfield, AK, The localization of spectra with respect to homology, Topology, 18, 4, 257-281 (1979) · Zbl 0417.55007 [12] Bousfield, AK, On the homology spectral sequence of a cosimplicial space, Am. J. Math., 109, 2, 361-394 (1987) · Zbl 0623.55009 [13] Baker, A.; Richter, B., Invertible modules for commutative $${\mathbb{S}}$$-algebras with residue fields, Manuscripta Math., 118, 1, 99-119 (2005) · Zbl 1092.55007 [14] Barnes, D.; Roitzheim, C., Monoidality of Franke’s exotic model, Adv. Math., 228, 6, 3223-3248 (2011) · Zbl 1246.55009 [15] Bell, JL; Slomson, AB, Models and Ultraproducts: An Introduction (1969), Amsterdam, London: North-Holland Publishing Co., Amsterdam, London [16] Chang, C. C., Keisler, H. J.: Model theory, third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam (1990) [17] Devinatz, ES; Hopkins, MJ, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Am. J. Math., 117, 3, 669-710 (1995) · Zbl 0842.14034 [18] Franke, Jens: Uniqeness theorems for certain triangulated categories with an Adams spectral sequence. http://www.math.uiuc.edu/K-theory/0139/ [19] Goerss, P. G., Hopkins, M. J.: Moduli problems for structured ring spectra. www.math.northwestern.edu/pgoerss/spectra · Zbl 1086.55006 [20] Gepner, D.; Haugseng, R., Enriched $$\infty$$-categories via non-symmetric $$\infty$$-operads, Adv. Math., 279, 575-716 (2015) · Zbl 1342.18009 [21] Goerss, PG; Henn, H-W; Mahowald, M., The rational homotopy of the $$K(2)$$-local sphere and the chromatic splitting conjecture for the prime 3 and level 2, Doc. Math., 19, 1271-1290 (2014) · Zbl 1315.55009 [22] Goerss, P.; Henn, H-W; Mahowald, M.; Rezk, C., A resolution of the $$K(2)$$-local sphere at the prime 3, Ann. Math. (2), 162, 2, 777-822 (2005) · Zbl 1108.55009 [23] Glasman, S., A spectrum-level Hodge filtration on topological Hochschild homology, Sel. Math. (N.S.), 22, 3, 1583-1612 (2016) · Zbl 1371.18011 [24] Henn, Hans-Werner: On finite resolutions of $$K(n)$$-local spheres, Elliptic cohomology, London Math. Soc. Lecture Note Ser., vol. 342, Cambridge Univ. Press, Cambridge, pp. 122-169 (2007) · Zbl 1236.55015 [25] Heuts, Gijs: Goodwillie approximations to higher categories. http://arxiv.org/abs/1510.03304 · Zbl 1431.55013 [26] Hinich, V., Dwyer-Kan localization revisited, Homol. Homot. Appl., 18, 1, 27-48 (2016) · Zbl 1346.18024 [27] Michael, H., Jacob, L.: Ambidexterity in $$k(n)$$-local stable homotopy theory. http://www.math.harvard.edu/ lurie/ [28] Hopkins, M.: Complex oriented cohomology theories and the language of stacks. http://www.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf [29] Hovey, M.: Bousfield localization functors and Hopkins’ chromatic splitting conjecture, The Čech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Am. Math. Soc., Providence, RI, pp. 225-250 (1995) · Zbl 0830.55004 [30] 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, Amer. Math. Soc., Providence, RI, pp. 261-304 (2004) · Zbl 1067.18012 [31] Hopkins, MJ; Palmieri, JH; Smith, JH, Vanishing lines in generalized Adams spectral sequences are generic, Geom. Topol., 3, 155-165 (1999) · Zbl 0920.55020 [32] Hopkins, MJ; Smith, JH, Nilpotence and stable homotopy theory. II, Ann. Math. (2), 148, 1, 1-49 (1998) · Zbl 0927.55015 [33] 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 [34] Hovey, M.; Strickland, NP, Morava $$K$$-theories and localisation, Mem. Am. Math. Soc., 139, 666, 100 (1999) · Zbl 0929.55010 [35] Iyengar, S., André-quillen homology of commutative algebras, Contemp. Math., 436, 203 (2007) · Zbl 1136.13009 [36] Lurie, J.: Higher algebra. http://www.math.harvard.edu/ lurie/ · Zbl 1175.18001 [37] Lurie, J.: Spectral algebraic geometry. http://www.math.harvard.edu/ lurie/ [38] Lurie, J., Higher Topos Theory, Annals of Mathematics Studies (2009), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1175.18001 [39] Lurie, Jacob: Elliptic cohomology II: Orientations, http://www.math.harvard.edu/ lurie/, 2018 [40] Mathew, A., A thick subcategory theorem for modules over certain ring spectra, Geom. Topol., 19, 4, 2359-2392 (2015) · Zbl 1405.55009 [41] Mathew, A., The Galois group of a stable homotopy theory, Adv. Math., 291, 403-541 (2016) · Zbl 1338.55009 [42] Mazel-Gee, A.: Quillen adjunctions induce adjunctions of quasicategories. https://arxiv.org/abs/1501.03146 · Zbl 1346.18003 [43] Mitchell, S.A.: Hypercohomology spectra and Thomason’s descent theorem, Algebraic $$K$$-theory (Toronto, ON, : Fields Inst. Commun., vol. 16, Amer. Math. Soc. Providence, RI 1997, 221-277 (1996) · Zbl 0888.19003 [44] Morava, J., Noetherian localisations of categories of cobordism comodules, Ann. Math. (2), 121, 1, 1-39 (1985) · Zbl 0572.55005 [45] Mathew, A.; Stojanoska, V., The Picard group of topological modular forms via descent theory, Geom. Topol., 20, 6, 3133-3217 (2016) · Zbl 1373.14008 [46] Lee, S., Nave, The Smith-Toda complex $$V((p+1)/2)$$ does not exist, Ann. Math. (2), 171, 1, 491-509 (2010) · Zbl 1194.55017 [47] Patchkoria, I., On exotic equivalences and a theorem of franke, Bull. Lond. Math. Soc., 49, 6, 1085-1099 (2017) · Zbl 1388.55008 [48] Douglas, C., Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Pure and Applied Mathematics (1986), Orlando, FL: Academic Press Inc, Orlando, FL · Zbl 0608.55001 [49] Ravenel, D.C.: Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, Princeton, NJ, 1992, Appendix C by Jeff Smith · Zbl 0774.55001 [50] Rognes, J.: Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008), no. 898, viii+137 · Zbl 1166.55001 [51] Schwede, S., The stable homotopy category is rigid, Ann. Math. (2), 166, 3, 837-863 (2007) · Zbl 1151.55007 [52] Schoutens, H.: The use of ultraproducts in commutative algebra. Lecture Notes in Mathematics, vol. 1999. Springer-Verlag, Berlin (2010) · Zbl 1205.13002 [53] Schwede, S.; Shipley, B., Stable model categories are categories of modules, Topology, 42, 1, 103-153 (2003) · Zbl 1013.55005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.