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Completed power operations for Morava $$E$$-theory. (English) Zbl 1326.55018
The authors consider a completed version of the algebraic approximation functors which is due to C. Rezk [Homology Homotopy Appl. 11, No. 2, 327–379 (2009; Zbl 1193.55010)] by taking completions into account. By Morava $$E$$-theory $$E=E_h$$ is meant a commutative ring spectrum with coefficients $E_*=W\mathbb{F}_{p^h}[[u_1, \cdots, u_{h-1}]][u^\pm]$ for $$u$$ of degree 2 where $$W\mathbb{F}_{p^h}$$ is the ring of Witt vectors on $$\mathbb{F}_{p^h}$$. Let $$\text{Mod}_E$$ denote the category of $$E$$-modules. The functor $$\mathbb{P} : \text{Mod}_E \to \text{Mod}_E$$ is given by $$\mathbb{P}(M)=\bigvee_{n\geq 0}\mathbb{P}_n(M)=\bigvee_{n\geq 0} (E\Sigma_n)_+\wedge_{\Sigma_n} M^{\wedge n}$$, which defines a monad on its homotopy category $$h\,\text{Mod}_E$$. Let $$L_K$$ be the Bousfield localization with respect to Morava $$K$$-theory $$K(h)$$ and $$j : \text{id} \to L_K$$ be the localization functor. Let $$\pi_*$$ denote the natural functor $$h\, \text{Mod}_E \to \text{Mod}_{E_*}$$. Then $$\pi_*j$$ can be uniquely factorized into the composite $\pi_{*} @>{\eta}>> L_0\pi_* \to \pi_*L_K,$ $$L_0$$ denoting the 0th derived functor. The authors construct a new version of the algebraic approximation functors $$\mathbb{T}_n : \text{Mod}_{E_*} \to \text{Mod}_{E_*}$$, which is defined as the Kan extension of $$\pi_*i$$ along $$\pi_*L_K\mathbb{P}_ni$$ where $$i$$ denotes the natural functor $$\text{Mod}_E \to h\, \text{Mod}_E$$. Using these functors the main theorem of this paper can be stated as follows: The natural map $$L_0\mathbb{T}\eta$$ gives an isomorphism $L_0\mathbb{T}(M) \cong L_0\mathbb{T}L_0(M)$ for all $$E_*$$-modules $$M$$ where $$\mathbb{T}=\bigoplus_{n \geq 0}\mathbb{T}_n$$. Write $$\widehat{\text{Mod}}_{E_*}$$ for the subcategory of $$\text{Mod}_{E_*}$$ consisting of $$L$$-complete $$E_*$$-modules and denote by $$\iota$$ its inclusion, wherein $$M$$ is called $$L$$-complete if $$\eta : M \to L_0M$$ is an isomorphism. Then this result tells us that the completed algebraic approximation functor $$\widehat{\mathbb{T}} : \widehat{\text{Mod}}_{E_*} \to \widehat{\text{Mod}}_{E_*}$$, which can be written as $$\widehat{\mathbb{T}}=L_0\mathbb{T}\iota$$, admits a natural monad structure being compatible with that of $$L_K\mathbb{P}$$. The proof of this uses the fact that if $$M$$ is a flat $$E$$-module then there are natural isomorphisms $$\widehat{\mathbb{T}}_n\pi_*M \cong \pi_*L_K\mathbb{P}_nM$$, which exhibits an advantage of using $$\widehat{\mathbb{T}}$$. The authors say that “the completed algebraic approximation functor $$\widehat{\mathbb{T}}_n$$ resembles the structure on the homotopy groups of $$K(h)$$-local commutative $$E$$-algebras more closely than $$\mathbb{T}_n$$”. The second part of this paper discusses the case of $$E_1=K^\wedge_p$$ from the point of view of making the usefulness of $$\widehat{\mathbb{T}}$$ explicit and consequently obtains the following result: In the case $$h=1$$, $$\mathbb{T} : \text{Mod}_{E_*} \to \text{Mod}_{E_*}$$ becomes the free $$\mathbb{Z}/2$$-graded $$\theta$$-ring over the ground $$\theta$$-ring $$\mathbb{Z}_p$$.

##### MSC:
 55S25 $$K$$-theory operations and generalized cohomology operations in algebraic topology 55S12 Dyer-Lashof operations 13B35 Completion of commutative rings
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##### References:
 [1] J Adámek, J Rosický, E M Vitale, What are sifted colimits?, Theory Appl. Categ. 23 (2010) 251 · Zbl 1225.18002 · emis:journals/TAC/volumes/23/13/23-13abs.html · eudml:233342 [2] J Adámek, J Rosický, E M Vitale, Algebraic theories, Cambridge Tracts in Math. 184, Cambridge Univ. Press (2011) · Zbl 1209.18001 [3] M Ando, Notes on formal groups, unpublished notes based on a course taught by Michael Hopkins at MIT in the spring of 1990 [4] M Ando, Isogenies of formal group laws and power operations in the cohomology theories $$E_n$$, Duke Math. J. 79 (1995) 423 · Zbl 0862.55004 · doi:10.1215/S0012-7094-95-07911-3 [5] A Baker, $$L$$-complete Hopf algebroids and their comodules (editors C Ausoni, K Hess, J Scherer), Contemp. Math. 504, Amer. Math. Soc. (2009) 1 · Zbl 1192.55004 · doi:10.1090/conm/504/09873 [6] M Behrens, C Rezk, The Bousfield-Kuhn functor and topological André-Quillen cohomology (2012) · math.mit.edu [7] J Borger, B Wieland, Plethystic algebra, Advances Math. 194 (2005) 246 · Zbl 1098.13033 · doi:10.1016/j.aim.2004.06.006 [8] A K Bousfield, On $$\lambda$$-rings and the $$K$$-theory of infinite loop spaces, $$K$$-Theory 10 (1996) 1 · Zbl 0845.55010 · doi:10.1007/BF00534886 [9] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972) · Zbl 0259.55004 [10] R R Bruner, J P May, J E McClure, M Steinberger, $$H_\infty$$ ring spectra and their applications, Lecture Notes in Math. 1176, Springer (1986) · Zbl 0585.55016 · doi:10.1007/BFb0075405 [11] M Demazure, Lectures on $$p$$-divisible groups, Lecture Notes in Mathematics 302, Springer (1986) · Zbl 0247.14010 [12] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules and algebras in stable homotopy theory, Math. Surveys and Monographs 47, Amer. Math. Soc. (1997) · Zbl 0894.55001 [13] P G Goerss, H W Henn, M Mahowald, C Rezk, A resolution of the $$K(2)$$-local sphere at the prime $$3$$, Ann. of Math. 162 (2005) 777 · Zbl 1108.55009 · doi:10.4007/annals.2005.162.777 [14] P G Goerss, M J Hopkins, André-Quillen (co)homology for simplicial algebras over simplicial operads (editors D Arlettaz, K Hess), Contemp. Math. 265, Amer. Math. Soc. (2000) 41 · Zbl 0999.18009 · doi:10.1090/conm/265/04243 [15] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151 · Zbl 1086.55006 · doi:10.1017/CBO9780511529955.009 [16] J P C Greenlees, J P May, Derived functors of $$I$$-adic completion and local homology, J. Algebra 149 (1992) 438 · Zbl 0774.18007 · doi:10.1016/0021-8693(92)90026-I [17] M J Hopkins, $$K(1)$$-local $$E_\infty$$-ring spectra, Math. Surveys Monogr. 201, Amer. Math. Soc. (2014) 287 · Zbl 1328.55018 · doi:10.1090/surv/201/16 [18] M J Hopkins, M Mahowald, H Sadofsky, Constructions of elements in Picard groups (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 89 · Zbl 0799.55005 · doi:10.1090/conm/158/01454 [19] M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, II, Ann. of Math. 148 (1998) 1 · Zbl 0927.55015 · doi:10.2307/120991 · www.math.princeton.edu [20] M Hovey, Some spectral sequences in Morava $$E$$-theory (2004) · Zbl 1063.55003 · mhovey.web.wesleyan.edu [21] M Hovey, Morava $$E$$-theory of filtered colimits, Trans. Amer. Math. Soc. 360 (2008) 369 · Zbl 1128.55005 · doi:10.1090/S0002-9947-07-04298-5 [22] M Hovey, N P Strickland, Morava $$K$$-theories and localisation, Mem. Amer. Math. Soc. 139, Amer. Math. Soc. (1999) · Zbl 0929.55010 · doi:10.1090/memo/0666 [23] A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207 · Zbl 1015.18008 · doi:10.1016/S0022-4049(02)00135-4 [24] G Laures, An $$E_\infty$$ splitting of spin bordism, Amer. J. Math. 125 (2003) 977 · Zbl 1058.55001 · doi:10.1353/ajm.2003.0032 [25] J Lurie, Higher topos theory, Annals of Math. Studies 170, Princeton Univ. Press (2009) · Zbl 1175.18001 · doi:10.1515/9781400830558 [26] J Lurie, Higher algebra (2014) · www.math.harvard.edu [27] S Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998) · Zbl 0906.18001 [28] C Rezk, Power operations for Morava $$E$$-theory of height $$2$$ at the prime $$2$$, · arxiv:0812.1320 [29] C Rezk, Rings of power operations for Morava $$E$$-theories are Koszul, · Zbl 1193.55010 · arxiv:1204.4831 [30] C Rezk, The congruence criterion for power operations in Morava $$E$$-theory, Homology, Homotopy Appl. 11 (2009) 327 · Zbl 1193.55010 · doi:10.4310/HHA.2009.v11.n2.a16 · intlpress.com [31] C Rezk, Analytic completion (2013) · www.math.uiuc.edu [32] C Rezk, Power operations in Morava $$E$$-theory: Structure and calculations (2013) · www.math.uiuc.edu [33] B Richter, An Atiyah-Hirzebruch spectral sequence for topological André-Quillen homology, J. Pure Appl. Algebra 171 (2002) 59 · Zbl 1070.55003 · doi:10.1016/S0022-4049(01)00117-7 [34] E Riehl, Categorical homotopy theory, New Math. Monographs 24, Cambridge Univ. Press (2014) · Zbl 1317.18001 · doi:10.1017/CBO9781107261457 [35] A Salch, Approximation of subcategories by abelian subcategories, (2010) · arxiv:1006.0048 [36] , Stacks project (2014) · stacks.math.columbia.edu [37] N P Strickland, Morava $$E$$-theory of symmetric groups, Topology 37 (1998) 757 · Zbl 0912.55012 · doi:10.1016/S0040-9383(97)00054-2 [38] N P Strickland, Gross-Hopkins duality, Topology 39 (2000) 1021 · Zbl 0957.55003 · doi:10.1016/S0040-9383(99)00049-X [39] G Valenzuela, Homological algebra of complete and torsion modules, PhD thesis, Wesleyan University (2015) [40] D Yau, Lambda-rings, World Scientific (2010) · Zbl 1198.13003 · doi:10.1142/7664 [41] A Yekutieli, On flatness and completion for infinitely generated modules over Noetherian rings, Comm. Algebra 39 (2011) 4221 · Zbl 1263.13028 · doi:10.1080/00927872.2010.522159 [42] Y Zhu, The power operation structure on Morava $$E$$-theory of height $$2$$ at the prime $$3$$, Algebr. Geom. Topol. 14 (2014) 953 · Zbl 1310.55011 · doi:10.2140/agt.2014.14.953 · www.msp.warwick.ac.uk · arxiv:1210.3730
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