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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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