Algebraic & Geometric Topology

The algebraic duality resolution at $p=2$

Agnès Beaudry

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The goal of this paper is to develop some of the machinery necessary for doing K(2)–local computations in the stable homotopy category using duality resolutions at the prime p = 2. The Morava stabilizer group S2 admits a surjective homomorphism to 2 whose kernel we denote by S21. The algebraic duality resolution is a finite resolution of the trivial 2[[S21]]–module 2 by modules induced from representations of finite subgroups of S21. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial 3[[G21]]–module 3 at the prime p = 3. The construction was never published and it is the main result in this paper. In the process, we give a detailed description of the structure of Morava stabilizer group S2 at the prime 2. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

Article information

Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3653-3705.

Received: 19 December 2014
Revised: 30 March 2015
Accepted: 14 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q45: Stable homotopy of spheres
Secondary: 55T99: None of the above, but in this section 55P60: Localization and completion

finite resolution K(2)-local chromatic homotopy theory


Beaudry, Agnès. The algebraic duality resolution at $p=2$. Algebr. Geom. Topol. 15 (2015), no. 6, 3653--3705. doi:10.2140/agt.2015.15.3653.

Export citation


  • A Adem, R,J Milgram, Cohomology of finite groups, 2nd edition, Grundl. Math. Wissen. 309, Springer, Berlin (2004)
  • A Beaudry, The chromatic splitting conjecture at $n=p=2$, preprint (2015)
  • A Beaudry, Towards $\pi_*L_{K(2)}V(0)$ at $p=2$, preprint (2015)
  • M Behrens, The homotopy groups of $S\sb {E(2)}$ at $p\geq 5$ revisited, Adv. Math. 230 (2012) 458–492
  • M Behrens, T Lawson, Isogenies of elliptic curves and the Morava stabilizer group, J. Pure Appl. Algebra 207 (2006) 37–49
  • I Bobkova, Resolutions in the $K(2)$–local category at the prime $2$, PhD thesis, Northwestern University, Ann Arbor, MI (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • A Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966) 442–470
  • C Bujard, Finite subgroups of extended Morava stabilizer groups, preprint (2012)
  • D,G Davis, Homotopy fixed points for $L\sb {K(n)}(E\sb n\wedge X)$ using the continuous action, J. Pure Appl. Algebra 206 (2006) 322–354
  • E,S Devinatz, M,J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47
  • J,D Dixon, M,P,F du Sautoy, A Mann, D Segal, Analytic pro-$\!p$ groups, 2nd edition, Cambridge Studies Adv. Math. 61, Cambridge Univ. Press (1999)
  • P,G Goerss, H-W Henn, The Brown–Comenetz dual of the $K(2)$–local sphere at the prime $3$, Adv. Math. 288 (2016) 648–678
  • P,G Goerss, H-W Henn, M Mahowald, The rational homotopy of the $K(2)$–local sphere and the chromatic splitting conjecture for the prime $3$ and level $2$, Doc. Math. 19 (2014) 1271–1290
  • P 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) 777–822
  • P Goerss, H-W Henn, M Mahowald, C Rezk, On Hopkins' Picard groups for the prime $3$ and chromatic level $2$, J. Topol. 8 (2015) 267–294
  • P,G Goerss, M,J Hopkins, Moduli spaces of commutative ring spectra, from: “Structured ring spectra”, (Cambridge, editor), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • H-W Henn, Centralizers of elementary abelian $p$–subgroups and mod-$\!p$ cohomology of profinite groups, Duke Math. J. 91 (1998) 561–585
  • H-W Henn, N Karamanov, M Mahowald, The homotopy of the $K(2)$–local Moore spectrum at the prime $3$ revisited, Math. Z. 275 (2013) 953–1004
  • T Hewett, Finite subgroups of division algebras over local fields, J. Algebra 173 (1995) 518–548
  • T Hewett, Normalizers of finite subgroups of division algebras over local fields, Math. Res. Lett. 6 (1999) 271–286
  • M Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture, from: “The Čech centennial”, (M Cenkl, H Miller, editors), Contemp. Math. 181, Amer. Math. Soc. (1995) 225–250
  • A Huber, G Kings, N Naumann, Some complements to the Lazard isomorphism, Compos. Math. 147 (2011) 235–262
  • J Kohlhaase, On the Iwasawa theory of the Lubin–Tate moduli space, Compos. Math. 149 (2013) 793–839
  • M Lazard, Groupes analytiques $p$–adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965) 389–603
  • J Neukirch, A Schmidt, K Wingberg, Cohomology of number fields, Grundl. Math. Wissen. 323, Springer, Berlin (2000)
  • D,C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press, Orlando, FL (1986)
  • D,C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton Univ. Press (1992)
  • L Ribes, P Zalesskii, Profinite groups, 2nd edition, Ergeb. Math. Grenzgeb. 40, Springer, Berlin (2010)
  • J-P Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413–420
  • K Shimomura, The Adams–Novikov $E\sb 2$–term for computing $\pi\sb *(L\sb 2V(0))$ at the prime $2$, Topology Appl. 96 (1999) 133–152
  • K Shimomura, X Wang, The Adams–Novikov $E\sb 2$–term for $\pi\sb *(L\sb 2S\sp 0)$ at the prime $2$, Math. Z. 241 (2002) 271–311
  • K Shimomura, X Wang, The homotopy groups $\pi\sb *(L\sb 2S\sp 0)$ at the prime $3$, Topology 41 (2002) 1183–1198
  • K Shimomura, A Yabe, The homotopy groups $\pi\sb *(L\sb 2S\sp 0)$, Topology 34 (1995) 261–289
  • N,P Strickland, Gross–Hopkins duality, Topology 39 (2000) 1021–1033
  • P Symonds, T Weigel, Cohomology of $p$–adic analytic groups, from: “New horizons in pro-$\!p$ groups”, (M du Sautoy, D Segal, A Shalev, editors), Progr. Math. 184, Birkhäuser, Boston (2000) 349–410