On Hopkins’ Picard group Pic\(_2\) at the prime 3. (English) Zbl 1185.55010
If (\(\mathcal{C},\otimes,e)\) is any reasonable symmetric monoidal category, then we can define the Picard group to be the group of isomorphism classes of invertible objects; that is, those objects \(X\) of \(\mathcal{C}\) for which there is another object \(Y\) so that \(X \otimes Y \cong e\). If \(\mathcal{C}\) is the stable homotopy category under smash product, the only invertible objects are the spheres \(S^n\); however, if \(\mathcal{C}\) is the \(E\)-local stable homotopy category for some homology theory \(E\), then the Picard group can be quite rich. This observation and this subject was initiated about twenty years ago by Mike Hopkins; of particular interest is the \(K(n)\)-local category, where \(K(n)\) is the \(n\)th Morava \(K\)-theory at a prime \(p\). Basic calculation appeared in [M. J. Hopkins, M. Mahowald and H. Sadofsky, Contemp. Math. 158, 89–126 (1994; Zbl 0799.55005)].
In general, the calculation of the \(K(n)\)-local Picard group proceeds in two steps. The first is a calculation in group cohomology, and the second is a calculation of exotic elements by homotopy theoretic methods. The group cohomology calculation is \(H^1(G_n,R_n^\times)\). Here \(G_n\) is the automorphism group of a pair \((\bar{\mathbb{F}}_p,\Gamma_n)\), with \(\Gamma_n\) a \(1\)-parameter formal group over the algebraic closure of the finite field of \(p\) elements and \(R_n^\times\) is the group of units in the Lubin-Tate deformation ring for \(\Gamma_n\). If \(n=1\), the group cohomology calculations are routine; however, there is a non-trivial exotic element if \(n=1\) and \(p=2\). If \(n=2\) and \(p > 3\), the group cohomology calculation was made in unpublished work by Hopkins; in this case, there are no exotic elements.
In this paper, the author makes the algebraic calculation at \(n=2\) and \(p=3\). This is harder than the previous case, as the group \(G_2\) contains \(3\)-torsion and, indeed, the delicate part of this paper is to analyze the contribution of that torsion. In the end, we have that for all primes \(p > 2\), \(H^1(G_2,R_2^\times) \cong \mathbb{Z}_p^2 \times \mathbb{Z}/2(p^2-1)\). Here \(\mathbb{Z}_p\) is the \(p\)-adic integers. If \(n=2\) and \(p=3\) there are exotic elements as well.
In general, the calculation of the \(K(n)\)-local Picard group proceeds in two steps. The first is a calculation in group cohomology, and the second is a calculation of exotic elements by homotopy theoretic methods. The group cohomology calculation is \(H^1(G_n,R_n^\times)\). Here \(G_n\) is the automorphism group of a pair \((\bar{\mathbb{F}}_p,\Gamma_n)\), with \(\Gamma_n\) a \(1\)-parameter formal group over the algebraic closure of the finite field of \(p\) elements and \(R_n^\times\) is the group of units in the Lubin-Tate deformation ring for \(\Gamma_n\). If \(n=1\), the group cohomology calculations are routine; however, there is a non-trivial exotic element if \(n=1\) and \(p=2\). If \(n=2\) and \(p > 3\), the group cohomology calculation was made in unpublished work by Hopkins; in this case, there are no exotic elements.
In this paper, the author makes the algebraic calculation at \(n=2\) and \(p=3\). This is harder than the previous case, as the group \(G_2\) contains \(3\)-torsion and, indeed, the delicate part of this paper is to analyze the contribution of that torsion. In the end, we have that for all primes \(p > 2\), \(H^1(G_2,R_2^\times) \cong \mathbb{Z}_p^2 \times \mathbb{Z}/2(p^2-1)\). Here \(\mathbb{Z}_p\) is the \(p\)-adic integers. If \(n=2\) and \(p=3\) there are exotic elements as well.
Reviewer: Paul Goerss (Evanston)
MSC:
| 55P42 | Stable homotopy theory, spectra |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
| 55Q51 | \(v_n\)-periodicity |
Citations:
Zbl 0799.55005References:
| [1] | K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer (1994) |
| [2] | P Goerss, H W Henn, M Mahowald, The homotopy of \(L_2V(1)\) for the prime 3 (editors G Arone, J Hubbuck, R Levi, M Weiss), Progr. Math. 215, Birkhäuser (2004) 125 · Zbl 1052.55010 |
| [3] | P Goerss, H W Henn, M Mahowald, C Rezk, A resolution of the \(K(2)\)-local sphere at the prime 3, Ann. of Math. \((2)\) 162 (2005) 777 · Zbl 1108.55009 · doi:10.4007/annals.2005.162.777 |
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| [5] | H W Henn, N Karamanov, M Mahowald, The homotopy of the \(K(2)\)-local Moore spectrum at the prime 3 revisited, to appear in Math. Zeit. · Zbl 1294.55003 · doi:10.1007/s00209-013-1167-4 |
| [6] | 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 |
| [7] | N Karamanov, À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de \(\pi_*{L_{K(2)}V(0)}\) et du groupe \(\mathrm{Pic}_2\) de Hopkins, PhD thesis, Université Louis Pasteur (2006) |
| [8] | N P Strickland, On the \(p\)-adic interpolation of stable homotopy groups (editors N Ray, G Walker), London Math. Soc. Lecture Note Ser. 176, Cambridge Univ. Press (1992) 45 · Zbl 0754.55008 |
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