Constructions of elements in Picard groups

Abstract

AasTRACT. We discuss the first author's PiC&l'd groups of stable homotopy. We give 1I. detailed description of the calculation of Pie" and go on to describe geometric constructions for lifts of the elements of PiCl' We also construct a 15 ceU complex that localizes to what we speculate ill an interesting element of Pic 2. For all n we describe an algebraic approxima.tion to Picn using the Ada.ms-Novikov spectral sequence. We also show that the p-adic integers embed in the group Pic n for all n and p. 1991 Mu.thematic~Subject ClaBificalion. Primary 55P42; Seconda.ry 55QI0. The authors were parLially supporLed by the National Science Founda.tion. This article is in final form and no version of it will be subm.itted for pubiicalion elsewhere.

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

“…Then it is shown in [4] (cf. [3]) that Pic(L 1 ) 0 = Z/2, whose generator is represented by the E(1)-localization of the question mark complex QM = V (0) ∪ η e 3 , where V (0) = S 0 ∪ 2 e 1 is the mod 2 Moore spectrum. Let E * r (W ) for a spectrum W denote the E r -term of the …”

On Hopkins’ Picard group Pic2at the prime 3

“…Pic.S/W n 7 ! S n defines an isomorphism between the integers and the Picard group of S , the stable homotopy category, by Hopkins, Mahowald and Sadofsky [6].…”

“…Hopkins' Picard group is the group Pic.K n / which we denote by Pic n . The first account of it appears in Strickland [8] and the case n D 1 is treated in detail in [6] where also some examples of elements of Pic 2 at the prime p D 2 are given.…”

“…Recall that E n is acted on by the (big) Morava stabilizer group G n D S n Ì Gal.F p n = F p / by E 1 -maps; see Goerss and Hopkins [4]. Let EG n be the category of profinite E n OEOEG n -modules, ie E nmodules with a continuous G n -action compatible with the action of G n on E n (see Goerss, Henn, Mahowald and Rezk [3] or Hopkins, Mahowald and Sadofsky [6] for details). The tensor product (over .E n / ) gives a monoidal structure on EG n .…”

“…The tensor product (over .E n / ) gives a monoidal structure on EG n . Proposition 1.1 [6] Let X 2 K n . Then the following conditions are equivalent:…”