On Hopkins’ Picard group Pic₂at the prime 3

A Mini-Course on Morava Stabilizer Groups and Their Cohomology

“…This defines a one-dimensional faithful representation of C 8 over W F9 which we denote it by l 1 , and its k-th tensor power by l k . Then the l k are naturally Z 3 [SD 16 ]-modules and l 4 splits as l 4,+ ⊕ l 4,− . Furthermore l 4,− is the representation χ of 3.4.…”

On Hopkins’ Picard groups for the prime 3 and chromatic level 2

“…The case p = 2 is considerably harder and not discussed here at all. The following algebraic results are due to Hopkins [13] if p > 3 (see also Lader [20]) and to Karamanov if p = 3 (see [19]). Theorem 2.9.…”

“…Part (2) follows from the observation [14] that the extension 0 −→ (Pic n ) 0 alg −→ (Pic n ) alg −→ Z/2 −→ 0 cannot be split. For part (3) and for the image of the elements (E 2 ) 0 S 2 and (E 2 ) 0 det under the isomorphism of part (a) of this theorem we refer the reader to [19,20]. Part (4) follows from Proposition 2.7 and from part (3).…”

The centralizer resolution of the 𝐾(2)-local sphere at the prime 2

“…These resolutions complement each other and they have been crucial in recent progress of our understanding of K(2)-local homotopy theory at the prime 3. In particular they have been used for proving the chromatic splitting conjecture for n = 2 [13], for determining Hopkins' Picard group of K(2)-local spectra [22], [15] and for identifying the Brown-Comenetz dual of the K(2)-local sphere [16].…”