Question

In his notes on elliptic cohomology, Lurie defines the multiplicative group $\mathbb{G}_m$ over a ring spectrum $A$ as $\operatorname{Spec} A[\mathbb{Z}]$. What is the value $\mathbb{G}_m(B)$ of the represented functor at an $A$-algebra $B$? If this is too hard to say in general: Are there at least any specific examples, other than Eilenberg-MacLane spectra, where one does know the answer?

Answer 1

It is slightly complicated.

One has a number of adjunctions: $$ \begin{eqnarray*} \mathbb{G}_m(B) &=& Hom_{A-alg}(A[\mathbb{Z}],B) \\ &\simeq& Hom_{E_\infty-rings}(\mathbb{S}[\mathbb{Z}],B)\\ &\simeq& Hom_{E_\infty-spaces}(\mathbb{Z},GL_1(B))\\ &\simeq& Hom_{spectra}(H\mathbb{Z},gl_1(B)). \end{eqnarray*} $$ (Note these adjunctions are weak equivalences of spaces, and the last two adjunctions require a fair amount of theory to make rigorous.)

The problem is that it is usually quite difficult to compute the maps out of the Eilenberg-Mac Lane spectrum $H\mathbb{Z}$ unless the target is also an Eilenberg-Mac Lane space. In the case where the algebra $B$ comes from a simplicial commutative ring, this is true and so one at least knows that the set of homotopy classes of maps $[H\mathbb{Z}, gl_1(B)]$ surjects onto $\pi_0(B)^\times$. Even for complex K-theory, the calculation is somewhat involved (but doable), but the only method that I can immediately think of involves the Bousfield-Kuhn functor.