Values of the multiplicative group over a ring spectrum - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:05:50Z http://mathoverflow.net/feeds/question/35488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35488/values-of-the-multiplicative-group-over-a-ring-spectrum Values of the multiplicative group over a ring spectrum e c 2010-08-13T12:41:38Z 2010-08-16T15:17:59Z <p>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?</p> http://mathoverflow.net/questions/35488/values-of-the-multiplicative-group-over-a-ring-spectrum/35493#35493 Answer by Tyler Lawson for Values of the multiplicative group over a ring spectrum Tyler Lawson 2010-08-13T13:37:20Z 2010-08-13T13:37:20Z <p>It is slightly complicated.</p> <p>One has a number of adjunctions: <code>$$\begin{eqnarray*} \mathbb{G}_m(B) &amp;=&amp; Hom_{A-alg}(A[\mathbb{Z}],B) \\ &amp;\simeq&amp; Hom_{E_\infty-rings}(\mathbb{S}[\mathbb{Z}],B)\\ &amp;\simeq&amp; Hom_{E_\infty-spaces}(\mathbb{Z},GL_1(B))\\ &amp;\simeq&amp; Hom_{spectra}(H\mathbb{Z},gl_1(B)). \end{eqnarray*}$$</code> (Note these adjunctions are weak equivalences of spaces, and the last two adjunctions require a fair amount of theory to make rigorous.)</p> <p>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 <code>$[H\mathbb{Z}, gl_1(B)]$</code> surjects onto <code>$\pi_0(B)^\times$</code>. 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.</p>