Let $R$ be a commutative ring spectrum (interpret this as you will; as an $E_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}_1(R)$ as usual denote its space of units. If $\tilde R$ is the connective cover of $R,$ is there a simple relationship between $\operatorname{GL}_1(\tilde R)$ and $\operatorname{GL}_1(R)$?
$\begingroup$
$\endgroup$
2
asked Sep 30, 2016 at 14:44
-
3$\begingroup$ They are equal by definition. $\endgroup$– Oscar Randal-WilliamsCommented Sep 30, 2016 at 14:58
-
2$\begingroup$ It might be worthwhile to remark that the space of units $GL_1(R)$ of a commutative ring spectrum has nevertheless a nontrivial delooping $Pic(R)$ that does depend on the negative homotopy of $R$, and that this construction can be somehow "iterated" to get more and more exotic deloopings. See for example arxiv.org/abs/1005.5370 $\endgroup$– Denis NardinCommented Sep 30, 2016 at 15:15
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
1
By definition, the space $GL_1(R)$ is the subspace of $\Omega^\infty R$ consisting of those elements whose path component $\alpha \in \pi_0(\Omega^\infty R) = \pi_0(R)$ is a unit in $\pi_0(R)$.
If $\tilde R \to R$ is a connective cover, then the map of spaces $\Omega^\infty \tilde R \to \Omega^\infty R$ is an isomorphism on all homotopy groups, and hence we get a weak equivalence $GL_1(\tilde R) \to GL_1(R)$.
answered Sep 30, 2016 at 14:59
-
$\begingroup$ Thank you a lot - indeed, obviously that the two $0$-th spaces of the $\Omega$-spectrum of $R$ and $\tilde R$ agree. Cheers for having the patience to spell out something so basic for me; it is greatly appreciated! $\endgroup$ Commented Sep 30, 2016 at 15:06
$\begingroup$
$\endgroup$
For symmetric ring spectra $R$ there is also a definition of the graded group of units, $GL_1^J(R)$, which retains information about the negative homotopy groups of $R$. See Sagave-Schlichtkrull, "Diagram spaces and symmetric spectra", Advances in Mathematics (2012). Here $J = \Sigma^{-1} \Sigma$ is a specific category, and the graded units can be viewed as spaces over $BJ \simeq QS^0$. For instance, $GL_1^J(ku)$ is different from $GL_1^J(KU)$, where $ku$ is the connective cover of the periodic complex $K$-theory spectrum $KU$.
