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Ricardo Andrade
Ricardo Andrade
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Is there a good notion of the spectrum of units $R^*$$R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?

A standard definition (see section 1.2 in http://arxiv.org/pdf/0810.4535v3.pdfhttp://arxiv.org/abs/0810.4535) seems to output a connective spectrum $gl_1(R)$: the underlying space is the fibrepullback of $\Omega^\infty(R) \to \pi_0(R)$ over the units $\pi_0(R)^\times \subset \pi_0(R)$. While this is completely reasonable if $R$ is itself connective, is there is a better operation when one does not assume $R$ to be connective?

Is there a good notion of the spectrum of units $R^*$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?

A standard definition (see section 1.2 in http://arxiv.org/pdf/0810.4535v3.pdf) seems to output a connective spectrum $gl_1(R)$: the underlying space is the fibre of $\Omega^\infty(R) \to \pi_0(R)$ over the units $\pi_0(R)^\times \subset \pi_0(R)$. While this is completely reasonable if $R$ is itself connective, is there is a better operation when one does not assume $R$ to be connective?

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?

A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems to output a connective spectrum $gl_1(R)$: the underlying space is the pullback of $\Omega^\infty(R) \to \pi_0(R)$ over the units $\pi_0(R)^\times \subset \pi_0(R)$. While this is completely reasonable if $R$ is itself connective, is there a better operation when one does not assume $R$ to be connective?

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unit
unit
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Units of a ring spectrum

Is there a good notion of the spectrum of units $R^*$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?

A standard definition (see section 1.2 in http://arxiv.org/pdf/0810.4535v3.pdf) seems to output a connective spectrum $gl_1(R)$: the underlying space is the fibre of $\Omega^\infty(R) \to \pi_0(R)$ over the units $\pi_0(R)^\times \subset \pi_0(R)$. While this is completely reasonable if $R$ is itself connective, is there is a better operation when one does not assume $R$ to be connective?