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Fernando Muro
Fernando Muro
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Fiber Fibre sequence of module spectra induces a Fiberfibre sequence of K$K$-theory spectra?

Let $A$ be an $\mathbb{E}_\infty$-ring spectraspectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras. We assume there is a homotopy fibre sequence $$ R_1\to R_2 \to R_3 $$ in athe stable infinity category $\text{Mod}_{A}(\text{Sp})$. Then, is there a homotopy fibre sequence $$ K(R_1) \to K(R_2) \to K(R_3) $$ in $\text{Mod}_{K(A)}(\text{Sp})$?

Fiber sequence of module spectra induces a Fiber sequence of K-theory?

Let $A$ be an $\mathbb{E}_\infty$-ring spectra, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras. We assume there is a homotopy sequence $$ R_1\to R_2 \to R_3 $$ in a stable infinity category $\text{Mod}_{A}(\text{Sp})$. Then is there a homotopy sequence $$ K(R_1) \to K(R_2) \to K(R_3) $$ in $\text{Mod}_{K(A)}(\text{Sp})$?

Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?

Let $A$ be an $\mathbb{E}_\infty$-ring spectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras. We assume there is a homotopy fibre sequence $$ R_1\to R_2 \to R_3 $$ in the stable infinity category $\text{Mod}_{A}(\text{Sp})$. Then, is there a homotopy fibre sequence $$ K(R_1) \to K(R_2) \to K(R_3) $$ in $\text{Mod}_{K(A)}(\text{Sp})$?

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user145752
user145752
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Fiber sequence of module spectra induces a Fiber sequence of K-theory?

Let $A$ be an $\mathbb{E}_\infty$-ring spectra, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras. We assume there is a homotopy sequence $$ R_1\to R_2 \to R_3 $$ in a stable infinity category $\text{Mod}_{A}(\text{Sp})$. Then is there a homotopy sequence $$ K(R_1) \to K(R_2) \to K(R_3) $$ in $\text{Mod}_{K(A)}(\text{Sp})$?