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I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an analogue question in the classic setting it would be appreciated.

Let $X$ be a scheme, in Voevodsky motivic homotopy theory we have an unstable homotopy category of $\mathbf{H}(X)$, a stable homotoy category $\mathbf{SH}(X)$ and an infinite suspension $\Sigma^\infty \colon \mathbf{H}(X) \to \mathbf{SH}(X)$. We have a looping given by the internal Hom object $\underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )$ in $\mathbf{H}(X)$ and also a looping $\underline{\mathrm{Hom}}_{\mathbf{SH}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )=\mathrm{Th}(-\mathcal{O})\wedge \underline{\phantom{a}}\ $ in $\mathbf{SH}(X)$. I would like to understand the relation between $\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )$ and $\mathrm{Th}(-\mathcal{O})\wedge \underline{\phantom{a}}\ $ and if for some simple objects they are the same.

To be more precise, let $L$ be a line bundle in $X$.

  1. What is the relation between $\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L))$ and $\mathrm{Th}(-\mathcal{O})\wedge \mathrm{Th}(L) $? And between $\mathrm{Th}(\mathcal{O})\wedge \Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L))$ and $ \mathrm{Th}(L)$? Are they isomorphic?
  2. If they are not isomorphic, it seems to me that they do have the same cohomology. To fix ideas: is their $K$-theory $K_0(\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L)))$ and $K_0(\mathrm{Th}(L-\mathcal{O}))$ isomorphic? What about $K_0(\mathrm{Th}(\mathcal{O})\wedge \Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L)))$ and $K_0(\mathrm{Th}(L))$, are they isomorphic?
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