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I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and denote $\xi_n $ its universal bundle. The cobordism spectrum $\mathrm{MGL}$ is made out of the universal Thom spaces.That is to say, it is the spectrum $$ (S\, ,\, \mathrm{Th}(\xi_1)\, ,\, \mathrm{Th}(\xi_2)\, ,\, \ldots \, ,\, \mathrm{Th}(\xi_i)\, ,\, \ldots \ ) $$ with the natural bounding maps. My question is

Is $\mathrm{MGL}[2r](r)$ isomorphic, in $\mathbf{SH}$, to the spectrum $(\, \mathrm{Th}(\xi_r)\, ,\, \mathrm{Th}(\xi_{r+1})\, ,\, \ldots \, ,\, \mathrm{Th}(\xi_{r+i})\, ,\, \ldots \ )$? And if so, why?

It seems to me that this should be right and simple, but since $\mathrm{MGL}$ is not an $\Omega$-spectrum I fail to see a direct reason.

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A simple way of seeing it is to explicitly spell out what we mean when we say that a spectrum is "presented" by a prespectrum. To say that that a spectrum $E$ is presented by $$(E_0,E_1,...)$$ means that, in whatever model for motivic spectra you are using, $$E\cong \mathrm{colim}_k \Sigma^{-2k,-k}\Sigma^{\infty}E_k$$ (here the colimit is a homotopy colimit). Since suspensions commute with colimits (being equivalences) we have that $$\Sigma^{2n,n}E\cong \mathrm{colim}_k \Sigma^{2(n-k),(n-k)}\Sigma^{\infty}E_k\cong \mathrm{colim}_k \Sigma^{-2k,-k}\Sigma^\infty E_{k+n}$$ where the last step is just reindexing the colimit. In particular, $\Sigma^{2n,n}E$ is presented by $$(E_n,E_{n+1},\dots)\,.$$

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  • $\begingroup$ Thanks again for the answer, Denis. Could you encourage me a reference to learn these basics? thanks $\endgroup$
    – Tintin
    Commented Apr 4, 2019 at 8:46
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    $\begingroup$ @Tintin Unfortunately references are known to be annoyingly spread out in homotopy theory. The trick I used in this answer is quite widespread whenever some version of spectra is used, I first learned it in the equivariant setting in the Hill-Hopkins-Ravenel paper (however it is useful even in the classical setting of spectra on topological spaces!). The best I can do is to encourage you to read some good introduction to spectra (possibly the best currently available is Schwede's book), although even there you won't find everything. $\endgroup$
    – Denis Nardin
    Commented Apr 4, 2019 at 8:51

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