# Connection between complex orientations and R-orientations for a ring spectrum R?

We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion $\mathbb{C}P^1\hookrightarrow \mathbb{C}P^\infty$. Is there a connection between this notion of "complex orientation" and the notion of a Thom spectrum being $R$ oriented with respect to a ring spectrum $R$ in the sense of ABGHR? That is, for a Thom spectrum $Mf$ associated to a map $f:X\to BGL_1\mathbb{S}$, and a ring spectrum $R$, we say that $Mf$ is $R$-oriented if the composition $X\to BGL_1\mathbb{S}\to BGL_1R$ is null. If we consider $MU$ to be the Thom spectrum associated to some map (I'm not sure which it should be, but I suspect this is the way it's done) $BU\to BGL_1\mathbb{S}$, can we rephrase the notion of complex orientation in this language?

Thanks!

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What is ABGHR? I guess you already know that a complex orientation on $E$ is the same as a map of ring spectra $MU\to E$. –  Mark Grant Feb 12 '13 at 19:57
Sorry, ABGHR is the paper: arxiv.org/abs/0810.4535 –  Jon Beardsley Feb 12 '13 at 20:49
And yeah, thanks Mark. I'm trying to figure out how that's related. –  Jon Beardsley Feb 12 '13 at 20:51

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