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Consider $R=\mathbb{C}[[x]][\frac{d}{dx}]$ as a graded ring (assign $x$ zero weight, and $\frac{d}{dx}$ some non-zero weight). Are there some $A_n$-ring spectra naturally arising in homotopy theory such that $\pi_*X\approx R$?

P.S.: the answer should depend on the weight assigned to $\frac{d}{dx}$. I am not sure what the weight should be (most probably either 1 or 2). I also do not mind adjoining the inverse of $\frac{d}{dx}$ if that is necessary.

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    $\begingroup$ Do you really want $\mathbb C$? This ring is a bit large from an algebraic perspective. The ring spectrum $[MU,MU]$ is likely close to what you want. $\endgroup$ – Phil Tosteson Feb 21 at 18:21
  • $\begingroup$ @PhilTosteson honestly I only recently began studying this so I am not completely clear about what I want. If switching the field helps, I do not mind (I would prefer to stay in char 0, but if that is not possible, char p is ok) $\endgroup$ – rori Feb 21 at 18:29
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    $\begingroup$ The ring of differential operators is not a graded ring when x is placed in degree 0 and d/dx in some non-zero degree. The commutation relation [d/dx, x] = 1 is not homogenous. It is a filtered ring though, and one can consider the corresponding Rees algebra where you adjoin an element h so that [d/dx, x] = h. This is naturally a graded ring. $\endgroup$ – Sam Gunningham Feb 21 at 18:42
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    $\begingroup$ You might find the paper "Differential operators and the Steenrod algebra" by R.M.W Wood to be relevant. It seems related to Phil Tosteson's comment as well. $\endgroup$ – Sam Gunningham Feb 21 at 23:20
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    $\begingroup$ If you want to work in characteristic zero then the homotopy category of rational spectra is just equivalent to the category of graded rational vector spaces, so there isn't really anything interesting to say. $\endgroup$ – Neil Strickland Feb 22 at 7:21

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