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Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?

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  • $\begingroup$ Do you mean smooth function to $\mathbb{R}$ or holomorphic functions to $\mathbb{C}$... or what? $\endgroup$ – Mattia Talpo Dec 15 '15 at 17:13
  • $\begingroup$ Sorry, I mean complex valued smooth functions. $\endgroup$ – Ying Xie Dec 16 '15 at 1:59
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Yes, it is. First of all, the ring of germs of holomorphic functions $\mathcal{O}^h_x$ is flat over the ring of germs regular functions $\mathcal{O}_x$ at some point $x \in X$, see for example Taylor, "Several complex variables with Connections to Algebraic Geometry and Lie Groups", Theorem 13.3.5.

Secondly, $\mathcal{O}^r_x$ is flat over the ring $\mathcal{O}^h_x$ of real analytic functions. This is so, since germs of real analytic functions in $\mathbb{C}^n$ is isomorphic to germs of holomorphic functions in $\mathbb{C}^{2n}$, and $\mathcal{O}^h_{\mathbb{C}^{2n},0}$ is flat over $\mathcal{O}^h_{\mathbb{C}^n,0}$, see for example Fischer, "Complex Analytic Geometry", Proposition 3.17.

Finally, the ring of germs of smooth functions is flat over the ring of real analytic functions, which can be found in Malgrange, "Ideals of Differentiable Functions", Corollary VI.1.12.

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  • $\begingroup$ this was really cool $\endgroup$ – pro Dec 16 '15 at 19:21
  • $\begingroup$ I guess that you mean to write "$\mathcal{O}^r_x$ is flat over the ring $\mathcal{O}^h_x$" at the start of the second paragraph, since the inclusion goes the other way. Also, it looks like you meant to refer to Corollary VI.1.12 of Malgrange at the end of the last paragraph. $\endgroup$ – Michael Oct 13 '18 at 23:46
  • $\begingroup$ You were right @Michael, corrected now. $\endgroup$ – Richard L Oct 14 '18 at 7:14

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