# Looking for a reference for a generalization of the Weierstrass preparation theorem

I am looking for a reference for the following generalization of the Weierstrass preparation theorem for formal power series. Suppose that $A$ is a noetherian complete local ring with residue field $k$, and let $f_1, \dots, f_n$ be non-invertible elements of the ring $A[[x_1, \dots, x_n]]$. Assume that the ring $k[[x_1, \dots, x_n]]/(g_1, \dots, g_n)$ is finite over $k$, where $g_i$ is the image of $f_i$ in $k[[x_1, \dots, x_n]]$. There is a unique continuous homomorphism of $A$-algebras $A[[y_1, \dots, y_n]] \to A[[x_1, \dots, x_n]]$ sending $y_i$ into $f_i$. Then $A[[x_1, \dots, x_n]]$ is a finite free $A[[y_1, \dots, y_n]]$-module.

Notice that I am not looking for a proof, I have one. I assume that this result must be known, but it is not in any of the sources I am familiar with.

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You don't mention what sources you have looked at. Have you checked books on rigid analytic geometry? For example, there is a discussion of the Weierstrass Preparation Theorem in Non-Archimedean Analysis by Bosch, Guntzer, Remmert. I don't have the book in front of me to check if their treatment easily covers the generalization you mention. –  KConrad Mar 20 '11 at 22:59
Dear Keith, thanks. I have not looked in books on rigid analytic geometry, because this subject is outside of my radar screen, but I will. –  Angelo Mar 21 '11 at 8:20
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