# Quotient Algebras, Ideals and Isomorphisms/Biholomorphisms

Consider a multivariate (complex) analytic function $f \colon (\mathbb{C}^{n}, \mathbf{0}) \to (\mathbb{C},0)$. Let $J_{f} = \langle \partial_{1} f, \dots, \partial_{n} f \rangle$ denote the Jacobi ideal (over $\mathbb{C}$) of $f$ generated by its partial derivatives. Define the Milnor Algebra of $f$ as the quotient $A_{f} = O_{n} / J_{f}$, where $O_{n} = \mathbb{C} \{z_1, \dots, z_n \}$ denotes the polynomial ring of convergent power series (about the origin).

For example, take $f = z^3$. Then $A_{f} = \mathbb{C}\{z\}/\langle z^2 \rangle \simeq \{a + b z \ | \ a, b \in \mathbb{C} \}$, which is two dimensional.

Given another multivariate (complex) analytic function $g \colon (\mathbb{C}^{m}, \mathbf{0}) \to (\mathbb{C},0)$, it is clear that if $O_{n} \simeq O_{m}$ and $J_{f} \simeq J_{g}$, then one has the algebra isomorphism $A_{f} \simeq A_{g}$.

If only the following isomorphisms $J_{f} \simeq J_{g}$ and $A_{f} \simeq A_{g}$ hold, is it necessarily true that $O_{n} \simeq O_{m}$ and $n = m$? Are such isomorphism necessarily biholomorphisms or coordinate changes? What pairs of isomorphisms imply a third among $J$, $A$ and $O$?

Any help, hints or references are certainly appreciated. Thanks!

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What about the Milnor algebra of $z^2 + w^3$, compared with that of $z^3$? Won't they be isomorphic? – Akhil Mathew Jul 11 '11 at 13:20
Sure. Milnor algebras are invariant under suspension: $A_{f} \simeq A_{f + z^{2}}$, provided that $f$ is not a function of $z$. – user02138 Jul 11 '11 at 22:00
The question above is concerned with the necessary and sufficient conditions on $f$ and $g$ to ensure $A_{f} \simeq A_{g}$. – user02138 Jul 12 '11 at 9:03