Help with understanding the notation $\mathbb{C}\{f\}$

Question

I am reading an article "Relative Cohomology and volume forms" of J. P. Francoise. Here the author considers the germ of a function $f\colon(\mathbb{C}^n,0) \to (\mathbb{C},0)$, and he speaks of the ring $\mathbb{C}\{f\}$, but he doesn't define this ring.

Does anyone know the definition of this ring $\mathbb{C}\{f\}$? Can someone recommend me a book or something similar?

Thank you.

Answer 1

If you look at his paper Le théorème de M. Sebastiani pour une singularité quasi-homogène isolée in which he discusses the theorem he's talking about at the start of the article you refer to, he states (with my poor translation):

Let $\mathcal{O}$ be the sheaf of germs of analytic functions at $0 \in \mathbb{C}^n$ and $\Omega^k$ the sheaf of $k$-forms with coefficients in $\mathcal{O}$.

M. Sébastiani completed the theory of E. Brieskorn and established that if $P$ denotes an element of $\mathcal{O}$ with an isolated singularity for which the Milnor number is $\mu$, then $G = \frac{\Omega^n}{dP\wedge d\Omega^{n-2}}$ is a $\mathbb{C}\{t\}$-module of rank $\mu$, where the action of $t$ is by multiplication by $P$.

Notice that $P$ here is $f$ in the article you're reading, and $G$ is apparently the relative cohomology under discussion.

So $\mathbb{C}\{f\}$ is an alternative notation for $\mathbb{C}\{t\}$, with it made explicit that the element $t$ acts by multiplication by $f$.

$\mathbb{C}\{t\}$ is a common notation for the ring of convergent power series in $t$, i.e. the subring of $\mathbb{C}[[t]]$ (the ring of formal power series $\sum_{i=0}^\infty c_k t^k$) consisting of those which have positive radius of convergence (i.e. elements $\sum_{i=0}^\infty c_k t^k$ which have $\mathrm{limsup}_{k\rightarrow\infty} {|c_k|^{1/k}}$ finite). Notice that $\mathbb{C}\{t\}$ is the ring of germs of holomorphic functions at $0 \in \mathbb{C}$.