Several Complex Variables (VII): The Local Ring of Germs

13 Sep 2018

• several complex variables

In the previous posts, we have used the tools of analysis to study holomorphic functions of several complex variables. From this point on, we will bring in powerful algebraic and geometric tools to help us.

In this post, we introduce the concept of the germ of a function, which captures its local behaviour about a given point. We will study the algebraic properties of the ring of germs of holomorphic functions, and prove that it is local and Noetherian.

Germs of functions

Let $X$ be a topological space, and fix $x\in X$. We want to capture the local behaviour of functions at $x$: if $f,g$ are functions that agree near $x$, then they should be considered the “same.”

Formally, if $f,g$ are defined on neighbourhoods $U,V$ of $x$, respectively, we say that $f$ and $g$ are equivalent at $x$ if there exists a neighbourhood $W\subseteq U\cap V$ of $x$ such that $f\rvert_W=g\rvert_W$. This is indeed an equivalence relation:

$$\begin{aligned} f|_W&=g|_W\\ f|_{W'}&=g'|_{W'} \end{aligned}\implies g|_{W\cap W'}=g'|_{W\cap W'},$$

and if $W,W’$ are neighbourhoods of $x$ then so is $W\cap W’$. Now the germ of $f$ at $x$, written as $f_x$, is defined as the equivalence class of $f$ under this relation.

Algebraically, the above construction is an example of a direct limit, with the following data:

• The collection of neighbourhoods of $x$, partially ordered by $\subseteq$;
• For each neighbourhood $U$ of $x$, a ring or algebra $R^U$ of functions $U\to R$;

• For each pair of neighbourhoods $U\supseteq V$ of $x$, a restriction map $\operatorname{res}^U_V:R^U\to R^V$, such that

  $$\operatorname{res}^V_W\circ\operatorname{res}^U_V=\operatorname{res}^U_W$$

  for neighbourhoods $U\supseteq V\supseteq W$.

We are interested in the special case of holomorphic functions $f$ defined on a neighbourhood of $\lambda\in\bb C^n$; the set of all germs $f_x$ of these functions is denoted $\mc H_\lambda$, or ${}_n\mc H _\lambda$ to emphasise the dimension $n$.

In order to work with elements of $\mc H_\lambda$, which are “holomorphic functions defined on an arbitrarily small neighbourhood of $\lambda$,” we usually have to pass to a representative in the equivalence class, ie. an actual holomorphic function defined on an actual neighbourhood of $\lambda$. As an illustrative example, we will perform this procedure in some detail to define the $\bb C$-algebra structure on $\mc H_\lambda$.

Given $\varphi,\psi\in\mc H_\lambda$, take holomorphic functions $f:U\to\bb C$, $g:V\to\bb C$ defined in neighbourhoods of $\lambda$ so that

$$\varphi=f_\lambda,\quad\psi=g_\lambda.$$

Let $W=U\cap V$. By replacing $f,g$ by $f\rvert_W,g\rvert_W$, which does not change their germs at $\lambda$, we may assumet that $f,g:W\to\bb C$ have a common domain. Then we define

$$\begin{aligned} \varphi+\psi&\coloneqq(f+g)_\lambda,\\ \alpha\varphi&\coloneqq(\alpha f)_\lambda,&(\alpha\in\bb C)\\ \varphi\psi&\coloneqq(fg)_\lambda. \end{aligned}$$

It remains to check that the above definitions are independent of the choice of representatives $f,g$, and domains of definition $U,V$. The main idea is that for any other choice, all functions are equal on some intersection of all relevant neighbourhoods.

From here on, we denote the germ of $f$ at $\lambda$ as $\bf f\coloneqq f_\lambda$, if the point $\lambda$ is clear from context. This notation makes sense now because, for example, we have $f_\lambda+g_\lambda=(f+g)_\lambda$, where $f+g$ is understood to be defined on the common domain of definition of $f$ and $g$, so there is no ambiguity in the notation $\bf f+\bf g$.

Local rings

Now we turn to elementary ring theoretic properties of $\mc H_\lambda$. First, note that if $f$ is nonzero at $\lambda$ then $f$ is nonzero on a neighbourhood of $\lambda$, so $\bf f$ is a unit in $\mc H_\lambda$ if and only if $f(\lambda)\neq0$.

A commutative ring is a local ring if it has a unique maximal ideal.

Proposition: A commutative ring $A$ is local if and only if the set of non-units forms an ideal, in which case it is the unique maximal ideal of $A$.

Proof: For each $x\in A$, consider the ideal $(x)=Ax$ generated by $x$.

• If $x$ is a non-unit then $(x)$ is contained in some maximal ideal, by Zorn’s lemma.
• If $x$ is a unit then $(x)=(1)=A$, so $x$ is not contained in any proper ideal.

Hence if $A$ is local then the unique maximal ideal contains precisely all the non-units. Conversely, every proper ideal lies in the set of non-units, so if this set forms an ideal then it is the unique maximal ideal. $\qed$

But in $\mc H_\lambda$, the set of non-units is the set of germs vanishing at $\lambda$, which clearly form an ideal. Hence we have:

Corollary: $\mc H_\lambda$ is a local ring. $\qed$

Local analysis

It is clear that the $\bb C$-algebras ${}_n\mc H _\lambda$ and ${}_n\mc H_0$ are isomorphic, by a suitable translation. Hence without loss of generality we can restrict our attention to the latter.

Many of the analytic results we have derived in previous posts are local in nature, and can be restated in terms of germs. For instance, since holomorphic functions are precisely the analytic functions, we have immediately:

Proposition: The algebra ${}_n\mc H_0$ is isomorphic to $\bb C\{z_1,\ldots,z_n\}$, the algebra of power series in $n$ variables with positive polyradius of convergence. $\qed$

Note that $\bb C\{z_1,\ldots,z_n\}$ is a proper subset of $\bb C[[z_1,\ldots,z_n]]$, the algebra of formal power series.

The vanishing order of $\bf f$ is the vanishing order of any representative $f$; this is well-defined since the vanishing order is a local concept, ie. does not change when restricting $f$ to a smaller neighbourhood of 0.

A Weierstrass polynomial of degree $k$ in $z_n$ is a polynomial $\bf h\in{}_{n-1}\mc H_0[z_n]$ of the form

$$\bf h(z)=z_n^k+\bf a_1(z')z_n^{k-1}+\cdots+\bf a_k(z'),$$

where $z=(z’,z_n)$ and each $\bf a_i$ is a non-unit in ${}_{n-1}\mc H_0$.

The Weierstrass theorems can now be stated more cleanly, as follows.

Weierstrass preparation theorem: If $\bf f\in{}_n\mc H_0$ has finite vanishing order $k$ in $z_n$, then $\bf f$ has a unique factorisation as

$$\bf f=\bf u\bf h,$$

where $\bf h$ is a Weierstrass polynomial of degree $k$ in $z_n$, and $\bf u$ is a unit in ${}_n\mc H_0$. $\qed$

Weierstrass division theorem: If $\bf f\in{}_n\mc H_0$ and $\bf h\in{} _{n-1}\mc H_0[z_n]$ is a Weierstrass polynomial of degree $k$, then $\bf f$ can be uniquely written as

$$\bf f=\bf g\bf h+\bf q,$$

where $\bf g\in{}_n\mc H_0$ and $\bf q\in{} _{n-1}\mc H_0[z_n]$ is a Weierstrass polynomial of degree less than $k$. $\qed$

Noetherian rings

A commutative ring is a Noetherian ring if each of its ideals are finitely generated. This acts as a strong finiteness condition, similar to compactness in topology.

Our target is to prove that ${}_n\mc H_0$ is Noetherian, which we will eventually use to define the dimension of a variety. We now need some results from commutative algebra, whose proofs we will only sketch.

A module $M$ over a ring $A$ is a Noetherian module if each of its submodules are finitely generated as $A$-modules. By definition, if $A$ is a Noetherian ring, then $A$ is a Noetherian $A$-module.

Proposition: If $M_1,M_2$ are Noetherian $A$-modules, then so is $M=M_1\oplus M_2$.

Proof: If $N$ is an $A$-submodule of $M$, then $N\cap(M_1\oplus\{0\})\subseteq M_1\oplus\{0\}\cong M_1$ has a finite set of generators $S_1$. Similarly, $N/M_1\subseteq M/M_1\cong M_2$ has a finite set of generators; pick a preimage under the quotient map for each generator to get $S_2$. Now check that $S_1\cup S_2$ generates $N$. $\qed$

Corollary: If $A$ is a Noetherian ring, then $A^n$ is a Noetherian $A$-module.

Proof: Induction on $n$. $\qed$

Hilbert basis theorem: If $A$ is Noetherian, then so is the polynomial ring $A[x]$.

Proof: Given an ideal $I$ in $A[x]$, consider the ideal $J$ in $A$ of all leading coefficients of elements of $I$, which has a finite set of generators $\{a_1,\ldots,a_n\}$. For each $i$, pick some $f_i=a_ix^{r_i}+g_i\in I$, where $\deg g_i< r_i$.

Let $r=\max_ir_i$, and let $M$ be the set of polynomials of degree less than $r$. Note that $M$ is isomorphic to $A^r$ as an $A$-module, and hence is Noetherian. In particular, the $A$-submodule $N=M\cap I$ has a finite generating set $S$.

For any element $f=ax^m+g\in I$ with $\deg g< m$ and $m\geq r$, write $a=\sum_ib_ia_i$ for some $b_i\in\mc I$, so $f-\sum_ib_if_ix^{m-r_i}$ belongs to $I$ and has degree less than $m$. Repeating this process, we see that $f\in\overline f+(f_1,\ldots,f_n)$ for some $\overline f\in I$, $\deg\overline f< r$. Hence $I$ is generated by $S\cup\{f_1,\ldots,f_n\}$. $\qed$

Now we are ready to prove the main theorem.

Theorem: The ring ${}_n\mc H_0$ is Noetherian.

Proof: We induct on $n$. For $n=0$, ${}_n\mc H_0$ is the base field $\bb C$, which has no nontrivial ideals, and is hence trivially Noetherian.

Assume that ${}_{n-1}\mc H_0$ is a Noetherian ring, so by Hilbert’s basis theorem ${} _{n-1}\mc H_0[z_n]$ is also a Noetherian ring. For any nonzero ideal $\mc I$ in ${}_n\mc H_0$, fix a nonzero element $\bf h\in\mc I$. After a coordinate change, we may assume that $\bf h$ has finite vanishing order; by the Weierstrass preparation theorem, after multiplying by a unit, we may assume that $\bf h$ is a Weierstrass polynomial.

Since $\mc I\cap{}_{n-1}\mc H_0[z_n]$ is an ideal in the Noetherian ring ${} _{n-1}\mc H_0$, it has finite generating set $S$. Now by the Weierstrass division theorem, any $\bf f\in\mc I$ can be written as

$$\bf f=\bf g\bf h+\bf q,$$

with $\bf g\in{}_n\mc H_0$ and $\bf q\in{} _{n-1}\mc H_0[z_n]$. Now $\bf f,\bf h\in\mc I$ implies $\bf q\in\mc I$, so $\bf h,\bf q\in\mc I\cap{}_n\mc H_0[z_n]$. Hence $\bf f$ lies in the ideal generated by $S$, so $S$ generates $\mc I$. $\qed$