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## Noetherianity of group rings of polycyclic groups

Today I learned the proof of a classic theorem of P. Hall from his paper

Hall, P.
Finiteness conditions for soluble groups.
Proc. London Math. Soc. (3) (1954), 419–436.

This theorem says that if $$G$$ is a polycyclic group, then the category of $$\mathbb{Z}[G]$$-modules is locally Noetherian, i.e. if $$M$$ is a finitely generated $$\mathbb{Z}[G]$$-module and $$N \subset M$$ is a submodule, them $$N$$ is finitely generated.

The proof is surprisingly easy, so I thought I would record it here.  The proof is by induction on the Hirsch length of $$G$$, i.e. the length of a subnormal series whose associated graded groups are cyclic.  The base case is where the Hirsch length is zero, so $$G$$ is the trivial group and the result is trivial.  Assume, therefore, that the Hirsch length of $$G$$ is positive and that the theorem is true whenever it is smaller.  Let $$M$$ be a finitely generated $$\mathbb{Z}[G]$$-module and let $$N \subset M$$ be a submodule.  Since $$G$$ is polycyclic, there exists a normal subgroup $$H$$ of $$G$$ such that $$G/H$$ is cyclic.  By induction, the category of $$\mathbb{Z}[H]$$-modules is locally Noetherian.  Let $$x \in G$$ be an element that projects to a generator for $$G/H$$.  Let $$S$$ be a finite generating set for $$M$$ and let $$M_H$$ be the $$H$$-submodule of $$M$$ generated by $$S$$.

Every element of $$m \in M$$ can be written as

$$m = \sum_{i=r}^s x^i c_i$$

for some $$-\infty < r \leq s < \infty$$ and some $$c_i \in M_H$$.  Of course, this expression is far from unique, but we will not dwell on this.  A polynomial of degree $$k \geq 0$$ is an element of the form

$$m = \sum_{i=0}^k x^i c_i$$

with $$c_i \in M_H$$ and with $$c_k \neq 0$$.  The leading term of this polynomial is $$c_k$$.  Let $$L_{N,k} \subset M_H$$ be the set of leading terms of element of $$N$$ that can be expressed as polynomials of degree $$k$$ together with $$0$$.  I claim that $$L_{N,k}$$ is an $$H$$-submodule of $$M_H$$.  We must check two things.

• It is closed under sums.  This is obvious.
• It is closed under multiplication by elements of $$\mathbb{Z}[H]$$.  Consider some $$h \in \mathbb{Z}[H]$$ together with an element $$n = \sum_{i=0}^k x^i c_i$$ of $$N$$ satisfying $$c_k \neq 0$$, so $$c_k$$ is an arbitrary element of $$L_{N,k}$$.  We want to show that $$h c_k \in L_{N,k}$$.  This is trivial if $$h c_k = 0$$, so assume that it is nonzero.  For $$0 \leq i \leq k$$, set $$c_i’ = x^{-i} x^k h x^{-k} x^i$$, so $$c_k’ = h c_k$$.  Since $$H$$ is a normal subgroup of $$G$$, we have $$c_i’ \in \mathbb{Z}[H]$$.  Moreover, $$x^k h x^{-k} n = \sum_{i=0}^k x^i c_i’ \in N$$.  It follows that $$c_k’ = h c_k \in L_{N,k}$$, as desired.

It follows from our induction hypothesis that each $$L_{N,k}$$ is a finitely generated $$\mathbb{Z}[H]$$-module.  Moreover, since a polynomial of degree $$k$$ can be multiplied by $$x$$ to get a polynomial of degree $$k+1$$, we see that

$$L_{N,0} \subset L_{N,1} \subset L_{N,2} \subset \cdots \subset M_H.$$

Again using our induction hypothesis, this increasing sequence of submodules of $$M_H$$ must stabilize.  We can thus choose a finite set $$\{f_1,f_2,\ldots,f_r\}$$ of elements of $$N$$ that for all $$i \geq 0$$ contains a set of polynomials of degree at most $$i$$ whose leading terms generate $$L_{N,i}$$.  Let $$N’$$ be the $$\mathbb{Z}[G]$$-submodule of $$N$$ generated by the $$f_i$$.  We claim that $$N’ = N$$.  Indeed, consider some $$n \in N$$.  For some $$\ell \in \mathbb{Z}$$, we can write

$$x^{\ell} n = \sum_{i=0}^k x^i c_i$$

with each $$c_i \in M_H$$.  By subtracting appropriate multiples of the $$f_i$$ to first kill off the terms of degree $$k$$, then the terms of degree $$k-1$$, etc., we can reduce this to $$0$$.  It follows that $$x^{\ell} n \in N’$$, and thus that $$n \in N’$$, as desired.