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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.


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