A Fitting remark

In this post I want to talk about an innocent commutative algebra lemma. Let R be a DVR with uniformizer \pi, and let M be a finite torsion R-module, so M\simeq \oplus_{i=1}^{n}R/\pi^{k_i} for some uniquely determined sequence k_1 \geq \dots \geq k_n > 0. I’ll somewhat abusively refer to the k_i‘s as the “elementary divisors” of M.

Lemma.  If N \subseteq M is an R-submodule generated by j elements, then \ell(N) \leq \sum_{1\leq i \leq j} k_i. Furthermore, if equality holds, then N is a direct summand of M.

(Here and throughout, \ell denotes R-module length.)

Proof. We first prove the inequality by induction on n. Fix a surjection f:M\to R/\pi^{k_{1}} such that \ker f\simeq\oplus_{2\leq i\leq n}R/\pi^{k_{i}}.  Choose a minimal basis n_{1},\dots,n_{j} of N such that f(n_{1}) generates f(N)\subseteq R/\pi^{k_{1}}. Choose some a_{i}\in R with f(n_{i})=a_{i}f(n_{1}) for all 2\leq i\leq j, and make the substitution n_{i}\to n_{i}-a_{i}n_{1} for i\geq2. Having done this, we get a basis n_{1},\dots,n_{j}\in N with f(n_{1}) generating f(N) and n_{i}\in\ker f for all 2\leq i\leq j. Let N'\subseteq\ker f\cap N denote the R-submodule generated by n_{2},\dots,n_{j}; applying our induction hypothesis to the modules N' \subseteq \ker f, we get the inequality \ell(N')\leq k_{2}+\cdots k_{j}. Since N' and Rn_{1}\subset N generate N, we get an inequality \ell(N)=\ell(N'+Rn_{1})\leq\ell(N')+\ell(Rn_{1}). But \ell(Rn_{1})\leq k_{1}, since \pi^{k_{1}} annihilates M, so
\ell(N)\leq\ell(N')+\ell(Rn_{1})\leq k_{2}+\cdots+k_{j}+k_{1},
as desired.

For the second claim, we argue by induction on j; the case j=1 is easy (argument: N must project isomorphically onto a direct summand of M of the form R/\pi^{k_1}). Maintain the previous notation, and assume we have an equality \ell(N)=k_{1}+\cdots+k_{j}. Since \ell(N')\leq k_{2}+\cdots+k_{j} and \ell(Rx_{1})\leq k_{1}, the chain of inequalities
\ell(N)=\ell(N'+Rn_{1})\leq\ell(N')+\ell(Rn_{1})\leq k_{1}+\cdots+k_{j}=\ell(N)
then forces \ell(N')=k_{2}+\cdots+k_{j} and \ell(Rn_{1})=k_{1}. Since N' and Rx_{1} are generated by j-1 and 1 element, respectively, they are both direct summands of M by the induction hypothesis. Finally, the equality \ell(N'+Rn_{1})=\ell(N')+\ell(Rn_{1}) implies that N'\cap Rn_{1}=0, so N\cong N'\oplus Rn_{1}\subseteq M is a direct summand of M. \square

This lemma really really really looks like it should be well-known, but I couldn’t find it stated in the literature.  Presumably I was just typing the wrong things into google.  Can some reader provide a reference?  If you can find a reference in a textbook (not a research paper), this will settle a bet between me and AJdJ.  Also, it would be really nice if there were a “coordinate-free” proof which didn’t involve choosing a basis for N.  Before finding the argument given above, I spent a while trying to make a proof based on the theory of Fitting ideals; the latter seem quite natural in this context, since one has the equality \mathrm{Fitt}(N)=(\pi^{\ell(N)}) for any finite R-module.  Can the reader make such a proof work?

OK literally while writing the previous two sentences I hit upon the following argument for the first part of the lemma. It clearly suffices to show the complementary inequality \ell(M/N)\geq\sum_{j<i\leq n}k_{i}. For this we use Fitting ideals as follows. Recall that for any finite torsion module Q over R with elementary divisors k_{i}, we have an equality \mathrm{Fitt}_{j}(Q)=(\pi^{\sum_{j<i}k_{i}}); in particular, \mathrm{Fitt}(Q)=\mathrm{Fitt}_{0}(Q)=(\pi^{\ell(Q)}), and \mathrm{Fitt}_{m}(Q)=R if Q is generated by \leq m elements. Returning to the situation at hand, we have an inclusion \mathrm{Fitt}_{j}(N)\mathrm{Fitt}(M/N)\subseteq\mathrm{Fitt}_{j}(M)=(\pi^{\sum_{j<i\leq n}k_{i}}) (this is a special case of Proposition XIII.10.7 in Lang’s Algebra). But \mathrm{Fitt}_{j}(N)=R since by assumption N is generated by j elements, so we get
(\pi^{\ell(M/N)})=\mathrm{Fitt}(M/N)\subseteq\mathrm{Fitt}_{j}(M)=(\pi^{\sum_{j<i\leq n}k_{i}}),
and this immediately implies the desired inequality.

Great! Is there a Fitting ideal proof of the second part?

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