Fitting ideals, order, and diagonal specialization of the multivariable Alexander module

By [F10], choose a finite CW structure on the compact smooth exterior $E_L$. By [F5], its lift gives a finite free complex $C_2\stackrel{d_2}{\to }{C}_1\stackrel{d_1}{\to }{C}_0$ with first homology $A_L$. Since ${\mathrm{\Lambda }}_r$ is Noetherian by [F2], every submodule of a finite free module is finitely generated: induct on the rank $n$ of ${\mathrm{\Lambda }}_r^n$; project a submodule $N$ to the first coordinate, choose lifts of finitely many generators of its image ideal, and apply the induction hypothesis to the kernel in ${\mathrm{\Lambda }}_r^{n-1}$. Thus $Z_1=\ker d_1$ is finitely generated, and $B_1=\mathrm{im}{d}_2$ is generated by the finitely many columns of $d_2$. The quotient $A_L=Z_1\mathrm{/}{B}_1$ is finitely generated. Choose a surjection ${\mathrm{\Lambda }}_r^m\to A_L$ with finite $m$; its kernel is a submodule of ${\mathrm{\Lambda }}_r^m$ and hence finitely generated by the same argument. This is a finite presentation of $A_L$.

Let $\bar{\eta }:H_1(E_L;\mathbb{Z})\to \mathbb{Z}$ be the total-linking map. By [F1] the positive meridians are the coordinate basis, so the coordinate-sum definition [F6] sends each one to $1$. Define $\eta :{\mathrm{\Lambda }}_r\to {\mathrm{\Lambda }}_1$ to be its induced ring map $t_i\mapsto t$. By [F7], quotienting the maximal abelian cover of $E_L$ by $\ker \bar{\eta }$ gives the connected cover classified by $\ker (\bar{\eta }\circ {\varphi }_E)$; this quotient cover is regular with deck group $H_1(E_L;\mathbb{Z})\mathrm{/}\ker \bar{\eta }\cong \mathbb{Z}$ by [F9]. The equivariant deformation retraction [F8] descends to a deformation retraction of the corresponding quotient of the open-complement cover onto this compact-exterior quotient cover; by [F6] these are the one-variable total-linking covers of the open complement and exterior, respectively. For every lifted cell orbit, tensoring its free rank-one ${\mathrm{\Lambda }}_r$ summand with ${\mathrm{\Lambda }}_1$ along $\eta$ identifies the quotient-orbit generator with a free rank-one ${\mathrm{\Lambda }}_1$ generator, and sends each boundary coefficient to its image under $\eta$. Therefore $C_{\ast }^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E_L^{\mathrm{a}\mathrm{b}};\mathbb{Z}){\otimes }_{{\mathrm{\Lambda }}_r,\eta }{\mathrm{\Lambda }}_1$ is the cellular chain complex of the total-linking cover of $E_L$.

Use the resulting finite presentation to define each $E_j(A_L)$ as in [F3]. Any other triangulation of the same marked exterior lifts to another CW structure on the same topological cover; [F5] identifies the homology of either lifted cellular complex with the same singular module $A_L$, and the deck action in both identifications is the same. A different lift of a cell multiplies its basis element by a deck monomial, also leaving the resulting singular module unchanged. Hence the modules computed from the two choices are isomorphic. By [F4] all their elementary ideals agree.

Put $K=\mathrm{Frac}({\mathrm{\Lambda }}_r)$ and $q={\dim }_K(A_L{\otimes }_{{\mathrm{\Lambda }}_r}K)$. For a finite presentation matrix with $n$ target generators, tensoring with $K$ shows its rank is $n-q$. Therefore at least one $(n-q)\times (n-q)$ minor is nonzero (with the empty minor equal to $1$ when $n=q$), so $E_q(A_L)\mathrm{\ne }0$. This ideal has finitely many generators by its finite presentation, and in the UFD [F2] their gcd exists and is unique up to a unit. Since [F4] makes the ideal presentation-independent, the gcd is well-defined up to a unit of ${\mathrm{\Lambda }}_r$, which by [F2] is $\pm t_1^{k_1}\cdots t_r^{k_r}$.

Homology does not in general commute with this coefficient extension. For $r=1$, $\eta$ is the identity; for $r\ge 2$, consider the finite free chain complex $C_1={\mathrm{\Lambda }}_r\stackrel{t_1-t_2}{\to }{C}_0={\mathrm{\Lambda }}_r$ with all other chain groups zero. The ring ${\mathrm{\Lambda }}_r$ is a domain and $t_1-t_2\mathrm{\ne }0$, so $H_1(C_{\ast })=0$. After diagonal base change $\eta$, the differential is multiplication by $t-t=0$, hence $H_1(C_{\ast }{\otimes }_{{\mathrm{\Lambda }}_r,\eta }{\mathrm{\Lambda }}_1)={\mathrm{\Lambda }}_1$, while $H_1(C_{\ast }){\otimes }_{{\mathrm{\Lambda }}_r,\eta }{\mathrm{\Lambda }}_1=0$. The multiplication map ${\mathrm{\Lambda }}_r\stackrel{t_1-t_2}{\to }{\mathrm{\Lambda }}_r$ is injective before tensoring and becomes the zero map afterward, so ${\mathrm{\Lambda }}_1$ is nonflat over ${\mathrm{\Lambda }}_r$ for $r\ge 2$. Thus for $r\ge 2$ the chain specialization does not in general identify the specialized chain-complex homology with the tensor product of $A_L$, and no equality of specialized homology orders follows without additional hypotheses.