The Alexander module of a link complement is finitely presented

The equivariant lifted finite model. Classify a cover $\tilde{Z}\to Z$ by $h_{\ast }K$. The lifting criterion gives normalized lifts $\tilde{h},\tilde{g}$ of $h,g$ between the two covers, since $h_{\ast },g_{\ast }$ are inverse and carry their subgroups to each other. The composite $\tilde{h}\tilde{g}$ lifts ${\mathrm{i}\mathrm{d}}_Z$ and fixes the chosen lift, so equals ${\mathrm{i}\mathrm{d}}_{\tilde{Z}}$ by uniqueness. Lift the exterior-fixing homotopy $gh\simeq {\mathrm{i}\mathrm{d}}_{X_L}$ starting at $\tilde{g}\tilde{h}$; its endpoint is the normalized lift of the identity, hence the identity. The maps induce the identity on the transported quotient deck group $\mathbb{Z}$; path lifting therefore makes them send the level-one chosen fibre point to the level-one point. Uniqueness of connected-domain lifts now gives $\tilde{h}t=t\tilde{h}$ and the corresponding identity for $\tilde{g}$. Conjugating the lifted homotopy by deck translation preserves its already-equivariant initial map, so homotopy-lift uniqueness makes that homotopy equivariant as well. Thus these are equivariant homotopy inverses and induce a $\mathrm{\Lambda }$-module isomorphism $A_L\cong H_1(\tilde{Z};\mathbb{Z})$.

Finite generation and presentation. Lift the cells of $Z$ to $\tilde{Z}$. Each cell has one deck orbit of lifts, so its cellular chains form a bounded complex $C_{\ast }$ of finite free $\mathrm{\Lambda }$-modules, one basis element per base cell, with deck-linear differentials. Cellular comparison identifies $A_L$ with $\ker d_1\mathrm{/}\mathrm{im}{d}_2$. By Noetherianity [F4], the kernel in the finite free module $C_1$ is finitely generated, and so is its quotient $A_L$. The kernel of a finite free surjection onto $A_L$ is again finitely generated by [F4], giving a finite presentation.

Specialization and ranks. Suppose $L$ is a knot. Put $F=\mathbb{Q}(t)$ and let $C(1)=C_{\ast }{\otimes }_{\mathrm{\Lambda },t\mapsto 1}\mathbb{Q}$. Sending all lifts of a cell to that base cell gives exactly the ordinary rational cellular complex of $Z$: the augmented coefficients are the ordinary incidence coefficients. Hence its homology dimensions are $(1,1,0,\dots )$ by [F6] and step 1.1. For each differential matrix, any minor nonzero at $t=1$ is a nonzero Laurent polynomial and thus nonzero over $F$. Consequently its rank over $F$ is at least its rank after specialization. Writing $c_i={\mathrm{rank}}_{\mathrm{\Lambda }}{C}_i$, the identity $\dim H_i=c_i-\mathrm{rank}{d}_i-\mathrm{rank}{d}_{i+1}$ gives ${\dim }_F{H}_i(C_{\ast }\otimes F)\le {\dim }_{\mathbb{Q}}{H}_i(C(1))$, so the generic groups in degrees $i\ge 2$ vanish.

The degree-zero and Euler arguments. The cover is connected; [F7] makes its integral $H_0$ equal to $\mathbb{Z}$ with trivial deck action, namely $\mathrm{\Lambda }\mathrm{/}(t-1)$. Exact localization [F5] therefore gives $H_0(C_{\ast }\otimes F)=0$, since $t-1$ is invertible in $F$. In either finite vector-space complex, summing the dimension identity of step 3.1 with alternating signs cancels all differential ranks. Thus the Euler sum ${\sum }_i(-1{)}^i{c}_i$ equals the corresponding alternating homology dimension sum. At $t=1$ this is $1-1=0$; over $F$ only the degree-one group could survive by step 3.1 and the degree-zero computation, so $0=-{\dim }_F{H}_1(C_{\ast }\otimes F)$, forcing that group to vanish.

Torsion and conclusion. Exact localization [F5] also gives $H_1(C_{\ast }\otimes F)\cong A_L{\otimes }_{\mathrm{\Lambda }}F$, so step 4.1 proves the knot clause. Localization at all nonzero elements vanishes exactly when each module element is killed by some nonzero element of the domain; this is the asserted $\mathrm{\Lambda }$-torsion condition. Rational homology of the cover may remain nonzero before inverting Laurent polynomials. Finite generation and presentation were proved in step 2.1 for every link, and the knot clause follows from the local specialization argument.