L²-dimension estimates for amenable D(R) groups

First replace the term above degree $n$ by a finite free module if necessary. The image of $d_{n+1}$ after tensoring with $R$ is the kernel of $d_n$ over $R$, a finite-dimensional space. Choose finitely many tensors in $C_{n+1}\otimes R$ whose boundaries span it, and expand each tensor as a finite sum $c\otimes r$. The finitely many resulting elements $c\in C_{n+1}$ define a finite free module $P_{n+1}$ with a map to $C_{n+1}$; compose this map with $d_{n+1}$. Keep $P_i=C_i$ for $l\le i\le n$ and set all other terms to zero. Since $d_n{d}_{n+1}=0$, this defines a bounded finite-free complex $P_{\ast }$ whose $R$-homology vanishes through degree $n$. Below $n$ its $A$-homology is the original one, and in degree $n$ its $A$-homology surjects onto the original one, since it has the same cycles and possibly fewer boundaries. It suffices by [F3] to prove the result for $P_{\ast }$.

Over the field $R$, split the exact complex through degree $n$: choose vector-space complements of the kernels in each of its finitely many finite-dimensional terms, and invert each boundary map onto its image on the chosen complement. This gives maps $s_i:P_i\otimes R\to P_{i+1}\otimes R$, $l\le i\le n$, with $ds+sd=I$ through degree $n$. For $R={\mathbb{F}}_p$, lift the finitely many matrix entries to integers and then to constant coefficients in $\mathbb{Z}\mathrm{\Gamma }$, obtaining maps $D_i$. For $R=\mathbb{Q}$, choose a single positive integer $a$ clearing the denominators of all these finitely many matrices, and lift $a{s}_i$ instead; in the finite-field case put $a=1$. Set $D_i=0$ outside the indicated range. Then $f=dD+Dd$ is a chain map over $\mathbb{Z}\mathrm{\Gamma }$, chain homotopic to zero, and its augmented matrix in degree $i\le n$ is $aI$, which is injective over $R$.

Apply [F1] to $f_i{\otimes }_{\mathbb{Z}}R$ on the free $R\mathrm{\Gamma }$-module $P_i{\otimes }_{\mathbb{Z}}R$; its augmentation is injective by step 2.1, so this map is injective. Hence $f_i$ is injective over $\mathbb{Z}\mathrm{\Gamma }$. For $R=\mathbb{Q}$ this follows by viewing a nonzero integral vector as a nonzero rational vector. For $R={\mathbb{F}}_p$, an integral kernel vector has finitely many coefficients; divide them all by the largest common power of $p$. Torsion-freeness of the integral target preserves the kernel equation, and the resulting nonzero vector modulo $p$ contradicts injectivity. Passing to $\mathbb{C}\mathrm{\Gamma }$ preserves this injection: any finite complex relation can be decomposed using a basis of the finite-dimensional rational span of its coefficients into rational relations, each of which clears denominators to an integral one.

Resolve the cokernel of the injective square matrix $f_i$ over $\mathbb{C}\mathrm{\Gamma }$. The degree-one Tor group is the kernel of $f_i:A^{r_i}\to A^{r_i}$ after tensoring, where $r_i$ is the rank of $P_i$; this is the same free-presentation computation used to prove [F2]. Consequently [F2] makes that kernel dimension zero. Its projection has trace zero and is therefore zero by faithfulness in [F3]. Thus $f_i$ is injective on the Hilbert completion. Let $Z_i$ be the Hilbert kernel of $d_i$. The chain identity makes $f_i$ preserve $Z_i$. Its restriction to $Z_i$ has zero kernel, and polar decomposition in [F3] makes its Hilbert-closed image have the same trace dimension as $Z_i$; since that image lies in $Z_i$, faithfulness makes it dense in $Z_i$.

The algebraic $A$-cycle module is the projection module $P_{Z_i}{A}^{r_i}$ and is Hilbert-dense in $Z_i$ by [F3]. On these cycles, $f_i=d_{i+1}{D}_i+D_{i-1}{d}_i=d_{i+1}{D}_i$, so its image is contained in the algebraic boundary module. Step 4.1 shows that these boundaries are Hilbert-dense in $Z_i$. The dense-quotient assertion in [F3] therefore gives dimension-zero $A$-homology of $P_{\ast }$ through degree $n$. The identity/surjection comparison of step 1.1 transfers this conclusion to $C_{\ast }$. The proof establishes vanishing of the dimension and does not assert algebraic vanishing of the homology module.