Finite generation is equivalent to uniform representation stability

$(\Rightarrow )$ Assume that $V$ is finitely generated, say generated in degree $\le g$. By [F3] $\mathrm{surj-deg}(V)\le g$, and by [F2] there is a finite surjection $M:={⨁}_{i=0}^{g}M(i{)}^{\oplus b_i}\twoheadrightarrow V$; let $K$ be its kernel. The module $M$ is finitely generated (each $M(i)$ is generated in degree $\le i$ by [F2]), so $K$ is finitely generated by [F5] applied over the field $k$; say $K$ is generated in degree $\le r$. These data will bound the injectivity degree of $V$ below.

$(\Leftarrow )$ Conversely, assume that $(V_n,{\phi }_n)$ is uniformly representation stable with range $n\ge N$. For $n\ge N$ condition (II) says that the $S_{n+1}$-span of ${\phi }_n(V_n)$ is all of $V_{n+1}$, and every injection $f\text{\hspace{0.17em}⁣}:[n]\to [n+1]$ equals $\sigma \circ f_0$ for the standard inclusion $f_0$ and some $\sigma \in S_{n+1}$, so $V_{n+1}$ is spanned by the images $f_{\ast }(V_n)$ over all injections $f\text{\hspace{0.17em}⁣}:[n]\to [n+1]$. The values in degrees $\le N$ lie in ${\mathrm{span}}_V(V_{\le N})$ by [F2], and if $V_m\subseteq {\mathrm{span}}_V(V_{\le N}{)}_m$ for some $m\ge N$, then every $f_{\ast }(V_m)$ lies in ${\mathrm{span}}_V(V_{\le N}{)}_{m+1}$ because the span is a sub-FI-module; the displayed spanning property of condition (II) then gives $V_{m+1}\subseteq {\mathrm{span}}_V(V_{\le N}{)}_{m+1}$. By induction $V_m\subseteq {\mathrm{span}}_V(V_{\le N}{)}_m$ for every $m\ge 0$, hence $V={\mathrm{span}}_V(V_{\le N})$. Each $V_{[m]}$ with $m\le N$ is finite-dimensional, hence generated by finitely many elements; let $\mathrm{\Sigma }$ be the finite union of these finitely many finite generating sets, so that ${\mathrm{span}}_V(\mathrm{\Sigma })\supseteq V_{[m]}$ for every $m\le N$. For a finite set $S$ with $\mathrm{\mid }S\mathrm{\mid }=m\le N$ and a bijection $\beta \text{\hspace{0.17em}⁣}:[m]\to S$ one has $V_S={\beta }_{\ast }(V_{[m]})\subseteq {\mathrm{span}}_V(\mathrm{\Sigma }{)}_S$, so ${\mathrm{span}}_V(\mathrm{\Sigma })\supseteq V_{\le N}$ and therefore ${\mathrm{span}}_V(\mathrm{\Sigma })=V$; thus $V$ is finitely generated.

Returning to $(\Rightarrow )$: for every $a$, the exactness of ${\mathrm{\Phi }}_a$ over the characteristic-zero field $k$ gives an exact sequence $0\to {\mathrm{\Phi }}_a(K)\to {\mathrm{\Phi }}_a(M)\to {\mathrm{\Phi }}_a(V)\to 0$ by [F6]. The module $M$ is a finite direct sum of copies of the free modules $M(i)$, each of injectivity degree $0$ by [F4], so $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(M{)}_n\to {\mathrm{\Phi }}_a(M{)}_{n+1}$ is injective for all $n$ by [F6] and [F7]; and $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(K{)}_n\to {\mathrm{\Phi }}_a(K{)}_{n+1}$ is surjective for $n\ge r$ by [F3]. Hence $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$ is injective for $n\ge r$: a lift $\tilde{x}\in {\mathrm{\Phi }}_a(M{)}_n$ of $x$ with $Tx=0$ has $T\tilde{x}$ in the image of ${\mathrm{\Phi }}_a(K)$, so $T\tilde{x}=T\tilde{y}$ for some $\tilde{y}\in {\mathrm{\Phi }}_a(K{)}_n$, and injectivity on ${\mathrm{\Phi }}_a(M)$ gives $\tilde{x}=\tilde{y}$, whence $x=0$. Therefore $\mathrm{inj-deg}(V)\le r$ by [F7], and $\mathrm{stab-deg}(V)\le \max (g,r)$ since $\mathrm{surj-deg}(V)\le g$.

Under the assumption of step 1.1 the weight is finite, indeed $\mathrm{weight}(V)\le g$ by [F3] over the characteristic-zero field $k$, and $\mathrm{stab-deg}(V)\le \max (g,r)<\mathrm{\infty }$ by step 2.1. By [F8] the sequence $(V_n,{\phi }_n)$ is uniformly representation stable, with stable range $n\ge g+\max (g,r)$.

Steps 1.1, 2.1 and 3.1 prove that finite generation implies uniform representation stability (through finiteness of stability degree and weight and the preceding proposition), and step 1.2 proves the converse; hence for an FI-module with finite-dimensional values over a characteristic-zero field the two properties are equivalent.