Span, finite generation and generation degree of an FI-module

Let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring) and let $\mathrm{\Sigma }$ be a set of supported elements $(S,v)$, where $S$ is a finite set and $v\in V_S$. The support $S$ is part of the generator data, even when distinct values of $V$ have overlapping underlying sets. Write ${\mathrm{\Sigma }}_S:=\{v\in V_S:(S,v)\in \mathrm{\Sigma }\}$; a sub-FI-module $W$ contains $\mathrm{\Sigma }$ when $v\in W_S$ for every $(S,v)\in \mathrm{\Sigma }$. The sub-FI-module generated by $\mathrm{\Sigma }$, written ${\mathrm{span}}_V(\mathrm{\Sigma })$, is the intersection of all sub-FI-modules of $V$ that are subfunctors and contain $\mathrm{\Sigma }$ (Submodule of a module, The one-step submodule criterion; intersections and sums of submodules are submodules for the objectwise intersection of submodules). The family is nonempty because $V$ itself qualifies, and the intersection is a sub-FI-module, since each transition map $f_{\ast }$ preserves every member and hence their intersection; thus ${\mathrm{span}}_V(\mathrm{\Sigma })$ is the smallest sub-FI-module of $V$ containing $\mathrm{\Sigma }$.

Explicitly, ${\mathrm{span}}_V(\mathrm{\Sigma }{)}_T$ is the $k$-span of all elements $f_{\ast }(v)$ with $(S,v)\in \mathrm{\Sigma }$ and $f\text{\hspace{0.17em}⁣}:S\to T$ a morphism of $FI$: the set of all finite $k$-linear combinations of such elements is a sub-FI-module of $V$ containing $\mathrm{\Sigma }$, because $g_{\ast }(f_{\ast }(v))=(g\circ f{)}_{\ast }(v)$, and every sub-FI-module containing $\mathrm{\Sigma }$ contains each $f_{\ast }(v)$; hence it equals the intersection. Call $V$ finitely generated when ${\mathrm{span}}_V(\mathrm{\Sigma })=V$ for some finite $\mathrm{\Sigma }$, and generated in degree $\le d$ when ${\mathrm{span}}_V(V_{\le d})=V$, where $V_{\le d}=\{([m],v):0\le m\le d,v\in V_{[m]}\}$ uses the finite skeleton of FI; transport by bijections shows that its span also contains every $V_S$ with $\mathrm{\mid }S\mathrm{\mid }\le d$. The generation degree of $V$ is the least $d\ge 0$ such that $V$ is generated in degree $\le d$, and $\mathrm{\infty }$ when no such $d$ exists (Generated submodule, cyclic and finitely generated modules, module basis and free module).

Finite generation of $V$ and pointwise finite generation of the individual $k$-modules $V_S$ are different conditions: the pointwise property is implied by finite generation (as shown below by Generation in bounded degree presents an FI-module as a quotient of free FI-modules) but does not imply it, and no pointwise finite generation or finite-dimensionality is built into either notion. A finite generating set gives a finite degree bound: take the maximum cardinality of its supporting sets (and $0$ for the empty generating set). When a generator is written simply as $v\in V_S$, its specified support $S$ is understood, so ${\mathrm{span}}_V(v)$ means ${\mathrm{span}}_V(\{(S,v)\})$. Generation in bounded degree alone does not imply finite generation. No choice principle is used: ${\mathrm{span}}_V(\mathrm{\Sigma })$ is defined by an intersection of subfunctors, and its explicit description involves only finite $k$-linear combinations.