The coinvariant functors ${\mathrm{\Phi }}_a$ and the transition maps $T$

A graded $k[T]$-module consists of $k$-modules $U_n$, $n\ge 0$, together with $k$-linear maps $T=T_n\text{\hspace{0.17em}⁣}:U_n\to U_{n+1}$ for all $n\ge 0$. A morphism $U\to U^{\mathrm{\prime }}$ of graded $k[T]$-modules is a family of $k$-linear maps $u_n\text{\hspace{0.17em}⁣}:U_n\to U_n^{\mathrm{\prime }}$ with $u_{n+1}\circ T_n=T_n^{\mathrm{\prime }}\circ u_n$ for all $n$.

Fix $a\ge 0$, a set $\mathbf{a}$ of cardinality $a$, and a functorial disjoint union of finite sets. Let $V$ be an FI-module over $k$ (The category FI and FI-modules over a commutative ring) and let $n\ge 0$. Put

${\mathrm{\Phi }}_a(V{)}_n:=(V_{\mathbf{a}\bigsqcup n}{)}_{S_n},$

the coinvariants of the $k{S}_n$-module $V_{\mathbf{a}\bigsqcup n}$: the quotient by the span of the elements $\sigma \cdot v-v$, $\sigma \in S_n$, $v\in V_{\mathbf{a}\bigsqcup n}$ (The coinvariants functor, Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets). Equivalently ${\mathrm{\Phi }}_a(V{)}_n=k{\otimes }_{k{S}_n}{V}_{\mathbf{a}\bigsqcup n}$, the two descriptions being identified by the balancing argument of The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums and the corresponding identification for $\mathbb{Z}$ in The coinvariants functor; the group ring $k{S}_n$ is a unital $k$-algebra with basis the group elements (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$).

To define the transition map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$, choose an injection $f\text{\hspace{0.17em}⁣}:n\to n+1$. It induces the injection ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f\text{\hspace{0.17em}⁣}:\mathbf{a}\bigsqcup n\to \mathbf{a}\bigsqcup n+1$ and hence a $k$-linear map $(V_{\mathbf{a}\bigsqcup n}\to V_{\mathbf{a}\bigsqcup n+1})$. This map is $S_n$-equivariant, where $S_n$ acts on the target through the injection ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$, so it descends to the quotient map $T=T_f$ on coinvariants. The map $T_f$ is independent of $f$: for two injections $f,g\text{\hspace{0.17em}⁣}:n\to n+1$ the composite $g\circ f^{-1}$ is a bijection $f(n)\to g(n)$, which extends along the complement to a permutation $\tau \in S_{n+1}$ with $g=\tau \circ f$; then ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup g=({\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup \tau )\circ ({\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f)$, and the map induced by ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup \tau$ on $(V_{\mathbf{a}\bigsqcup n+1}{)}_{S_{n+1}}$ is the identity because it comes from the action of the group element $\tau$. Thus $T$ is well defined and $({\mathrm{\Phi }}_a(V),T)$ is a graded $k[T]$-module.

A map of FI-modules $\alpha \text{\hspace{0.17em}⁣}:V\to W$ gives $k$-linear maps ${\alpha }_{\mathbf{a}\bigsqcup n}\text{\hspace{0.17em}⁣}:V_{\mathbf{a}\bigsqcup n}\to W_{\mathbf{a}\bigsqcup n}$ commuting with the $S_n$-actions and with the maps induced by ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$, hence maps $({\mathrm{\Phi }}_a\alpha {)}_n$ of coinvariants commuting with $T$; so ${\mathrm{\Phi }}_a$ is a functor from FI-modules over $k$ to graded $k[T]$-modules.

Permutations of $\mathbf{a}$ commute with the moving-block action of $S_n$, so ${\mathrm{\Phi }}_a(V{)}_n$ also carries an $S_a$-action. Since ${\mathrm{i}\mathrm{d}}_{\mathbf{a}}\bigsqcup f$ commutes with these permutations, every $T$ is $S_a$-equivariant. Naturality has the same equivariance.

Exactness. Coinvariants of a group module are right exact (The coinvariants functor is right exact), so the functor $M\mapsto M_{S_n}\cong k{\otimes }_{k{S}_n}M$ is right exact and each ${\mathrm{\Phi }}_a$ is right exact in every degree. If $k$ contains $\mathbb{Q}$, then $1\mathrm{/}n!$ exists in $k$ and the averaging map $s\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to V_{\mathbf{a}\bigsqcup n}$, $s([v])=\frac{1}{n!}{\sum }_{\sigma \in S_n}\sigma \cdot v$, is a well-defined $k$-linear section of the quotient map that is natural in $V$; it identifies ${\mathrm{\Phi }}_a(V{)}_n$ with the invariants $V_{\mathbf{a}\bigsqcup n}^{S_n}$. Under this natural identification ${\mathrm{\Phi }}_a$ becomes the invariants functor, which is left exact (kernels of equivariant maps are computed pointwise), and therefore ${\mathrm{\Phi }}_a$ is exact; in that case the induced map ${\mathrm{\Phi }}_a(W)\to {\mathrm{\Phi }}_a(V)$ is injective for every sub-FI-module $W\subseteq V$ (Submodule of a module, Module homomorphism and isomorphism, kernel, image and cokernel), so ${\mathrm{\Phi }}_a(W)$ is identified with a submodule of ${\mathrm{\Phi }}_a(V)$. Finite averaging uses only the finite groups $S_n$; no choice principle is used.