Polynomial conditions and homology of FI-modules

We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable degree) and clarify their relationship. For one of these conditions, we give improved twisted homological stability ranges for the symmetric groups. As another application, we improve the representation stability ranges for congruence subgroups with respect to the action of an appropriate linear group by a factor of 2 in its slope.


Introduction
There are (at least) two classes of papers that deal in some depth with FI-modules: (1) In papers such as [3; 4; 6; 11; 14] the FI-module is the central object of study. They attach homological invariants to an FI-module by means such as FI-homology or local cohomology, and study the relationship of these invariants both with the stabilization behavior of the FI-module and/or between each other.
(2) Papers such as [12; 13; 16; 18; 22] might be thought of as stability machines. The sequence {S n } of symmetric groups is but one of many sequences of groups they deal with and FI-modules arise as the suitable notion of coefficient systems for {S n }. They declare a coefficient system to be polynomial with certain parameters in a recursive fashion: there is a base case, and above that, being polynomial demands a related coefficient system to be polynomial with some of the parameters lowered.
Notation. We write FI for the category of finite sets and injections. An FI-module is a functor V : FI → ‫-ޚ‬Mod and given a finite set S, we write V S for its evaluation; given an injection of finite sets α : S → T , we write V α : V S → V T for its induced map. For n ∈ ‫ގ‬ we set V n := V {1,...,n} . We write FI-Mod for the category of FI-modules. Throughout, our notation for FI-modules will be consistent with [6] and [1]. and call it the stable degree of V. In both polynomial conditions for FI-modules we shall consider, the stable degree will be in analogy with the usual degree of a polynomial. Also see [6,Proposition 2.14].
First polynomial condition and local degree. Suppose f is a function in n ∈ ‫ގ‬ which is equal to a polynomial of degree ≤ r in the range n ≥ L. We can consider its discrete derivative f , which is the function Note that f is equal to a polynomial of degree ≤ r − 1 in the same range n ≥ L.
The first polynomial condition we treat for FI-modules is a categorification of this recursion. See [22,Section 4.4 and Remark 4.19] for references to similar definitions in the literature.
Definition 1.1. For every pair of integers r ≥ −1, L ≥ 0, we define a class of FI-modules Poly 1 (r, L) recursively via Remark 1.2. Let V be an FI-module and r ≥ −1, L ≥ 0 be integers. The following can be seen to be equivalent by inspection: Our first main result is that the stable degree δ(V ) and the local degree h max (V ) together characterize the first polynomial condition.
Theorem A. For every pair of integers r ≥ −1, L ≥ 0, we have Second polynomial condition and regularity. The second polynomial condition we shall treat is, perhaps deceivingly, very similar to the first one. In fact the confusion between the two and the resulting need to clarify was what prompted this paper. Definition 1.3. For every pair of integers r ≥ −1, M ≥ 0, we define a class of FI-modules Poly 2 (r, M) recursively via Remark 1.4. Let V be an FI-module and r ≥ −1, M ≥ 0 be integers. The following can be seen to be equivalent by inspection: • V ∈ Poly 2 (r, M).
• In the sense of [22,Definition 4.10], V has degree r at M. 1 Note that [21] is an early preprint version of the published [22]. The authors switched from for every FI-module V. We say that V is generated in degrees ≤ g if t 0 (V ) ≤ g, and that V is presented in finite degrees if t 0 (V ) and t 1 (V ) are both finite. We call reg(V ) the regularity of V.
Our second main result is that the stable degree δ(V ) and the regularity reg(V ) together characterize the second polynomial condition. Twisted homological stability with FI-module coefficients. For any FI-module V and homological degree k ≥ 0, there is a sequence of maps between the homology groups of the symmetric groups twisted by V n 's. For the stabilization of this sequence, recently Putman [18, Theorems A and A ′ ] established explicit ranges for the class Poly 2 (r, M) in terms of r, M. We give ranges for the class Poly 1 (r, L) in terms of r, L.
Theorem C. Let V be an FI-module and r, L ≥ 0 be integers such that V ∈ Poly 1 (r, L). Then for every k ≥ 0, the map H k (S n ; and a surjection for n ≥ • an isomorphism for n ≥ max{2L + 1, 2k + 2r + 2}, • and a surjection for n ≥ max{2L + 1, 2k + 2r }. The ranges in Theorem C are improvements over these.

SL U
n -stability ranges for congruence subgroups. For every ring R, the assignment n → GL n (R) defines an FI-group (a functor from FI to the category of groups), for which we write GL • (R). If I is an ideal of R, as the kernel of the mod-I reduction we get a smaller FI-group We wish to extend the S n -action on H k (GL n (R, I ); A) to an action of a linear group and formulate representation stability over it, in accordance with [17, fifth Remark, page 990].
Special linear group with respect to a subgroup of the unit group. For a commutative ring A and a subgroup U ≤ A × , we write : det( f ) ∈ U}, so that we interpolate between SL n (A) ≤ SL U n (A) ≤ GL n (A) as we vary 1 ≤ U ≤ A × . Note that we are using the notation in [19], whereas in [13] and [12] this group is denoted GL U n (A). Hypothesis 1.6. In the triple (R, I, n 0 ), we have a commutative ring R, an ideal I of R, and an integer n 0 ∈ ‫ގ‬ such that the mod-I reduction SL n (R) → SL n (R/I ) for the special linear group is surjective for every n ≥ n 0 .
Stable rank of a ring. Let R be a nonzero unital (associative) ring. A column vector v ∈ Mat m×1 (R) of size m is unimodular if there is a row vector u ∈ Mat 1×m (R) such that uv = 1. Writing I r ∈ Mat r ×r (R) for the identity matrix of size r , we say a column vector v of size m is reducible if there exists A ∈ Mat (m−1)×m (R) with block form A = [I m−1 | x] such that the column vector Av (of size m − 1) is unimodular. We write st-rank(R) ≤ s if every unimodular column vector of size > s is reducible. Remark 1.7. We make a few observations about Hypothesis 1.6.
(1) It is straightforward to check that the triple (R, I, n 0 ) satisfies Hypothesis 1.6 if and only if setting U := {x + I : x ∈ R × }, there is a short exact sequence of groups in the range n ≥ n 0 where the epimorphism is the mod-I reduction.
Consequently, for every n ≥ n 0 and any coefficients A, the conjugation GL n (R)action on the homology groups H ⋆ (GL n (R, I ), A) descends to an SL U n (R/I )-action. It is this action for which we will obtain an improved representation stability range.
(2) For a Dedekind domain R and any ideal I of R, the triple (R, I, 0) satisfies Hypothesis 1.6; see [7, page 2].
(3) If SL n (R/I ) is generated by elementary matrices for n ≥ n 0 , then (R, I, n 0 ) satisfies Hypothesis 1.6.
(4) If the K -group SK 1 (R/I ) = 0 (equivalently, the natural map K 1 (R/I ) → (R/I ) × is an isomorphism) and st-rank(R/I ) ≤ s < ∞, then by (3)  for every homological degree k ≥ 1 and abelian group A, there is a coequalizer diagram of ‫ޚ‬G n -modules whenever

Homological algebra of FI-modules
Regularity in terms of local cohomology. We first recall a characterization of the regularity by Nagpal, Sam and Snowden [14].
The rest follows from Theorem 2.1. □ The derivative and local cohomology. In this section, we investigate the relationship between the local cohomology of an FI-module and that of its derivative. We write K := ker(id FI-Mod → ) so that we have an exact sequence There is nothing to show when h 0 (V ) = −1, so we consider two cases.
is nonempty and hence has a least element, say N . Noting N > d, let A be a finite set of size N −1 and f : S → A so by the minimality of N we have 0 There is a finite set T and an injection f : Given an FI-module V, the following are equivalent: (1) V is presented in finite degrees.
Proof. For (1), note that K V is certainly generated in degrees Now applying H 0 m to the short exact sequence the associated long exact sequence gives a short exact sequence for every j ≥ 1. Using these isomorphisms after applying H 0 m to the short exact sequence for every j ≥ 0, naturally in U . For (2), assume V is presented in finite degrees. Then deg(K V ) = h 0 (V ) < ∞ (so we have the long exact sequence from (1)) and V is presented in finite degrees by Lemma 2.4 and Proposition 2.5. Now invoke [6, Theorem 2.10] for V and V . □ Corollary 2.7. For every FI-module V presented in finite degrees, the following hold: Proof. By Proposition 2.6, for every j ≥ 0 we have and (1)  To prove (2), fix j ≥ 1 and set N := max{h j−1 ( V ), h j ( V )} so for every n > N , by Proposition 2.6 we have an isomorphism But H j m (V ) has finite degree, therefore the above isomorphisms in the entire range n > N have to be between zero modules so that h j Critical index and the regularity of derivative. In this section, we introduce the notion of critical index for an FI-module and use it to study how regularity interacts with the derivative functor. Remark 2.9. Let V be as in Definition 2.8 and set γ := crit(V ), ρ := reg(V ). The following will not be needed in our arguments but we note them for context.
In the former case, we have j ∈ J (V ) and by Theorem 2.1, and in the latter case, we have j + 1 ∈ J (V ) and by Theorem 2.1. Yet another application of Theorem 2.1 now yields reg( V ) ≤ ρ −1, which is exactly (1). To prove (2), we further assume that γ ≥ 1. We claim that To that end, by Proposition 2.6 we have an exact sequence which we evaluate at a finite set of size ρ − γ to get an exact sequence of S ρ−γ -modules. Here: Therefore we conclude that On the other hand, part (1) and Theorem 2.1 give the reverse inequality to the above, establishing our claim, the equation reg( V ) = ρ − 1 and the inequality crit( V ) ≤ γ − 1. To see in fact crit( V ) = γ − 1, we can take 0 ≤ j < γ − 1 and evaluate the exact sequence at a finite set of size ρ − 1 − j to get Identifying the polynomial conditions. In this section we prove Theorems A and B. Because FI-modules being presented in finite degrees is such a common assumption, we first incorporate it as a redundant hypothesis in Theorems 2.12 and 2.13, and then remove this redundancy using Theorem 2.11.
• Ind FI FB for the left adjoint of the restriction functor Res FI FB : FI-Mod → FB -Mod.
. Then there is a short exact sequence Thus applying H FI 0 to ( †), the associated long exact sequence splits into isomorphisms for every i ≥ 0. In particular, we have Thus V is presented in finite degrees by Corollary 2.3. Next, assume δ(V ) ≥ 0. We can apply Proposition 2.6 to V to conclude h max ( V ) < ∞. We also have δ( V ) ≤ δ(V ) − 1, therefore V is presented in finite degrees by the induction hypothesis. We conclude by applying Proposition 2.5. Proof. We fix L ≥ 0 and employ induction on r . For the base case r = −1, we first let V ∈ Poly 1 (−1, L), that is, deg(V ) ≤ L −1. Then V is torsion so δ(V ) = −1, and by Corollary 2.3 V is presented in finite degrees with h max (V ) ≤ L − 1. Conversely, suppose V is presented in finite degrees, δ(V ) ≤ −1, and h max (V ) ≤ L − 1. Then V is torsion, so H 0 m (V ) = V has degree ≤ L − 1. For the inductive step, fix r ≥ 0 and assume that we have U is presented in finite degrees, δ(U ) ≤ r − 1, and h max (U ) ≤ L − 1 .
By the induction hypothesis, we conclude the following.
• V is presented in finite degrees: it follows that V is presented in finite degrees by Proposition 2.5.
Conversely, let V be an FI-module which is presented in finite degrees, δ(V ) ≤ r , and h max (V ) ≤ L − 1. We observe: • By Proposition 2.5, V is presented in finite degrees. By the induction hypothesis, we conclude the following.
• V is presented in finite degrees: it follows that V is presented in finite degrees by Proposition 2.5. Conversely, let V be an FI-module which is presented in finite degrees, δ(V ) ≤ r , and reg(V ) ≤ M − 1 (in particular, h 0 (V ) ≤ M − 1 by Theorem 2.1). We observe: • r +1 V = r V is torsion, so δ( V ) ≤ r − 1.
• By Proposition 2.5, V is presented in finite degrees.
Thus by Theorem B, we have Consequently by [18,Theorem A], for every k ≥ 0 the map It remains to improve the bounds in the case L ≥ max{1, 2r − 1} to • n ≥ max{2k + 2r + 1, L} for the isomorphism range, • n ≥ max{2k + 2r, L} for the surjection range. To that end, we induct on r . For the base case r = 0, by [1,Theorem 2.11] there is an H FI 0 -acyclic I with δ(I ) ≤ 0 and a map V → I which is an isomorphism in degrees ≥ L. As I is torsion but also is H FI 0 -acyclic, we have I = K I = 0, in other words I → I is an isomorphism. Thus I n is the same trivial S n -representation for every n ≥ 0 (namely the abelian group I 0 with the trivial S n -action). Now by [15,Corollary 6.7], for every k ≥ 0 the map is an isomorphism for n ≥ 2k. Thus for every k ≥ 0, the map is an isomorphism for n ≥ max{2k, L} (which is better than what the base case demands: an isomorphism for n ≥ max{2k+1, L} and a surjection for n ≥ max{2k, L}).
In particular by [6, Proposition 2.9, part (7)], this applies to In degrees n ≥ L, writing I := L V , we have a short exact sequence 0 → V n → I n → U n → 0 of S n -modules, and the associated long exact sequence in H * (S n ; −) maps to that of H * (S n+1 ; −). More precisely, suppressing the symmetric groups in the homology notation, there is a commutative diagram
We now consider the groupoid G := SL U (R/I ) in order to follow the argument and notation in [12, proof   By [6,Proposition 3.3] we also have δ(V ) ≤ 2k. Thus by Theorem B and Remark 1.4, and in the sense of [12,Definition 2.40], V has polynomial degree ≤ 2k in ranks > 2s + 3 if k = 1, 4k + 2s if k ≥ 2.
(1) By [12,Remark 2.42], V has the same polynomial degree and rank bounds as a U G-module.