FI -modules over Noetherian rings

FIm -MODULES OVER NOETHERIAN RINGS arXiv:1705.00876v2 [math.RT] 23 May 2017 LIPING LI AND NINA YU Abstract. In this paper we study representation theory of the category FIm introduced in [6, 7] which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FIm in a natural way. In particular, we prove the representation stability property of finitely generated FIm -modules over fields of characteristic 0. 1. Introduction 1.1. Motivation. Representation theory of the category FI and its plentiful applications in various fields such as algebra, algebraic topology, algebraic geometry, number theory, rooted from the work of Church, Ellenberg, Farb, and Nagpal in [4, 5], have been actively studied. With their initial contributions and the followed explorations described in a series of papers [3, 8, 9, 10, 18, 13, 14, 16, 17, 21, 22, 23, 24], people now have a satisfactory understanding on representational and homological properties of FI-modules and their relationships with representation stability properties. Furthermore, quite a few combinatorial categories sharing similar structures as FI, including FIG , FId , FIm , FIM have been introduced and studied with viewpoints from representation theory, commutative algebra, and combinatorics; see [6, 7, 24, 26]. The main goal of this paper is to investigate representations of FIm , a generalization of FI introduced by Gadish in [6, 7]. Intuitively, FIm is a product of m copies of FI, so it is reasonable to expect that methods to study FI-modules and their outcomes can extend to FIm in a natural way. As the reader will see, this is indeed the case, although the techniques become more complicated and subtle. In particular, we show that FIm is locally Noetherian over any commutative Noetherian ring, and homological properties of relative projective FI-modules (or ♯-filtered modules in literature) still hold for FIm . As an important application, we show that finitely generated FIm -modules over fields of characteristic 0 have representation stability. 1.2. Notation. Throughout this paper let m be a positive integer. By definition (see [6, 7]), objects of the category FIm are m-tuples of finite sets T = (T1 , . . . , Tm ), and morphisms from an object T to another object T′ are maps f = (f1 , . . . , fm ) such that each fi : Ti → Ti′ is injective. It has a skeletal full subcategory C, whose objects are n = ([n1 ], . . . , [nm ]) where [ni ] = {1, 2, . . . , ni } and by convention [0] = ∅. When m = 1, FIm coincides with FI. Note that objects of C form a ranked poset isomorphic to Nm , where N is the set of all nonnegative integers. That is, t 6 n if and only if ti 6 ni for all i ∈ [m]. To simplify the notation, we identify objects in C with elements in Nm and hope that this simplification would not cause too much confusion to the reader. The degree of an object n = ([n1 ], . . . , [nm ]) is defined to be deg(n) = n1 + . . . + nm . The degree of a morphism f ∈ C(n, t) is defined to be deg(t) − deg(n). With this degree function the category FIm becomes a graded category. Let k be a unital commutative ring. A representation of C, or a C-module, is a covariant functor V from C to k -Mod, the category of k-modules. It is well known that C -Mod is an abelian category. Moreover, it has enough projectives. In particular, for n ∈ Nm , the k-linearization of the representable functor C(n, −) is projective. We call it a free module, and denote it M (n). The value of a representation V on an object n is denoted by Vn . A representation V of C is said to be finitely generated if there exists a finite subset S of V such that any submodule containing S coincides with V ; or equivalently, there exists a surjective homomorphism M M (n)⊕an → V n∈Nm The first author was supported by the National Natural Science Foundation of China 11541002, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037. The second author was supported by China NSF 11601452. 1 2 LIPING LI AND NINA YU P such that n∈Nm an < ∞. It is said to be generated in degrees 6 N if in the above surjection one can let an = 0 for all n with deg(n) > N . It is easy to see that V is finitely generated if and only if it is generated in degrees 6 N for a certain N ∈ N and the values Vn with deg(n) 6 N are finitely generated k-modules. The category C has m distinct self-embedding functors ι1 , . . . , ιm of degree 1. That is, for each i ∈ [m], ιi is a faithful functor C → C such that n = (n1 , . . . , ni , . . . , nm ) 7→ (n1 , . . . , ni + 1, . . . , nm ) = n + 1i where 1i = (0, . . . , 0, 1, 0, . . . , 0) and 1 is on the i-th position. They induce m distinct shift functors Σi : C -Mod → C -Mod. Let Σ be the direct sum of these shift functors Σi , i ∈ [m]. There exist natural maps V → Σi V , and hence a natural map V ⊕m → ΣV . The derivative functor D is defined to be the cokernel functor induced by these natural maps V ⊕m → ΣV . Therefore, for each C-module V , we have an exact sequence 0 → KV → V ⊕m → ΣV → DV → 0. 1.3. Noetherianity. By the ranked poset structure on the set of objects, the category C has a two-sided ideal of morphisms G C(i, j). 06i<j Denote the k-linearization of this ideal by m, which is a two-sided ideal of the category algebra kC (for a definition, see [25]). Given a C-module V and a nonnegative integer s, the s-th homology group of V is defined to be Hs (V ) = TorkC s (kC/m, V ). This is a kC-module since m is a (kC, kC)-bimodule. Accordingly, the s-th homological degree is hds (V ) = sup{deg(n) | (Hs (V ))n 6= 0}. By convention, we set hds (V ) to be -1 whenever the above set is empty. We call the zeroth homological degree generating degree, and denote it by gd(V ). When k is a commutative Noetherian ring, it is easy to see that a locally finite (see Definition 3.1) Cmodule V is finitely presented if and only if hd1 (V ) < ∞. Using this homological characterization, we can prove: Theorem 1.1. The category C is locally Noetherian over any commutative Noetherian ring k. That is, submodules of finitely generated C-modules are still finitely generated. Remark 1.2. For m = 1, this result was proved in [5, Theorem A]. For arbitrary m > 1, Gadish proved in [7] this theorem for fields of characteristic 0. 1.4. Relative projective modules. For each object n, its endomorphisms form a group Sn isomorphic to Sn1 × . . . × Snm , where Sni is a symmetric group on ni letters. Since the group algebra kSn is a subalgebra of the category algebra kC and M (n) is a (kC, kSn )-bimodule, given a kSn -module U , it induces a C-module M (n) ⊗kSn U . We call these modules basic relative projective modules. A C-module V is said to be relative projective if it has a filtration 0 = V −1 ⊆ V 0 ⊆ . . . ⊆ V s = V such that V i+1 /V i is isomorphic to a basic relative projective module for −1 6 i 6 s − 1. They are also called ♯-filtered modules or just filtered modules in literature such as [14, 16, 17, 19]. As the reader can see, they generalize projective modules, and have similar homological properties. Theorem 1.3. Let V be C-module over a commutative ring k, and suppose that gd(V ) < ∞. Then the following statements are equivalent: (1) (2) (3) (4) V is relative projective; Hs (V ) = 0 for all s > 1; H1 (V ) = 0; Hs (V ) = 0 for some s > 1. Remark 1.4. The case of m = 1 was independently proved by the authors in [17] and Ramos in [21]. FIm -MODULES OVER NOETHERIAN RINGS 3 1.5. Torsion theory. Let V be a C-module. An element v ∈ Vn is said to be torsion if there exist another object t and an injection f : n → t such that f · v = 0. Torsion elements in V generate a submodule of V , denoted by VT . If VT = 0, we say that V is torsion free; if VT = V , we call V a torsion module. In general, there is a short exact sequence 0 → VT → V → VF → 0, where VF and VT are the torsion free part and the torsion part of V respectively. The torsion degree td(V ) of V is defined in Definition 2.8. The following theorem generalizes the corresponding result of finitely generated FI-modules over commutative Noetherian rings. Theorem 1.5. Let V be finitely generated C-module over a commutative Noetherian ring k. There exists a complex F• : 0 → V → F0 → F1 → ... → Fl → 0 such that the following statements hold: (1) each F j is a relative projective module with gd(F j ) 6 gd(V ) − j; (2) l 6 gd(V ), (3) all homology groups H j (F • ) of this complex are finitely generated torsion modules. Consequently, Σn1 1 . . . Σnmm V is a relative projective module if ni > tdi (H j (F • )) + 1 for all 0 6 j 6 l and i ∈ [m]. By the theorem, we define Ni (V ) = max{tdi (H j (F • )) | j > 0} for i ∈ [m], which are finite numbers independent of the choice of a particular complex. Remark 1.6. For m = 1, this result was verified by Nagpal in [19, Theorem A]. Other proofs with explicit upper bounds were given by the authors in [14, 17] and Ramos in [21]. In this paper we will modify the techniques in [14, 17] to prove the above theorem. When m = 1, this complex has played a vital role in establishing upper bounds of homological invariants and developing a local cohomology theory for FI-modules; see [14, 16, 17]. 1.6. Representation stability. The above results can be used to prove certain representation stability patterns of finitely generated C-modules over fields. Let us recall some notation. A partition of a nonnegative integer n is a sequence λ = (λ1 > λ2 > . . . > λl ) such that |λ| = λ1 + . . . + λl = n. For t > n + λ1 , we define the padded partition to be λ(t) = (t − n > λ1 > . . . > λl ) which is a partition of m. For n = (n1 , . . . , nm ) ∈ Nm , an m-fold partition λ = (λ1 , . . . , λm ) such that λi is a partition of ni for m i ∈ [m]. Let λ1 = (λ11 , . . . , λm 1 ) ∈ N . For t > n + λ1 , one defines an m-fold padded partition λ(t) = (λ1 (t1 ), . . . , λm (tm )). When k is a field of characteristic 0, for an object n in C and its endomorphism group Sn ∼ = Sn1 ×. . .×Snm , from group representation theory we know that simple kSn -modules are parameterized by m-fold partitions of n. Moreover, when t > n + λ1 , m-fold padded partitions provide a uniform way to describe irreducible modules of distinct St . Denote by Lλ(t) the irreducible kSt -module parameterized by the m-fold partition λ(t). Motivated by [7], we define representation stability of FIm -modules as follows: Definition 1.7. Let k be a field of characteristic 0 and V be a finitely generated C-module. We say that V has representation stability if for all morphisms f : n → t, when ni ≫ 0 for i ∈ [m], one has: • the linear map V (f ) : Vn → Vt is injective; • the image of V (f ) generates Vt as a kSt -module; • there exist finitely many m-fold partitions λ(1) , . . . , λ(k) such that M ci Lλ(i) (n), Vn = i∈[k] where the multiplicity ci is independent of n. Our next theorem generalizes representation stability of FI-modules proved by Church, Ellenberg, and Farb in [4]. 4 LIPING LI AND NINA YU Theorem 1.8. Let V be a C-module over a field of characteristic 0. Then V is finitely generated if and only if it has representation stability. Moreover, in that case the numbers ni , i ∈ [m], in the above definition only need to satisfy ni > max{2 gd(V ), Ni (V ) + 1}. Remark 1.9. Gadish showed the representation stability for finitely generated projective C-modules in [7, Theorem 6.13]. 1 The conclusion of the above theorem follows immediately from his result as well as Theorem 1.5. Another important asymptotic behavior is the polynomial growth property. We have: Theorem 1.10. Let V be a finitely generated C-module over a field. Then there exist m polynomials Pi ∈ Q[X], i ∈ [m], with degrees not exceeding gd(V ) such that for objects n satisfying ni > max{gd(V ), Ni (V )+1} for all i ∈ [m], one has Y Pi (ni ). dimk Vn = i∈[m] 1.7. Projective dimensions. Homological properties of relative projective modules can also be used to classify finitely generated C-modules whose projective dimension is finite, extending [17, Theorem 1.5]. Theorem 1.11. Let k be a commutative Noetherian ring whose finitistic dimension findim k is finite, and let V be a finitely generated C-module. Then the projective dimension pdkC (V ) is finite if and only if there exist a finite set S of objects n and finitely generated kSn -modules Wn , such that V is a relative projective module whose filtration components are M (n) ⊗kSn Wn and pdk (Wn ) < ∞ for all n ∈ S. Moreover, in that case pd(V ) = max{pdk (Wn )}n∈S 6 findim k. 1.8. Remarks. The behaviors of FIG (see [16] for a definition) are very similar to those of FI, where G is a finite group. One can define product categories FIm G and shift functors, and establish analogue versions of the main results listed above for FIm G -modules. The category OI was introduced in [24]. Its objects are finite linearly ordered sets, and morphisms are order-preserving injections. One can define the product category OIm for m > 1 together with shift functors for OIm -modules. Using the techniques described in Section 3, we can prove a similar version of Theorem 1.1 for OIm . However, since Statement (1) of Lemma 2.1 fails for OIm , we cannot extend other main results of FIm in this paper to OIm . We would like to thank Wee Liang Gan for many valuable discussions and comments on our manuscript. After the paper was posted on arXiv, Sam told us that the Noetherianity of FIm is an immediately corollary of [24, Theorem 1.1.3 and Proposition 4.3.5] since FI is known to be quasi-Gröbner. The authors were also notified by Casto that in [1] he had independently proved Theorem 1.5 over fields of characteristic 0 and Theorem 1.8 of this paper, and described quite a few interesting applications of these results on arithmetics and geometry. The authors thank them for the discussions, comments, and communications. 2. Preliminary results Throughout this section let k be a unital commutative ring, C be the skeletal subcategory of FIm with objects parameterized by n ∈ Nm , and V be a C-module. 2.1. Some combinatorics. Recall that objects in C are elements n = (n1 , . . . , nm ) ∈ Nm (or more precisely, objects are parameterized by n ∈ Nm ), and morphisms from object n to t are maps f = (f1 , . . . , fm ) such that each fi : [ni ] → [ti ] is an injection, i ∈ [m]. Define n 6 t if the set C(n, t) of morphisms is nonempty. This is a well defined partial order on the set of objects in C and is compatible with the usual order on Nm ; that is, n 6 t if and only if ni 6 ti for all i ∈ [m]. We describe some combinatorial properties of C, most of which can be easily verified from the corresponding results of FI. Lemma 2.1. Let n, t and l be three distinct objects in C. One has: (1) If n 6 t, then the endomorphism group St acts transitively on C(n, t) from the left and the endomorphism group Sn acts freely on C(n, t) from the right. 1Note that free C-modules have different meanings in this paper and in [7]. Free C-modules defined in [7, Definition 1.8] are direct sums of basic relative projective modules in our sense, and when k is a field of characteristic 0, they coincide with projective C-modules. FIm -MODULES OVER NOETHERIAN RINGS 5 (2) If f and g are two distinct morphisms in C(n, t) and h ∈ C(t, l), then h ◦ f 6= h ◦ g. (3) For any sequence of objects n 6 n1 6 n2 6 . . . 6 ns 6 t, one has C(ns , t) ◦ C(ns−1 , ns ) ◦ . . . ◦ C(n1 , n2 ) ◦ C(n, n1 ) = C(n, t). Proof. As claimed, these statements can be deduced from the corresponding results of FI. We give a proof for the second statement as an explanation. Since f 6= g, we know that there exists an i ∈ [m] such that fi : [ni ] → [ti ] is different from gi : [ni ] → [ti ]. Therefore, by the corresponding result of FI, one knows that hi ◦ gi 6= hi ◦ fi . Consequently, h ◦ f 6= h ◦ g.  An important combinatorial property of C is the existence of self-embedding functors of degree 1. For i ∈ [m], we define a functor ιi : C → C as follows. For n ∈ Nm , one has n = ([n1 ], . . . , [ni−1 ], [ni ], [ni+1 ], . . . , [nm ]) 7→ ([n1 ], . . . , [ni−1 ], [ni + 1], [ni+1 ], . . . , [nm ]) = n + 1i . For a map f = (f1 , . . . , fm ) : n → t, the j-th component (ιi (f ))j of the map ιi (f ) : n + 1i → t + 1i coincides with fj for j 6= i, while the i-th component (ιi (f ))i : [ni + 1] → [ti + 1] is defined by ( 1, 1 = t ∈ [ni + 1]; (2.1) t 7→ fi (t − 1) + 1, 1 6= t ∈ [ni + 1]. The reader can check that each ιi is a faithful functor. Lemma 2.2. For i, j ∈ [m], one has ιi ◦ ιj = ιj ◦ ιi . Proof. It follows directly from the definition. Without loss of generality we assume that i < j. For an object n in C, one has ιi (ιj (n)) = n + 1j + 1i = n + 1i + 1j = ιj (ιi (n)). For a morphism f : n → t, both ιi (ιj (f )) and ιj (ιi (f )) equal (f1 , . . . , fi−1 , (ιi (f ))i , fi+1 , . . . , fj−1 , (ιj (f ))j , fj+1 , . . . , fm ).  For each ιi , there is a family of inclusions (2.2) {πn,i : n = ([n1 ], . . . , [ni ], . . . , [nm ]) → ([n1 ], . . . , [ni + 1], . . . , [nm ]) = n + 1i | n ∈ Nm } such that the j-th component of πn,i is the identity map for j 6= i, and the i-th component of πn,i maps t ∈ [ni ] to t + 1 ∈ [ni + 1]. This collection of maps gives a natural transformation πi between the identity functor IdC and the self-embedding functor ιi . 2.2. Shift functors. Each self-embedding functor ιi , i ∈ [m], induces a pull-back functor Σi : C -Mod → C -Mod by sending V to V ◦ ιi . This is an exact functor called the i-th shift functor. The natural transformations πi , i ∈ [m], induce natural transformations πi∗ between the identity functor on C -Mod and Σi . Consequently, there is a natural map V → Σi V for each i ∈ [m]. Let Σ be the direct sum of those pull-back functors Σi , which is exact as well. By taking direct sum, we obtain a natural map V ⊕m → ΣV . The derivative functor D of Σ is defined to be the cokernel of the natural map V ⊕m → ΣV . We also define KV to be the kernel of this natural map. With these definitions, we get an exact sequence 0 → KV → V ⊕m → ΣV → DV → 0. Note that both K and D are also direct sums of components. Explicitly, for each i ∈ [m], one has an exact sequence 0 → K i V → V → Σi V → D i V → 0 which is precisely the i-th component of the previous one. We list certain properties of these functors. Lemma 2.3. Let V be a C-module. Then one has: (1) Σi M (n) ∼ = M (n) ⊕ M (n − 1i )⊕ni . (2) Di M (n) ∼ = M (n − 1i )⊕ni . 6 LIPING LI AND NINA YU (3) gd(DV ) 6 gd(ΣV ) 6 gd(V ) = gd(DV ) + 1 whenever V is nonzero. (4) For i, j ∈ [m], Σi ◦ Σj = Σj ◦ Σj . (5) For i, j ∈ [m], Σi ◦ Dj ∼ = Dj ◦ Σi , and in particular, Σ ◦ D ∼ = D ◦ Σ. Proof. (1): The first statement follows from the m = 1 case. Indeed, for an object n, the free module M (n) is the external tensor product of free FI-modules. Explicitly, M (n) ∼ = M (n1 ) ⊠ . . . ⊠ M (nm ), where M (ni ) is the free FI-module associated with object [ni ] in FI. Now we have Σi M (n) = M (n1 ) ⊠ . . . ⊠ (ΣM (ni )) ⊠ . . . ⊠ M (nm ) and ΣM (ni ) ∼ = M (ni ) ⊕ M (ni − 1)⊕ni , where Σ is the shift functor of FI-modules defined in [4]. The conclusion follows. (2): Immediately follows from the previous statement. (3): Take a surjection P → V → 0 such that P is a direct sum of free C-modules with gd(P ) = gd(V ). Applying the shift functor one gets a surjection ΣP → ΣV → 0. Since gd(DP ) = gd(P ) by the Statement (1) of this lemma, one deduces that gd(ΣV ) 6 gd(V ). Since DV is a quotient module of ΣV , one also has gd(ΣV ) > gd(DV ). Now we show that gd(DV ) = gd(D(V )) + 1 whenever V 6= 0, the proof of which is similar to that of [15, Lemma 1.5] and [17, Proposition 2.4]. As explained in [15, Lemma 1.5], we only need to deal with the case that gd(V ) is finite. Since the conclusion holds for V with gd(V ) = 0, we assume that gd(V ) > 0. From the surjection DP → DV → 0 one deduces that gd(DV ) 6 gd(DP ) = gd(P ) − 1. On the other hand, there is an object n such that deg(n) = gd(V ) > 1 and (H0 (V ))n 6= 0. Now let V ′ be the submodule of V generated by Vt with t 6 n and t 6= n. Then V ′ is a proper submodule of V since otherwise one should have (H0 (V ))n = 0. The surjection V → V /V ′ → 0 induces a surjection DV → D(V /V ′ ) → 0, and hence gd(DV ) > gd(D(V /V ′ )). Note that the value of D(V /V ′ ) on an object t with deg(t) = deg(n) − 1 is nonzero, and the values of D(V /V ′ ) on all objects smaller than t is 0. Therefore, gd(DV ) > gd(D(V /V ′ )) > deg t = gd(V ) − 1. (4): Immediately follows from Lemma 2.2. (5): There are two cases: When i = j, by the definition, (Σi Di V )n = Vn+2i /πn+1i ,i · Vn+1i ; (Di Σi V )n = Vn+2i /ιi (πn,i ) · Vn+1i where ιi and πn,i are defined in (2.1) and (2.2) respectively. By these definitions, both πn+1i ,i and ιi (πn,i ) are maps: n + 1i = ([n1 ], . . . , [ni + 1], . . . , [nm ]) → ([n1 ], . . . , [ni + 2], . . . , [nm ]) = n + 2i . Moreover, for j 6= i, their j-th components are all identity maps. While their i-th components are [ni + 1] → [ni + 2], t 7→ t + 1; ( 1, t = 1; [ni + 1] → [ni + 2], t 7→ t + 1, 2 6 t 6 ni + 1. respectively. Now the reader can see that the family of bijections (e1 , . . . , ei−1 , g, ei+1 , . . . , em ) ∈ Sn+2i , where ej is the identity element in Sj , and g ∈ Sni +2 permutes 1 and 2 and fixes all other elements in [ni + 2], gives an isomorphism between Σi Di and Di Σi . When i 6= j, by the definition, (Σi Dj V )n = (Dj V )n+1i = Vn+1i +1j /πn+1i ,j · Vn+1i ; (Dj Σi V )n = Vn+1i +1j /ιi (πn,j ) · Vn+1i where ιi and πn,i are defined in (2.1) and (2.2) respectively. Now it is routine to check that πn+1i ,i and ιi (πn,i ) are the same map from n + 1i to n + 1i + 1j . Explicitly, for s 6= j, the s-th component of this map FIm -MODULES OVER NOETHERIAN RINGS 7 is the identity, while its j-th component sends elements in [nj ] to [nj + 1] by adding 1. Consequently, in this case Dj Σi = Σi Dj .  Although Σ is a direct sum of Σi for i ∈ [m], in general we do not have a good relationship between homological degrees of V and those of Σi V and Di V , except that gd(Di V ) < gd(V ) and gd(Di V ) 6 gd(Σi V ) 6 gd(V ) always hold. Here is an example. Example 2.4. Let m = 2, s be a positive integer, and V be a C-module satisfying the following conditions: gd(V ) = s, and Vn = 0 whenever n2 > 0. Then Σ2 V = D2 V = 0, so gd(V ) − gd(Σ2 V ) = s + 1 can be as large as we want. However, under a certain weak condition, we can still obtain a good control on the difference gd(V ) − gd(Di V ) for i ∈ [m]. Lemma 2.5. Let V be a nonzero C-module such that Vn = 0 for all objects n satisfying ni = 0. Then gd(Di V ) 6 gd(Σi V ) 6 gd(V ) = gd(Di V ) + 1. Proof. From the given condition we know that gd(V ) > 1. Now one can copy the proof of Statement (3) of Lemma 2.3, noting that in that argument the object n satisfies ni > 1, so one can choose t = n − 1i .  2.3. Torsion modules. Recall that an element v ∈ Vn for a certain object n is a torsion element if there exist another object t and an injection f ∈ C(n, t) such that f · v = 0. Let VT be the submodule of V generated by all these torsion elements, and denote VF = V /VT . We say that V is torsion if V = VT . If VT = 0, V is called torsion free. In particular, all free modules M (n) are torsion free by the second statement of Lemma 2.1. Lemma 2.6. Let V be a C-module. Then: (1) V is a torsion module if and only if for very object n and every v ∈ Vn , v is a torsion element. In particular, L of torsion modules are still torsion. L submodules and quotient modules (2) Ki V = n∈Nm {v ∈ Vn | πn,i · v = 0} = n∈Nm {v ∈ Vn | f · v = 0, ∀f ∈ C(n, n + 1i )}. (3) V is torsion free if and only if KV = 0. (4) If V is torsion free, so is ΣV . (5) In a short exact sequence 0 → U → V → W → 0, if both U and W are torsion free, so is V . (6) In a short exact sequence 0 → U → V → W → 0, if W is torsion free, then the sequence 0 → DU → DV → DW → 0 is exact as well. (7) If V is torsion, so is Σi V for i ∈ [m]. Proof. (1): One direction of the first part is trivial. For the other direction, suppose that V is a torsion module, so it is generated by torsion elements. Therefore, it suffices to prove the following claim: if v ∈ Vn is a torsion element and f : n → t is an injection, then f · v ∈ Vt is a torsion element as well. Since v is a torsion element, there exist an object l > n and an injection g ∈ C(n, l) such that g · v = 0. By Lemma 2.1, every injection in C(n, l) sends v to 0, so does every morphism in C(n, l + t). Taking an h ∈ C(t, l + t), we know (h ◦ f ) · v = 0. That is, f · v is a torsion element. The second part immediately follows from the first one. (2): By the definition, the natural map V → Σi V , while restricted to a fixed object n, is given by v 7→ πn,i ·v, and hence the first equality holds. For the second equality, one just notices that πn,i ∈ C(n, n+1i ) and this morphism set has only one orbit as an Sn+1i -set. (3): If KV 6= 0, then there exists a certain i ∈ [m] such that Ki V 6= 0. By the above description, V contains nonzero torsion elements, and hence is not torsion free. Conversely, if V is not torsion free, we can find two objects n < t, a nonzero element v ∈ Vn , and a morphism f ∈ C(n, t) such that f · v = 0. By a simple induction one can assume that deg(t) − deg(n) = 1; that is, there is a certain i ∈ [m] such that t = n + 1i . Clearly, v ∈ Ki V , so KV 6= 0. (4): Since V is torsion free, there is a short exact sequence 0 /V θV / ΣV / DV / 0. Applying the exact functor Σ one gets a short exact sequence 0 / ΣV ΣθV / ΣΣV / ΣDV / 0. 8 LIPING LI AND NINA YU Note that in the above sequence, the map ΣθV might not be the natural map θΣV : ΣV → Σ(ΣV ). However, since we just proved that DΣ ∼ = ΣD in Lemma 2.3, this sequence is actually isomorphic to the exact sequence determined by the map θΣV . In particular, KΣV ∼ = ΣKV = 0. That is, ΣV is torsion free as well. One can also directly check that ΣV is torsion free by applying the argument in the proof of [17, Proposition 2.1]. (5): Applying the exact functor Σ one gets a commutative diagram where all vertical rows represent natural maps: /V /W /0 /U 0 0  / ΣU  / ΣV  / ΣW / 0. According to the third statement, the maps U → ΣU and W → ΣW are injective, so is the map V → ΣV by the snake Lemma. (6): Follows from the above commutative diagram and the snake Lemma. (7): Suppose that Σi V 6= 0 and take an element v ∈ (Σi V )n = Vn+1i . Since V is torsion, there exist an object t > n + 1i and a morphism f ∈ C(n + 1i , t) such that f · v = 0. Consequently, every morphism in C(n + 1i , t) sends v to 0. Note that the self-embedding functor ιi maps morphisms in C(n, t − 1i ) into C(n + 1i , t) injectively. Therefore, every morphism in C(n, t − 1i ) sends v to 0, where v is regarded as an element in Σi V . That is, v is still a torsion element in Σi V .  Remark 2.7. Intuitively, an element v ∈ Vn is contained in Ki V if and only if it vanishes when moving one step along the i-th coordinate axis direction. Therefore, Ki V is a direct sum of several direct summands, each of which is supported on a “hyperplane” (when m = 1, a hyperplane is just a point) perpendicular to the i-th coordinate axis. One may give a more conceptual description for Ki V . Let Ii be the free k-module spanned by all morphisms in C(n, t), n, t ∈ Nm , such that ti > ni . The reader can check that Ii is a two-sided ideal of kC. Moreover, Ki V is precisely the trace of kC/Ii in V . That is, X Ki V = α(V ). α ∈ HomkC (M (n)/Ii M (n), V ) n ∈ Nm Now we define torsion degree, an invariant playing a crucial role in this paper. Definition 2.8. For i ∈ [m], let tdi (V ) = sup{s | ∃n such that (Ki V )n 6= 0 and ni = s}. The torsion degree of V is set to be td(V ) = sup{tdi (V ) | i ∈ [m]}. Whenever a set on the right side is empty, we set the number on the left side to be -1. Remark 2.9. For m = 1, this is not the usual definition of torsion degrees in [14, 16, 17]. However, as shown in [13], the above one is an equivalent definition. Explicitly, for i ∈ [m], tdi (V ) < ∞ if and only if the following two conditions hold: there is an object n with ni = tdi (V ) such that we can find an element 0 6= v ∈ Vn satisfying α · v = 0 for all α ∈ C(n, n + 1i ); for any object t with ti > tdi (V ) and any nonzero v ∈ Vt and α ∈ C(t, t + 1i ), α · v 6= 0. It is also clear that td(V ) = td(VT ) since V and VT have the same torsion elements, which completely determine the torsion degree. At this moment it is not clear that td(V ) is a finite number. However, we have: Lemma 2.10. Let V be a finitely generated nonzero C-module. One has: (1) If V ′ is a submodule, then td(V ) 6 max{td(V ′ ), td(V /V ′ )}. (2) If V is Noetherian, then td(V ) is finite and tdi (Σi V ) 6 tdi (V ) − 1 whenever tdi (V ) 6= 0. In particular, td(Σ1 . . . Σm V ) 6 td(V ) − 1 whenever td(V ) 6= 0. FIm -MODULES OVER NOETHERIAN RINGS 9 Proof. (1): Firstly suppose that td(V ) is finite, there exists a certain i ∈ [m] such that td(V ) = tdi (V ). Now by the previous remark, there is an object n with ni = tdi (V ) such that we can find an element 0 6= v ∈ Vn satisfying α · v = 0 for all α ∈ C(n, n + 1i ). If v ∈ Vn′ , then tdi (V ′ ) > tdi (V ), so td(V ′ ) > td(V ). Otherwise, the image 0 6= v̄ ∈ (V /V ′ )n of v satisfies α · v = 0 for all α ∈ C(n, n + 1i ), so by the same argument, tdi (V /V ′ ) > tdi (V ), and hence td(V /V ′ ) > td(V ). If td(V ) = ∞, there exists a certain i ∈ [m] such that tdi (V ) = ∞. In particular, we can find an infinite sequence of objects nj and an infinite sequence of nonzero elements v j ∈ Vnj such that α · v j = 0 for all α ∈ C(nj , nj + 1i ) and n1i < n2i < . . .. Using the same argument, one can show that either tdi (V ) = ∞ or tdi (V /V ′ ) = ∞. (2): To show the finiteness of td(V ), by Definition 2.8, it suffices to show that tdi (V ) < ∞ for i ∈ [m]. By Statement (2) of Lemma 2.6 and Remark 2.7, Ki V is a direct sum of a few direct summands, each of which is supported on a “hyperplane” in Nm which is perpendicular to the i-th coordinate axis. Since V is Noetherian, Ki V is finitely generated as well. Therefore, the number of such direct summands is finite. However, tdi (V ) is nothing but the maximal height of these “hyperplanes”, so must be finite as well. It is clear that tdi (Σi V ) = tdi (V )−1 whenever Ki V 6= 0. Moreover, one observes that tdi (Σj V ) 6 tdi (V ). These observations can be easily verified by Remark 2.7.  Remark 2.11. When m = 1, one has gd(KV ) = td(V ); see [14]. For m > 1, the equality might not hold, and the reader can easily find an example such that gd(KV ) > td(V ). However, we always have gd(Ki V ) > tdi (V ). To see this, consider the direct summand of Ki V which is supported on the highest hyperplane. Then tdi (V ) is precisely the height of this hyperplane, while the generating degree of this direct summand is greater than or equal to this height since the degree of every object in this hyperplane is at least the height. Therefore, as claimed, gd(Ki V ) > tdi (V ) for each i ∈ [m]. Consequently, gd(KV ) > td(V ). For m = 1, one has td(ΣV ) = td(V ) − 1 whenever V is not torsion free. This equality does not hold for m > 1 as well. For example, let m = 2, and V be a nonzero C-module such that its values on objects different from (3, 3) are all zeroes. Then td(ΣV ) = td(V ) = 3. One cannot strengthen the inequality in the above lemma to an equality, either. Indeed, let m = 2 and V be a nonzero module such that its values on all objects different from (3, 0) are zeroes. Then Σ1 Σ2 V = 0 and hence td(V ) − td(Σ1 Σ2 V ) = 4. 3. Noetherianity Throughout this section let k be a commutative Noetherian ring. We will show the following statement: every finitely generated C-module is also finitely presented. By a standard homological argument, this statement is equivalent to the locally Noetherian property of C. 3.1. Truncation functors. We introduce the following definition. Definition 3.1. A C-module V is locally finite if for each object n, Vn is a finitely generated k-module. Since k is Noetherian, the reader can see that the category of all locally finite C-modules is an abelian category. Furthermore, a locally finite C-module is finitely generated (resp., finitely presented) if and only if its generating degree is finite (resp., generating degree and first homological degree are finite). From now on we only consider locally finite C-modules. We have: Lemma 3.2. The following statements are equivalent: (1) The category C is locally Noetherian. (2) Every finitely generated C-module is finitely presented. (3) The first homological degree of every finitely generated module is finite. Proof. Obvious.  For each i ∈ [m], we define Ji = k G C(t, n), ni >0 spanned by morphism ending at objects n with ni > 0. This is a two-sided ideal of the category algebra kC. Furthermore, the reader can check that the quotient algebra kC/Ji is isomorphic to the category algebra kC′ , where C′ is the skeletal category of FIm−1 whose objects are parameterized by elements in Nm−1 . By convention, if m = 1, then kC/Ji ∼ = k. 10 LIPING LI AND NINA YU Let τi = kC/Ji ⊗kC −, which is a functor from the category of C-modules to the category of C′ -modules (by identifying kC/Ji with kC′ ). Conversely, every C′ -module can be viewed as a C-module by lifting. The behavior of τi has a very explicit description. That is, given a C-module V , one has ( Vn , if ni = 0; (τi V )n = 0, otherwise, and (Ji V )n = ( Vn , 0, if ni > 0; otherwise. Therefore, we obtain a short exact sequence 0 → Ji V → V → τi V → 0. Moreover, one has Σi V = Σi Ji V since Σi τi V = 0. 3.2. Finitely presented property. Let V be a finitely generated C-module. In this subsection we relate the finitely presented property of V to that of Σi V for each i ∈ [m]. Lemma 3.3. Suppose that FIm−1 is locally Noetherian over k, and let V be a finitely generated C-module. (1) If there is a certain i ∈ [m] such that Σi V is finitely presented, then V is finitely presented. (2) If the natural map V → Σi V is injective, and Di V is finitely presented, then V is finitely presented. Proof. By the exact sequence 0 → Ji V → V → τi V → 0, it suffices to show that both τi V and Ji V are finitely presented. Note that τi V is finitely generated, and so is Ji V because gd(Ji V ) 6 gd(Σi Ji V ) + 1 = gd(Σi V ) + 1 6 gd(V ) + 1 by Lemma 2.5 (the first inequality) and Lemma 2.3 (the second inequality). We claim that hd1 (τi V ) is finite. Let P → τi V → 0 be a surjection such that P is a finitely generated projective C-module with gd(P ) = gd(τi V ). It induces the following commutative diagram 0 0 /W / Ji P / W̃  P  P  / τi P  / τi V /W /0 / 0. Since gd(P ) is finite, so is gd(Ji P ) by the same argument as in the previous paragraph. Moreover, by viewing terms in the bottom row as FIm−1 -modules and using the given condition, one deduces that gd(W ) < ∞. Consequently, from the top row we conclude that gd(W̃ ) is finite, so τi V is finitely presented. Now we turn to Ji V . Let 0 → U → Q → Ji V → 0 be a short exact sequence such that Q is a finitely generated projective C-module with gd(Q) = gd(Ji V ). (1): Applying Σi we get 0 → Σi U → Σi Q → Σi Ji V = Σi V → 0. Since Σi V is supposed to be finitely presented, gd(Σi U ) < ∞. Moreover, by the structure of Ji V , we can suppose that Qn = 0 for any object n with ni = 0. Then by Lemma 2.5, we conclude that gd(U ) 6 gd(Σi U ) + 1. That is, Ji V is finitely presented as well. (2): Since the natural map V → Σi V is injective, so is the natural map Ji V → Σi Ji V = Σi V . Therefore, we have the following commutative diagram 0 0 / τi V / Ji V /V  Σi J i V  Σi V  / D i Ji V  / Di V / τi V /0 / 0. Note that Di V is finitely presented by the given condition, and we just proved that τi V is finitely presented. Consequently, Di Ji V is finitely presented, too. FIm -MODULES OVER NOETHERIAN RINGS 11 By Statement (6) of Lemma 2.6, the short exact sequence 0 → U → Q → Ji V → 0 induces a short exact sequence 0 → Di U → Di Q → Di Ji V → 0. Since Di Ji V is finitely presented, gd(Di U ) < ∞. By Lemma 2.5, gd(U ) < ∞. That is, Ji V is finitely presented.  3.3. Filtrations. In this subsection we construct a sequence of quotient modules for a C-module V using a fixed shift functor Σ1 . Let V 0 = V , V 1 be the image of the natural map V 0 → Σ1 V 0 , and V 2 be the image of the natural map V 1 → ΣV 1 , and so on. In this way we obtain a sequence of quotient maps V = V 0 → V 1 → V2 → .... Define V̄ = lim V i , → which is a quotient module of V . Lemma 3.4. Let V̄ be as define above. Then the natural map V̄ → Σ1 V̄ is injective. Moreover, if the kernel of V → V̄ is finitely generated, then the above sequence stabilizes after finitely many steps. Proof. The proof is similar to proofs of [11, Lemmas 3.1, 3.3 and 3.7].  3.4. Proof of Noetherianity. Now we are ready to prove the locally Noetherianity of C over commutative Noetherian rings. A proof of Theorem 1.1. We use a double induction on m and the generating degree gd(V ). For m = 0, kFI0 = k by our convention, so the conclusion holds. Suppose that the conclusion holds for FIs with s < m. For FIm , it suffices to show that for any finitely generated C-module V we have hd1 (V ) < ∞ by Lemma 3.2. Clearly, this is true for gd(V ) = −1, that is, V = 0, so we suppose that gd(V ) > 0. As described in the previous subsection, we obtain a sequence of quotient maps V = V 0 → V 1 → V 2 → ... and a limit V̄ . If gd(V̄ ) < gd(V ), then by the induction hypothesis on generating degrees, V̄ is finitely presented. If gd(V̄ ) = gd(V ), then the natural map V̄ → Σ1 V̄ is injective, and gd(D1 V̄ ) < gd(V̄ ) 6 gd(V ). Therefore, by the induction hypothesis on generating degrees, D1 V̄ is finitely presented, so does V̄ by Statement (2) of Lemma 3.3. In both cases, we know that V̄ is finitely presented, so the kernel of the quotient map V → V̄ is finitely generated. By Lemma 3.4, there exists an N ∈ N such that V n = V N for all n > N . By our construction, there is a short exact sequence 0 → V N → Σ1 V N −1 → D1 V N −1 → 0. The induction hypothesis tells us that D1 (V N −1 ) is finitely presented since gd(D1 V N −1 ) < gd(V N −1 ) 6 gd(V ). We just proved that V N is finitely presented, so is Σ1 V N −1 . By Statement (1) of Lemma 3.3, V N −1 is finitely presented as well. Replacing V N by V N −1 and carrying out this procedure, recursively one can show that all V i , 0 6 i 6 N are finitely presented. In particular, V is finitely presented. The conclusion follows by induction.  4. Relative projective modules In this section we consider relative projective modules. These special modules generalize projective modules, and have played a key role in constructing a homological computation machinery for FI-modules. For details, see [14, 16, 17]. Our goal is to extend most results in [17] from FI to FIm for arbitrary m > 1. Let k be an arbitrary commutative ring, and V be a C-module. 4.1. Basic properties. Given an object n and a kSn -module W , we can define a C-module M (n) ⊗kSn W as M (n) is a (kC, kSn )-bimodule. We call it a basic relative projective module. Definition 4.1. A C-module V is a relative projective module if it has a filtration 0 = V 0 ⊆ V 1 ⊆ . . . ⊆ V n = V such that each factor V i+1 /V i , 0 6 i 6 n − 1, is isomorphic to a basic relative projective module. The reader immediately sees that if V is a relative projective module, then gd(V ) < ∞. Of course, one can remove this restriction from the above definition. However, since we are mostly interested in finitely generated C-modules, it is reasonable to impose this condition. It is also obvious that projective modules with finite generating degrees are relative projective modules. Lemma 4.2. Let V be a C-module generated by its value on a certain object n. One has: (1) The following are equivalent: 12 LIPING LI AND NINA YU • V is a basic relative projective module; • Hs (V ) = 0 for all s > 1; • H1 (V ) = 0. (2) If V is relative projective, then it is torsion free. (3) If hd1 (V ) 6 gd(V ), then V is relative projective. (4) If V is relative projective, so is Σi V and Di V for i ∈ [m]. ∼ M (n) ⊗kS Vn . Take a short Proof. (1): Suppose that V is basic relative projective module. That is, V = n exact sequence 0 → W → P → Vn → 0 of kSn -modules such that P is projective. Applying the functor M (n) ⊗kSn − which is exact by Statement (1) of Lemma 2.1, we get a short exact sequence ∼ V → 0. 0 → M (n) ⊗kS W → M (n) ⊗kS P → M (n) ⊗kS Vn = n n n Applying the functor kC/m ⊗kC − we recover the original short exact sequence. That is, H1 (V ) = 0. Replacing V by V ′ = M (n) ⊗kSn W we deduce that H2 (V ) = 0. Recursively, for every s > 1, one gets Hs (V ) = 0. Conversely, suppose that H1 (V ) = 0. Since V is generated by Vn , there is a short exact sequence 0 → K → M (n) ⊗kSn Vn → V → 0. The long exact sequence . . . → H1 (V ) = 0 → H0 (K) → H0 (M (n) ⊗kSn Vn ) = Vn → H0 (V ) = Vn → 0 implies that H0 (K) = 0. That is, K = 0, and hence V is relative projective. (2): Since V is relative projective, without loss of generality we assume that V = M (n) ⊗kSn Vn . Take an arbitrary l > n, an element 0 6= v ∈ Vl , and a morphism f ∈ C(l, t) with t > l. We want to show that f · v 6= 0. Note that Vl = C(n, l) ⊗kSn Vn , and C(n, l) is a right free kSn -module by Statement (1) of Lemma 2.1. Therefore, v can be written as (g1 ⊗ u1 ) + . . . + (gs ⊗ us ) such that each ui ∈ Vn is nonzero, and moreover gi and gj are contained in distinct orbits of the right Sn -set C(n, l) if i 6= j. Consequently, f · v = (fg1 ⊗ u1 ) + . . . + (fgs ⊗ us ). We claim that fgi and fgj are contained in distinct orbits of the right Sn -set C(n, t) if i 6= j. Indeed, if fgi and fgj are in the same orbit, there exists an automorphism h ∈ Sn such that fgi h = fgj . By Statement (2) of Lemma 2.1, gi h = gj , contradicting the assumption imposed on gi and gj . Therefore, those fgi ⊗ ui are nonzero and lie in distinct direct summands of C(n, l) ⊗kSn Vn . Consequently, f · v 6= 0. (3): Again, consider the short exact sequence 0 → K → M (n) ⊗kSn Vn → V → 0 and its induced long exact sequence . . . → 0 → H1 (V ) → H0 (K) → H0 (M (n) ⊗kSn Vn ) = Vn → H0 (V ) = Vn → 0. Since hd1 (V ) 6 gd(V ), we know that H0 (K) is only supported on objects whose degrees are at most gd(V ) = deg n. But from the short exact sequence we see that K is only supported on objects strictly greater than n. The only possibility is that K = 0. (4): Consider a short exact sequence 0 → K → P → V → 0 such that P is a projective kC-module generated by Pn . By the previous arguments we know that all terms in it are basic relative projective modules generated by their values on n, and hence are torsion free. By Statement (6) of Lemma 2.6, for each i ∈ [m], we get a short exact sequence 0 → Di K → Di P → Di V → 0. Since Di V = 0 whenever ni = 0, without loss of generality we assume that ni > 0. Then Di V is generated by its value on the object n − 1i since so is Di P . Replacing V by K one knows that Di K is also generated by its value on n − 1i . Since Di P is projective, the long exact sequence of homology groups induced by 0 → Di K → Di P → Di V → 0 tells us that gd(Di V ) = gd(Di P ) > gd(Di K) > hd1 (Di V ). By the previous statement, Di V is relative projective. But Di V = Σi V /V , so Σi V is relative projective as well since so is V .  Remark 4.3. For m = 1, Statements (2) and (4) have been verified in [19, Lemma 2.2]. Actually, slightly modifying the argument there, one can prove the following stronger conclusion: If V is a basic relative projective module generated by its value on a certain object n, then for i ∈ [m], Vn ), Σi V ∼ = V ⊕ (M (n − 1i ) ⊗kS n−1i whenever ni > 0. For ni = 0, Σi V ∼ =V. FIm -MODULES OVER NOETHERIAN RINGS 13 From the proof we also observe the following fact: Let V be a basic relative projective C-module generated by Vn , and Wn be a kSn -module. Then H0 (M (n) ⊗kSn Wn ) ∼ =V. = Wn , and M (n) ⊗kSn H0 (V ) ∼ Since every relative projective module has a filtration whose factors are basic relative projective modules, from the above lemma one gets: Proposition 4.4. Let V be a C-module with gd(V ) < ∞. Then V is relative projective if and only if H1 (V ) = 0, and if and only if Hs (V ) = 0 for all s > 1. Moreover, if V is relative projective, then it is torsion free, and Σi V and Di V are relative projective as well for i ∈ [m]. Proof. The first statement is clear, and the first half of the second statement follows from Statement (5) of Lemma 2.6. To prove the second half of the second statement, suppose that V • : 0 = V 0 ⊆ V 1 ⊆ . . . ⊆ V n = V is a filtration of V such that each factor is a basic relative projective module. Applying the i-th shift functor Σi we get a filtration Σi V • of Σi V such that each factor is still a relative projective module (might not be a basic relative projective module) by Statement (4) of the previous lemma. Moreover, since relative projective modules are torsion free, by Statement (6) of Lemma 2.6, applying Di we get a filtration Di V • of Di V such that each factor is a basic relative projective module.  The following recursion lemma generalizes results in [17, Lemmas 3.12 and 4.2], and plays a vital role in the proofs of Theorems 1.3 and 1.5. Lemma 4.5. Let V be a torsion free C-module with gd(V ) < ∞. One has: (1) If DV is relative projective, so is V . (2) If V is relative projective, and V ′ is relative projective submodule, then V /V ′ is relative projective. Proof. The conclusions hold for V = 0 trivially, so we may assume that V 6= 0. (1): Since gd(V ) is finite, H0 (V ) is only supported on a set S of finitely many objects in Nm . We use induction on the cardinality |S|. The conclusion holds clearly if |S| = 0. For |S| > 1, we choose an object n ∈ S such that deg(n) is maximal; that is, deg(n) = gd(V ) (of course, this choice might not be unique). Let V ′ be the submodule of V generated by its values on objects in S \ {n}. Then V ′′ = V /V ′ is nonzero and is generated by its value on n. Consider the short exact sequence 0 → V ′ → V → V ′′ → 0. We claim that V ′′ is a basic relative projective module. To see this, from the long exact sequence of homology groups one has hd1 (V ′′ ) 6 max{gd(V ′ ), hd1 (V )} 6 max{gd(V ), hd1 (V )}. since gd(V ′ ) 6 gd(V ) by our construction of V ′ . Furthermore, let 0 → W → P → V → 0 be a short exact sequence of C-modules such that P is a free C-module satisfying gd(V ) = gd(P ). Applying D we get another short exact sequence 0 → DW → DP → DV → 0 such that DP is also free and satisfies gd(DV ) = gd(DP ). Then one has hd1 (V ) 6 gd(W ) 6 gd(DW ) + 1 6 max{hd1 (DV ), gd(DV )} + 1 6 gd(DV ) + 1 = gd(V ) since hd1 (DV ) = −1. Putting the above two inequalities together we conclude that hd1 (V ′′ ) 6 gd(V ) = gd(V ′′ ). By Statement (3) of Lemma 4.2, V ′′ is a basic relative projective module as claimed. The conclusion follows after we show that V ′ is relative projective. By the induction hypothesis, it suffices to show that DV ′ is relative projective. Since V ′′ is relative projective, and hence torsion free, by Statement (6) of Lemma 2.6, we get a short exact sequence 0 → DV ′ → DV → DV ′′ → 0. Note that DV is relative projective by the given condition, and so is DV ′′ by Statement (4) of Lemma 4.2. The long exact sequence of homology groups tells us that H1 (DV ′ ) = 0, so DV ′ is relative projective by Proposition 4.4. (2): We use induction on gd(V ). The conclusion holds if gd(V ) = −1; that is, V = 0. For gd(V ) > 0, consider the following commutative diagram: 0 / V′ /V / V /V ′ 0  / ΣV ′  / ΣV  / Σ(V /V ′ ) /0 δ / 0. The kernel of δ is a torsion module, and is isomorphic to a submodule of DV ′ by the snake Lemma. But V ′ is relative projective, so DV ′ is also relative projective, and hence is torsion free. This happens if and only if this kernel is 0. That is, V /V ′ is torsion free. 14 LIPING LI AND NINA YU By Statement (6) of Lemma 2.6, we get a short exact sequence 0 → DV ′ → DV → D(V /V ′ ) → 0. By Proposition 4.4, both DV ′ and DV are relative projective. Since gd(V ) > gd(DV ), by the induction hypothesis, D(V /V ′ ) is relative projective as well. The previous statement tells us that V /V ′ is relative projective.  4.2. A proof of Theorem 1.3. Now we are ready to prove various homological characterizations of relative projective C-modules. A proof of Theorem 1.3. The equivalence of the first three statements is implied by Proposition 4.4. Clearly, (2) implies (4). The argument showing that (4) implies (1) is completely the same as that of [17, Proposition 4.4]. For the convenience of the reader, we include it here. Of course, we can assume that s > 1. Take a projective resolution P • → 0 and let Z i be the i-th cycle. Then we have 0 = Hs (V ) = H1 (Z s−1 ). Consequently, Z s−1 is a relative projective module. Applying the previous lemma to the short exact sequence 0 → Z s−1 → P s−2 → Z s−2 → 0 one deduces that Z s−2 is relative projective as well. The conclusion follows by recursion.  A useful corollary is: Corollary 4.6. Let 0 → U → V → W → 0 be a short exact sequence of C-modules with finite generating degrees. Then if two terms in it are relative projective, so is the third term. Proof. Consider the long exact sequence of homology groups and apply the above theorem.  4.3. Projective dimensions. In this subsection we give an application of the previous theorem, classifying finitely generated C-modules with finite projective dimension over commutative Noetherian rings. Recall that the finitistic dimension of a commutative Noetherian ring k, denoted by findim k, is defined to be sup{pdk (T ) | T is a finitely generated k-module and pdk (T ) < ∞}. A proof of Theorem 1.11. The proof for FI described in [17, Subsection 4.2] actually works for arbitrary m > 1 with small modifications. We give a sketch. First, if pdkC (V ) is finite, then it has a finite projective resolution. By the previous theorem, V must be a relative projective module. Let S be the set of objects n such that (H0 (V ))n 6= 0. Then V has a filtration such that each factor is a basic relative projective module M (n)⊗kSn Wn , n ∈ S. As the proof of [17, Lemma 4.7] (replacing n in that proof by a maximal element in S), an induction on the size of S asserts that pdkC (V ) > pdkC (M (n) ⊗kSn Wn ) = pdkSn (Wn ) = pdk (Wn ) for n ∈ S, where the two equalities can be shown as in [17, Lemmas 4.5 and 4.6]. But from a standard homological argument one also has pdkC (V ) 6 max{pdkC (M (n) ⊗kSn Wn )}n∈S . Therefore, the description of V holds if we assume that pdkC (V ) is finite. The other direction is obvious.  For semisimple rings or finite dimensional local algebras, one has: Corollary 4.7. Let k be a commutative Noetherian ring whose finitistic dimension is 0. Then a finitely generated C-module has finite projective dimension if and only if it is projective. 4.4. A proof of Theorem 1.5. In this subsection we prove Theorem 1.5. Let k be a commutative Noetherian ring, and all C-modules considered here are finitely generated. By Theorem 1.1, the category of finitely generated C-modules is abelian. Recall that tdi (V ) for i ∈ [m] and td(V ) are defined in Definition 2.8. Moreover, by Lemma 2.10, these numbers are finite. Lemma 4.8. Let V be finitely generated C-module over a commutative Noetherian ring and i ∈ [m]. Then Ki Σni i V = 0 for ni > tdi (V ) + 1. In particular, for n > td(V ), the shifted module Σn1 . . . Σnm V is torsion free. FIm -MODULES OVER NOETHERIAN RINGS 15 Proof. From the proof of Lemma 2.6 and Remark 2.7 one easily sees that Σni i Ki V = 0 for ni > tdi (V ). As explained in the proof of Statement (4) of Lemma 2.6, one has Ki Σni i V ∼ = Σni i Ki V = 0. n If n > td(V ), since td(V ) > tdi (V ) for all i ∈ [m], one has Ki Σi V = 0 for all i ∈ [m]. Therefore, K i Σn . . . Σn V ∼ = Σn . . . Σn Σn . . . Σn K i Σn V = 0 1 m 1 i−1 i+1 m i by Statement (4) of Lemma 2.3. Consequently, KΣn1 . . . Σnm V = 0.  However, Σn V might not be torsion free for any n ∈ N. Here is an example. Example 4.9. Let m = 2 and n = (0, 0). Let V be the C-module M (n)/I2 M (n). Then for an object t = (t1 , t2 ), we have: ( k, if m2 = 0; Vt ∼ = 0, otherwise. ∼ V . Consequently, one has ΣV ∼ Clearly, V is a torsion module. Moreover, one has Σ2 V = 0 and Σ1 V = =V, so Σn V can never be torsion free. However, since td(V ) = 0, one has Σ1 Σ2 V = 0. Proposition 4.10. Let V be a finitely generated C-module over a commutative Noetherian ring k. Then for n ≫ 0, the shifted module Σn1 . . . Σnm V is relative projective. Proof. Suppose that V 6= 0. By the previous lemma, Σn1 . . . Σnm V is torsion free. Therefore, by Statement (1) of Lemma 4.5, it suffices to show that DΣn1 . . . Σnm V is relative projective. By Statement (5) of Lemma 2.3, one has DΣn1 . . . Σnm V ∼ = Σn1 . . . Σnm DV. Since gd(DV ) = gd(V ) − 1 by Statement (3) of Lemma 2.3, the induction hypothesis on generating degrees asserts that Σn1 . . . Σnm DV is relative projective. The conclusion follows by induction.  A proof of Theorem 1.5. The proof is almost the same as that of [17, Theorem 4.14], so we only give a brief explanation. For n ≫ 0, applying the functor Σn1 . . . Σnm to the short exact sequence 0 → VT → V → VF → 0 one gets a short exact sequence 0 → 0 → F 0 → F 0 → 0, where Σn1 . . . Σnm VT = 0 by Lemma 2.10 and F 0 = Σn1 . . . Σnm VF is a relative projective module by the previous proposition. The map V → F 0 in the complex can be defined as the composite of the quotient V → VF and the natural injection VF → F 0 . Let V 1 be the cokernel of the map V → F 0 and repeat the above procedure for it. Eventually one gets a complex 0 → V → F 0 → F 1 → . . .. n−1 Dm V . By (3) of Lemma 2.3, gd(DV ) < gd(V ), so gd(Dm V ) < gd(V ). Note that V 1 ∼ = Σn1 . . . Σnm−1 Σm Moreover, it also tells us that functors Σi , i ∈ [m], do not increase the generating degree. Therefore, gd(V 1 ) < gd(V ). Consequently, gd(V ) > gd(F 0 ) > gd(V 1 ) > gd(F 1 ) > . . . and gd(V ) > gd(V 1 ) > . . . . Consequently, l 6 gd(V ) and gd(F j ) 6 gd(V ) − j. From the definition of this complex one sees that the image of V → F 0 is VF and the kernel is VT . The map F i → F i+1 is the composite of F j → V j+1 → VFj+1 → F j+1 , whose image is VFj+1 . As in the proof of [17, Theorem 4.14], one can show that the homology at F j is VTj+1 , a finitely generated torsion module. Suppose that ni > tdi (H j (F • )) + 1 for all 0 6 j 6 l and i ∈ [m]. Applying Σn1 1 . . . Σnmm to the above complex, we get a shifted complex 0 → Σn1 1 . . . Σnmm V → Σn1 1 . . . Σnmm F 0 → Σn1 1 . . . Σnmm F → . . . → Σn1 1 . . . Σnmm F l → 0. By Lemma 2.10, all homology groups of F • vanish under the shift. Therefore, the above shifted complex is exact. Now applying Corollary 4.6 we deduce that Σn1 1 . . . Σnmm V is a relative projective module.  Remark 4.11. By Corollary 4.6 and the construction of F • , V is a relative projective module if and only if F • is an exact sequence, if and only if all Hj (F • ) = 0 for 0 6 j 6 l. With this observation, one can obtain the following stronger conclusion: Σn1 1 . . . Σnmm V is a relative projective module if and only if ni > max{tdi (Hj (F • )) | 0 6 j 6 l}. 16 LIPING LI AND NINA YU Define N (V ) = max{td(H j (F • )) | j > 0}, and for each i ∈ [m], let Ni (V ) = max{tdi (Hj (F • )) | j > 0} . These are finite numbers independent of the choice of particular relative projective complexes since different ones are quasi-isomorphic. 5. Representation stability As a main application of the theorems previously established, in this section we deduce representation stability properties of finitely generated FIm -modules over fields, as we did for FI-modules in [11, 14, 16, 17]. 5.1. Representation stability. Representation stability of finitely generated FI-modules was first observed in [4]. We extend their result to finitely generated FIm -modules. A proof of Theorem 1.8. Clearly, if V is representation stable, then for each object n, Vn is finite dimensional by the third condition in Definition 1.7. Moreover, the second condition in that definition tells us that V is generated by its values on finitely many objects. Therefore, V is finitely generated. Conversely, if V is finitely generated, then the second condition in Definition 1.7 must hold. To check the first and the third conditions, consider the complex F• : 0 → V → F0 → F1 → ... → Fl → 0 where each F i is a finitely generated projective C-module since k is a field of characteristic 0. For an object n with ni > max{2 gd(V ), Ni (V ) + 1} for all i ∈ [m], as explained in Remark 4.11, we get an exact sequence 0 → Vn → Fn0 → Fn1 → . . . → Fnl → 0 since the values of all homology groups of F • on n vanish, so the first condition in Definition 1.7 holds. Moreover, from Theorem 1.5 we know that gd(F j ) 6 gd(V ). Now as Gan and the first author did for FIG in [11, Proof of Theorem 1.12], the reader can deduce the third condition in Definition 1.7 from the corresponding results of finitely generated projective modules, which was verified in [7, Theorem 6.13].2  5.2. Hilbert functions. Throughout this subsection let k be a filed of arbitrary characteristic, and V be a finitely generated C-module over k. The Hilbert function of V is defined by Nm → N, n 7→ dimk Vn . Lemma 5.1. Let F be a finitely generated relative projective modules. Then there exist m polynomials Pi ∈ Q[X] with degrees not exceeding gd(V ) such that for objects n satisfying ni > gd(V ) for all i ∈ [m], one has dimk Fn = P1 (n1 ) . . . Pm (nm ). Proof. Since F has a filtration by basic relative projective modules, it suffices to show that each filtration component satisfies the conclusion. That is, without loss of generality we can assume that F is a basic relative projective module. Suppose that F is generated by its value Fl on an object l with deg(l) = gd(F ). Consequently, F ∼ = M (l) ⊗kSl Fl . Now let n be an object such that ni > gd(F ) for all i ∈ [m]. Clearly, n > l. Therefore, Vn = kC(l, n) ⊗kSl Vl and hence dimk Vn = dimk Vl · Y i∈[m] which implies the conclusion. (ni )! , (ni − li )!(li )!  We prove the polynomial growth property of V , extending [5, Theorem B]. A proof of Theorem 1.10. Again, consider the complex F• : 0 → V → F 0 → F 1 → . . . → F l → 0. For an object n with ni > max{gd(V ), Ni (V ) + 1} for all i ∈ [m], we get an exact sequence 0 → Vn → Fn0 → Fn1 → . . . → Fnl → 0. Moreover, by Theorem 1.5 gd(F i ) 6 gd(V ). Now the reader only needs to check that those relative projective modules have the desired property, which is established in the previous lemma.  2We use the condition that n > 2 gd(V ) for i ∈ [m] to intrigue the conclusion of that theorem. i FIm -MODULES OVER NOETHERIAN RINGS 17 6. Questions and further remarks So far the reader can see that many representational and homological properties of FI extend to FIm for arbitrary m > 1. However, compared to the rich results of representation theory of FI, there are still many interesting parallel parts for FIm deserving to be established. In this section we list a few questions. 6.1. Regularity. Church and Ellenberg have shown that reg(V ) 6 gd(V ) + hd1 (V ) − 1 for FI-modules over any commutative ring. For arbitrary m > 1, Gan and the first author proved in [12] that reg(V ) < ∞ for finitely generated FIm -modules over commutative Noetherian rings. However, for m > 1, there is no known upper bound for the regularity. Therefore, we wonder whether there exists an upper bound of reg(V ) in terms of hdi (V ), 0 6 i 6 m, for any FIm -modules. If this is true, in particular, the category of FIm -modules satisfying hdi (V ) < ∞, 0 6 i 6 m, is an abelian category containing all finitely generated FIm -modules, and one can work in this large category without worrying about the ground ring k. 6.2. Coinduction functor. The shift functors Σi , i ∈ [m], are restriction functors, and hence have left and right adjoint functors. The left adjoint functors are inductions which can be easily defined, while the right adjoint functors, called coinduction functors, are more delicate. In [10] Gan and the first author defined coinduction functor for FI-modules, explored its properties, and proved a few useful results. We hope to extend the nice properties of coinduction functor and their outcomes described in [10, 16] to FIm -modules. In particular, when k is a field of characteristic 0, is it still true that every finitely generated projective FIm -module is also injective? 6.3. Local cohomology theory. Motivated by the results described in Subsection 1.4 of this paper, we believe that there exists a local cohomology theory for FIm -modules, extending the work of Ramos and the first author for FI-modules in [16]. In particular, the following statement might hold: Let V be a finitely generated C-module over a commutative Noetherian ring. Then V is a relative projective module if and only if all local cohomology groups vanish. Moreover, the homology groups in the complex F • coincide with local cohomology groups of V . In a recent paper [20], Nagpal, Sam, and Snowden proved that for a finitely generated FI-module over a commutative Noetherian ring, one has reg(V ) = max{td(Hi (V )) + i | i > 0}, confirming a conjecture posted in [16]. We wonder whether there exists a similar equality for FIm . 6.4. Applications in algebra, topology, and geometry. Representation theory of FI has rich applications in representation stability theory. Indeed, the (co)homology groups of many sequences of mathematical objects have been found to be equipped with an FI-module structure. Recently, Gadish described applications of finitely generated projective FIm -modules on a generalization of configuration spaces in [6, 7]. We believe that those results on finitely generated FIm -modules shall have more interesting applications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] K. Casto, Representation theory and arithmetic statistics of generalized configuration spaces, in preparation. T. Church, Bounding the homology of FI-modules, arXiv:1612.07803. T. Church and J. Ellenberg, Homology of FI-modules, to appear in Geom. Topol., arXiv:1506.01022. T. Church, J. Ellenberg, B. Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833-1910, arXiv:1204.4533. T. Church, J. Ellenberg, B. Farb, R. Nagpal, FI-modules over Noetherian rings, Geom. 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Snowden, GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc. 368 (2016), 1097-1158. [24] S. Sam, A. Snowden, Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), 159-203, arXiv:1409.1670. [25] P. Webb, Standard stratifications of EI categories and Alperin’s weight conjecture, J. Algebra 320 (2008), 4073-4091. [26] J. Wilson, FIW -modules and stability criteria for representations of classical Weyl groups, J. Algebra 420 (2014), 269-332. Key Laboratory of Performance Computing and Stochastic Information Processing (Ministry of Education), College of Mathematics and Computer Science, Hunan Normal University; Changsha, Hunan 410081, China. E-mail address: [email protected] School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China. E-mail address: [email protected].