Castelnuovo-Mumford regularity of representations of certain product categories
aa r X i v : . [ m a t h . R T ] J a n CASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAINPRODUCT CATEGORIES
WEE LIANG GAN AND LIPING LI
Abstract.
We show in this paper that representations of a finite product of categories satisfying certaincombinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented infinite degrees, and hence the category consisting of them is abelian. These results apply to examples suchas the categories FI m and FI mG . Introduction
Representation theory of infinite categories is a relatively new research area with applications in topology,geometric group theory, algebraic geometry, and commutative algebras; see for instance [1, 2, 3, 4]. Tostudy representation theoretic properties of an infinite category C , usually the first step is to find a suitableabelian category of representations in which homological algebra can be carried out. Since the category ofall representations of C (also called C -modules) is too large for practical purpose, one naturally considersthe category of finitely generated C -modules. However, this idea does not always work well because of thefollowing reasons: (1), there are many infinite categories (for example, the poset of positive integers equippeda partial ordering induced by division) which are not locally Noetherian even when the coefficient ring is afield;, that is, submodules of finitely generated C -modules might not be finitely generated; (2), it is not aneasy task to show the locally Noetherian property of C over a commutative Noetherian coefficient ring, see forinstance [17]; (3), for some applications in topology, people frequently have to consider infinitely generated C -modules over arbitrary coefficient rings. Therefore, in many cases we expect to find intermediate abeliansubcategories between the category of all C -modules and the category of finitely generated C -modules.A significant resolution of this question was established by Church and Ellenberg in [1] for the categoryFI of finite sets and injections. They proved that if a representation V of FI over an arbitrary commutativecoefficient ring is presented in finite degrees (see Definition 2.2), then its Castelnuovo-Mumford regularity (or regularity for short, see Definition 2.4) is finite. Furthermore, they obtained an explicit and simpleupper bound for the regularity in [1, Theorem A]. This result immediately implies that representations ofFI presented in finite degrees form an abelian category.The main goal of this paper is to extend the above result of Church and Ellenberg to finite products C × · · · × C m of categories C , . . . , C m satisfying certain combinatorial conditions (see Subsections 2.2 and3.1). An example of such finite products is the category FI m , the product of m copies of FI for every positiveinteger m , which was introduced by Gadish in [4, 5] and whose representation theoretic properties were alsostudied in [11, 13]. Let C = C × · · · × C m . Under the combinatorial assumptions, objects of C form a rankedposet, so its representations (or C -modules, see the definition in Subsection 2.3) have a graded structure, andhence we can define regularity for them. Based on an inductive machinery developed by the authors in [8]as well as a key observation that a numerical invariant of an C -module V is finite whenever V is presented infinite degrees, we prove the following result, partially answered a question proposed in [13, Subsection 6.1]. Theorem 1.1.
Let C = C × · · ·× C m where C , . . . , C m are categories satisfying the combinatorial conditionsspecified in Subsections 2.2 and 3.1. Then a C -module V over an arbitrary commutative coefficient ring hasfinite Castelnuovo-Mumford regularity if and only if it is presented in finite degrees. The basic strategy to prove the above theorem is as follows. The category C is equipped with m self-embedding functors , which induce m shift functors Σ i , i ∈ [ m ], in the module category (see Subsection 3.1). The second author was supported by the National Natural Science Foundation of China (Grant No. 11771135), HuXiangHigh-Level Talents Gathering Project by the Science and Technology Department of Hunan Province (Grant No. 2019RS1039),and the Research Foundation of Education Bureau of Hunan Province (Grant No. 18A016). Both authors appreciate theanonymous referee for carefully checking the manuscript and providing many very helpful and insightful comments.
When V is a torsion-free C -module (see Subsection 2.6), there is a short exact sequence of C -modules:0 → V ⊕ m → M i ∈ [ m ] Σ i V → M i ∈ [ m ] D i V → , where D i ’s are the cokernel functors, and one can deduce the finiteness of regularity of V from that of thethird term via induction. For an arbitrary C -module V , we construct a tree T ( V ) of quotient modules of V recursively. We prove that it is a finite tree when V is presented in finite degrees, and furthermore themodules on the lowest level are torsion free, turning to the special case previously handled. We also provethe fact that if all modules on a level of T ( V ) have finite regularity, so are all members lying on the levelabove it. This recursive method allows us to verify the finiteness of regularity of all members in T ( V ), inparticular that of V .A cornerstone of the above strategy is the finiteness of the tree T ( V ). In [8], the authors consideredfinitely generated representations of several infinite combinatorial categories over commutative Noetherianrings, where the finiteness of this tree follows directly from the locally Noetherian property of these categories.In this paper we introduce a new idea; that is, for C -modules V presented in finite degrees, we assign anintegral numerical invariant to each member in T ( V ), and show that if a member is a child (see Definition4.1) of another member, then the numerical invariant of the former is strictly smaller than that of the laterone. Now the finiteness of T ( V ) is guaranteed by the fact that the numerical invariants of all C -modulesshare a common lower bound.This paper is organized as follows. In Section 2, we describe the setting for our results and recall somebasic definitions. In Section 3, we discuss shift functors. In Section 4, for every C -module, we construct atree of quotient modules and use it to give a proof for the main theorem. The last section contains corollariesof the main theorem. 2. Preliminaries
Notations.
Denote by N the set of non-negative integers. For each n ∈ N , we set [ n ] = { , . . . , n } ; inparticular, [0] = ∅ . We fix a commutative ring k with identity.2.2. The categories C , . . . , C m and their product C . Throughout this paper, we let C be the product C × · · · × C m where each C i is a category satisfying the following conditions: • Ob( C i ) = N . • For each r, n ∈ Ob( C i ), the morphism set C i ( r, n ) is nonempty if and only if r n . • If r < m < n , every morphism f : r → n in C i is a composition f = hg for some morphisms g : r → m and h : m → n . • For each n ∈ Ob( C i ), every endomorphism of n in C i is an isomorphism. • If r < n , the group C i ( n, n ) acts transitively on the set C i ( r, n ).In the terminology of [6], C i is an EI category of type A ∞ satisfying the transitivity condition. Example 2.1.
The skeleton of FI whose objects are [ n ] for all n ∈ N satisfies the above conditions.The objects of the category C = C × · · · × C m are m -tuples n = ( n , . . . , n m ) where each n i ∈ N , andthe morphisms are m -tuples f = ( f , . . . , f m ) : r → n where each f i ∈ C i ( r i , n i ). There is a partial orderingon Ob( C ) defined as follows: r n if and only if r i n i for each i ∈ [ m ], or equivalently, the morphismset C ( r , n ) is nonempty. We note that C is a graded category . Explicitly, for an object n and a morphism f : r → n , one defines the rank | n | = P i ∈ [ m ] n i and the degree | f | = | n | − | r | . It is clear that every morphismof degree s in C can be written as a composition of s morphisms of degree 1.2.3. C -modules. By definition, a C -module (or a representation of C ) is a covariant functor from C to k -Mod,the category of all k -modules, and a homomorphism between two C -modules is a natural transformation.Since k -Mod is an abelian category, so is C -Mod, the category of all C -modules. Furthermore, the abeliancategory C -Mod has enough projective objects (see, for example, [18, Exercise 2.3.8]). In particular, the k -linearization of representable functors M ( n ) = k C ( n , − ) are projective C -modules. By a free C -module , wemean a C -module isomorphic to one of the form L i ∈ I M ( n i ) where I is any indexing set and n i ∈ Ob( C ).For every C -module V , there is a surjective C -module homomorphism M n ∈ N m M ( n ) ⊕ c n → V → ASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAIN PRODUCT CATEGORIES 3 where the multiplicities c n could be infinity. It follows that every projective C -module is a direct summandof a free C -module. Definition 2.2. A C -module V is generated in finite degrees if there exists a natural number N V , dependenton V , and a surjective homomorphism of the form (2.1) such that the multiplicity c n can be taken to be 0for objects n satisfying | n | > N V . The C -module V is presented in finite degrees if there exists a surjectivehomomorphism of the above form such that both V and the kernel are generated in finite degrees. Remark 2.3.
The above definition can be restated in a more intuitive way. That is, a C -module V isgenerated in finite degrees if there exists a natural number N V such that for every object n with | n | > N V ,the value V n of V on the object n satisfies the following equality: V n = X r (cid:8) nf ∈ C ( r , n ) f · V r , where f · V r means V ( f )( V r ), the image of V r under the k -linear map V ( f ).2.4. The category algebra.
There is another way to study C -modules from the traditional ring theoreticviewpoint. The category algebra k C is set to be the free k -module whose basis is the set of morphisms in C . Multiplication is defined by the following rule: given two morphisms f : r → n and g : q → t , let f · g be the composition (from right to left) f ◦ g if t = r , and 0 otherwise. Then k C is a non-unital associative k -algebra. Furthermore, the graded structure of C induces a graded structure on k C , and it is generated indegrees 0 and 1. Explicitly, one has the following decomposition: k C = M s ∈ N (cid:16) M n , r ∈ N m | n | − | r | = s k C ( r , n ) (cid:17) . In particular, the direct sum m over all s > k C ; it is precisely the free k -modulespanned by all non-invertible morphisms.Let k C -Mod be the category of all k C -modules. By [14, Theorem 7.1], the category C -Mod may beidentified with a full subcategory of k C -Mod. In particular, the free C -modules k C ( n , − ) are identified withprojective k C -modules of the form k C e n , where e n is the identity morphism on n and is also viewed as anidempotent in the algebra k C .2.5. Homology groups of C -modules. Let V be a C -module. The zeroth homology group H ( V ) of V isdefined as follows. For each n ∈ Ob( C ), let( H ( V )) n = V n (cid:30) X r (cid:8) n k C ( r , n ) · V r . Then H ( V ) is a C -module on which every non-invertible morphism in C acts as the zero map.The functor H : V H ( V ) is right exact, so we make the following definition: Definition 2.4.
For each s ∈ N , we define H s : C -Mod → C -Mod to be the s -th left derived functor of H .For any C -module V , the s -th homological degree hd s ( V ) is defined byhd s ( V ) = sup {| n | | ( H s ( V )) n = 0 } or − V is:reg( V ) = sup { hd s ( V ) − s | s ∈ N } . Suppose V is a C -module. We call hd ( V ) and max { hd ( V ) , hd ( V ) } the generation degree and presen-tation degree of V , and denote them by gd( V ) and prd( V ) respectively. It is easy to see that there exists asurjective morphism P → V such that P is a free C -module with gd( P ) = gd( V ). Remark 2.5.
In [8, 13], the authors provided another abstract version for the above definitions. Recallthat the graded category algebra k C has a two-sided ideal m spanned by all non-invertible morphisms in C .Then one can check that H ( V ) ∼ = ( k C / m ) ⊗ k C V , and hence H s ( V ) ∼ = Tor k C s ( k C / m , V ). Moreover, using thelanguage of homological degrees, Definition 2.2 can be restated as follows: a k C -module V is generated infinite degrees if gd( V ) is finite, and is presented in finite degrees if prd( V ) is finite. WEE LIANG GAN AND LIPING LI
Torsion theory of C -modules. Let V be a C -module. An element v ∈ V r for some r ∈ Ob( C ) is torsion if for some n ∈ Ob( C ) and f ∈ C ( r , n ), one has f · v = 0. Note that the group C ( n , n ) acts transitivelyon C ( r , n ), so if f · v = 0 for some f ∈ C ( r , n ), then all morphisms in C ( r , n ) send v to 0. We say that V is a torsion module if for every n ∈ Ob( C ), all elements v ∈ V n are torsion. For an arbitrary C -module V , thereis a canonical short exact sequence 0 → V T → V → V F → V T is torsion, and V F is torsion-free ;that is, V F contains no nonzero torsion elements.We now define a key invariant t ( V ) recording certain information of the torsion part of V . Definition 2.6.
Suppose V is a C -module. For each i ∈ [ m ], define t i ( V ) = sup { s | ∃ n and 0 = v ∈ V n such that k C ( n , n + i ) · v = 0 and n i = s } , where i is the m -tuple with 1 at the i -th coordinate and zeroes at other coordinates; if the above setis empty, we set t i ( V ) = − t ( V ) of V to be the m -tuple( t ( V ) , . . . , t m ( V )), and define t ( V ) = t ( V ) + · · · + t m ( V ) . It is easy to see that t ( V ) has a lower bound − m , and t ( V ) = − m if and only if V is torsion-free. Ingeneral, t ( V ) might not be a finite number. Remark 2.7.
Loosely speaking, we can view objects in C as “integral points” in the free module k m . For i ∈ [ m ], if t i ( V ) > t i ( V ) is infinite), then there exist a “hyperplane” perpendicularto the i -th coordinate axis, an object n lying in this hyperplane, and a nonzero element v ∈ V n such that v becomes 0 when it moves one step along the i -th direction. In this situation, t i ( V ) is precisely the supremumof the “heights” of these hyperplanes. (The torsion vector of an FI m -module V was introduced in [13,Definition 2.8] with a different version of definition. We would like to point out that t ( V ) is distinct fromthe torsion degree in [13, Definition 2.8], which is not the sum but the supremum of these t i ( V )’s.)3. Shift functors
Assumptions and definitions.
From now on, we make the following assumptions on the category C i for every i ∈ [ m ]:(i) We assume that there is a faithful functor ψ i : C i → C i such that ψ i ( n ) = n + 1 for every n ∈ Ob( C i ). Define the shift functor Ψ i on C i -Mod to be the pullbackfunctor ψ ∗ i , that is, Ψ i : C i -Mod → C i -Mod , V V ◦ ψ i . Note that for every C i -module V and n ∈ Ob( C i ), one has (Ψ i V ) n = V n +1 .(ii) We assume that there is a natural transformation ε i : Id C i → ψ i , where Id C i denotes the identity functor on C i . Note that for each n ∈ Ob( C i ), one has the morphism( ε i ) n ∈ C i ( n, n + 1). The natural transformation ε i induces a natural transformation µ i : Id C i -Mod → Ψ i , where Id C i -Mod denotes the identity functor on C i -Mod. Explicitly, for each C i -module V , the homo-morphism ( µ i ) V : V → Ψ i V is defined at each n ∈ Ob( C i ) by V n → V n +1 , v ( ε i ) n · v .(iii) For each n ∈ Ob( C i ), denote the free C i -module k C i ( n, − ) by M i ( n ). We assume that the homomor-phism ( µ i ) M i ( n ) : M i ( n ) → Ψ i M i ( n )is injective and its cokernel is a projective C i -module with generation degree n − Remark 3.1.
In the terminology of [7, Definition 1.2], the shift functor Ψ i is, in particular, a generic shiftfunctor.For any C i -module V , we denote by ∆ i V the cokernel of ( µ i ) V : V → Ψ i V . Thus, for each n ∈ N , ∆ i M i ( n )is a projective C i -module and gd( M i ( n )) n − ASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAIN PRODUCT CATEGORIES 5
Lemma 3.2.
Let i ∈ [ m ] . For every n ∈ Ob( C i ) , the C i -module Ψ i M i ( n ) is projective and has generationdegree n .Proof. There is a short exact sequence0 → M i ( n ) → Ψ i M i ( n ) → ∆ i M i ( n ) → . Since M i ( n ) and ∆ i M i ( n ) are projective C i -modules, it follows that Ψ i M i ( n ) is also projective. Sincegd( M i ( n )) = n and gd(∆ i M i ( n )) n −
1, we have gd(Ψ i M i ( n )) n . (cid:3) Remark 3.3. If C i is the skeleton of FI whose objects are [ n ] for all n ∈ N , then C i satisfies all the aboveassumptions (see [3, Proposition 2.12]).3.2. Shift functors for C . Recall that C = C × · · · × C m . Let i ∈ [ m ]. We define the functor ρ i : C → C by ρ i = (Id C , . . . , ψ i , . . . , Id C m )where ψ i is in the i -th coordinate. Note that ρ i ( n ) = n + i for each n ∈ Ob( C ). We define the i -th shiftfunctor Σ i on C -Mod to be the pullback functor ρ ∗ i , that is,Σ i : C -Mod → C -Mod , V V ◦ ρ i . Similarly, the natural transformation ε i : Id C i → ψ i induces a natural transformation Id C → ρ i , whichin turn induces a natural transformation Id C -Mod → Σ i . Thus, for each C -module V , we have a naturalhomomorphism V → Σ i V . Explicitly, for each n ∈ Ob( C ), the natural map V n → (Σ i V ) n is defined by v h · v where h ∈ C ( n , n + i ) with h i = ( ε i ) n i ∈ C i ( n i , n i + 1) and h j = Id n j ∈ C j ( n j , n j ) if j = i .Let K i V and D i V be, respectively, the kernel and cokernel of the natural homomorphism V → Σ i V , sothat we have a natural exact sequence0 → K i V → V → Σ i V → D i V → . Suppose n ∈ Ob( C ). Since the group C ( n + i , n + i ) acts transitively on C ( n , n + i ), it follows that( K i V ) n = { v ∈ V n | f · v = 0 for some f ∈ C ( n , n + i ) } = { v ∈ V n | f · v = 0 for every f ∈ C ( n , n + i ) } . For each i ∈ [ m ] and C -module V , we denote by V i the i -th copy of V in the direct sum V ⊕ m . For anysubset S ⊂ [ m ], let V S = M i ∈ S V i , Σ S V = M i ∈ S Σ i V, K S V = M i ∈ S K i V, D S V = M i ∈ S D i V. In particular, if S is the empty set, then these are the zero module. We have a natural exact sequence0 → K S V → V S → Σ S V → D S V → . It is plain that the functor Σ S is exact, and the functor D S is right exact. Lemma 3.4.
Suppose V is a C -module and S is a subset of [ m ] .(1) For every n ∈ Ob( C ) , the C -modules Σ S M ( n ) and D S M ( n ) are projective. Moreover, one has: gd(Σ S M ( n )) | n | , and gd( D S M ( n )) | n | − .(2) If V is nonzero, then one has: gd( D [ m ] V ) gd(Σ [ m ] V ) gd( V ) = gd( D [ m ] V ) + 1 . (3) If V is nonzero, then one has: prd(Σ S V ) prd( V ) , prd( D S V ) prd( V ) − . In particular, if V is presented in finite degrees, then Σ S V and D S V are also presented in finitedegrees.(4) The C -module K [ m ] V is zero if and only if V is torsion-free.(5) For every i ∈ [ m ] , one has: t i ( V ) gd( K i V ) .(6) For every i, j ∈ [ m ] , one has t i (Σ j V ) t i ( V ) . Moreover, t i (Σ i V ) < t i ( V ) whenever t i ( V ) ∈ N . WEE LIANG GAN AND LIPING LI
Proof. (1) For any i ∈ [ m ], one has:Σ i M ( n ) = M ( n ) ⊠ · · · ⊠ Ψ i M i ( n i ) ⊠ · · · ⊠ M m ( n m ) ,D i M ( n ) = M ( n ) ⊠ · · · ⊠ ∆ i M i ( n i ) ⊠ · · · ⊠ M m ( n m ) . By Lemma 3.2, Ψ i M i ( n i ) is a projective C i -module, so it is a direct summand of a free C i -module. It followsthat Σ i M ( n ) is a direct summand of a free C -module. Also, by Lemma 3.2, one has gd(Ψ i M i ( n i )) n i , sogd(Σ i M ( n )) | n | . Similarly for D i M ( n ).(2) Since D [ m ] V is a quotient of Σ [ m ] V , we have gd( D [ m ] V ) gd(Σ [ m ] V ).Let P → V be a surjective morphism where P is a nonzero free C -module with gd( P ) = gd( V ). SinceΣ [ m ] is exact and D [ m ] is right exact, we have surjective morphisms Σ [ m ] P → Σ [ m ] V and D [ m ] P → D [ m ] V .It follows, using (1), that gd(Σ [ m ] V ) gd(Σ [ m ] P ) gd( P ) = gd( V ) , gd( D [ m ] V ) gd( D [ m ] P ) gd( P ) − V ) − . It remains to prove that gd( D [ m ] V ) > gd( V ) −
1. Suppose there exists n ∈ Ob( C ) such that | n | > H ( V )) n = 0. Let V ′ be the C -submodule of V generated by V t for all t ∈ Ob( C ) such that | t | < | n | .Then we have ( V /V ′ ) t = 0 if | t | < | n | . Since | n | >
1, there exists i ∈ [ m ] such that n i >
1; set r = n − i ∈ Ob( C ). Since ( V /V ′ ) r = 0 but (Σ i ( V /V ′ )) r = ( V /V ′ ) n = 0, we have ( D i ( V /V ′ )) r = 0, and so( D [ m ] ( V /V ′ )) r = 0, hence D [ m ] ( V /V ′ ) is a nonzero C -module. But for every t ∈ Ob( C ) such that | t | < | r | ,we have (Σ [ m ] ( V /V ′ )) t = 0, so ( D [ m ] ( V /V ′ )) t = 0. Hence,gd( D [ m ] V ) > gd( D [ m ] ( V /V ′ )) > | r | = | n | − . It follows that gd( D [ m ] V ) > gd( V ) − → W → P → V → P is a free C -module withgd( P ) = gd( V ). We have: gd( W ) max { gd( P ) , hd ( V ) } = prd( V ) . Applying the exact functor Σ S , we obtain the short exact sequence 0 → Σ S W → Σ S P → Σ S V →
0. By (1),Σ S P is a projective C -module with gd(Σ S P ) gd( P ) = gd( V ). Using (2), we have:gd(Σ S V ) gd( V ) , hd (Σ S V ) gd(Σ S W ) gd( W ) prd( V ) . Therefore, prd(Σ S V ) prd( V ).Similarly, applying the right exact functor D S , we get an exact sequence D S W → D S P → D S V → D S V ) gd( V ) − , gd( D S W ) max {− , gd( W ) − } prd( V ) − . Noting that the kernel of D S P → D S V → D S W , we conclude thathd ( D S V ) gd( D S W ) prd( V ) − . Consequently, prd( D S V ) prd( V ) − K [ m ] V is nonzero, then K i V is nonzero for some i ∈ [ m ], in which case ( K i V ) n = 0 for some n ∈ Ob( C ), and so V n contains a nonzero torsion element.Conversely, suppose V is not torsion-free. Then there exists a morphism f ∈ C ( n , r ) of positive degreesuch that f · v = 0 for some nonzero v ∈ V n . Since every morphism is a composition of morphisms of degree1, there exists such a morphism f of degree 1, say f ∈ C ( n , n + i ). It follows that ( K i V ) n = 0, so K i V isnonzero, and hence K [ m ] V is nonzero.(5) We only need to consider the case that gd( K i V ) is finite. An element v ∈ V n is contained in K i V if and only if it vanishes when moving one step along the i -th direction. Thus K i V is either 0 or can bedecomposed into a direct sum of direct summands, each of which is supported on a hyperplane perpendicularto the i -th axis. If t i ( V ) > gd( K i V ), then t i ( V ) >
0. By Definition 2.6, there exist an object n and a nonzeroelement v ∈ V n such that n i > gd( K i V ) and v vanishes when it moves one step along the i -th direction.In this situation, v ∈ ( K i V ) n , so K i V has a direct summand supported on the hyperplane H consisting ofobjects r with r i = n i . Clearly, the zeroth homology group of this summand is also supported on H , so thegeneration degree of this summand must be at least n i . This forces gd( K i V ) > n i , which is a contradiction.(6) To prove the first inequality, we assume that t i ( V ) is finite. If the inequality does not hold, then t i (Σ j V ) >
0, so there exist an object n and a nonzero element v ∈ (Σ j V ) n such that n i > t i ( V ), and v vanishes when it moves along the i -th direction. That is, one has C ( n , n + i ) · v = 0 where v is viewed as an ASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAIN PRODUCT CATEGORIES 7 element in Σ j V . But (Σ j V ) n = V n + j . By the definition of shift functors, one has ρ j ( C ( n , n + i )) · v = 0where v is regarded as an element in V . Note that ρ j ( C ( n , n + i )) ⊆ C ( n + j , n + j + i ) , so v ∈ V n + j also vanishes when it moves one step along the i -th direction. Consequently, t i ( V ) > ( n + j ) i > n i , which is a contradiction.Now we turn to the second inequality. Since t i ( V ) >
0, we know that K i V is nonzero. The secondinequality can be proved similarly by noting that if i = j and n i > t i ( V ), then one can deduce that t i ( V ) > ( n + i ) i = n i + 1, which is also a contradiction. (cid:3) Remark 3.5.
The above results have been verified when C is a skeleton of the category FI in [3, 9, 12]. Inparticular, for FI-modules V , one has t ( V ) = gd( KV ). This fact plays a crucial role in [9], where the secondauthor provided an alternative proof for the upper bound of Castelnuovo-Mumford regularity of FI-modulesappearing in [1, Theorem A]. However, this equality in general does not hold when C is a skeleton of FI m for m >
1, and the reader can easily find a counterexample.Recall that for a C -module V , we let t ( V ) be the sum of all t i ( V ) for i ∈ [ m ], and t ( V ) > − m . The aboveresults have two useful corollaries. Corollary 3.6.
Let V be a C -module and i ∈ [ m ] . If t i ( V ) > , then t ( V /K i V ) t ( V ) − .Proof. Without loss of generality we can assume that t ( V ) is finite, so t j ( V ) is also finite for every j ∈ [ m ].Since t i ( V ) >
0, we know that K i ( V ) is nonzero. Consider the short exact sequence0 → V /K i V → Σ i V → D i V → → K i V → V → Σ i V → D i V → . It is not hard to see that t j ( V /K i V ) t j (Σ i V ) for all j ∈ [ m ] by the definition of torsion vectors. But bythe previous lemma, t j (Σ i V ) t j ( V ) for j = i , and t i (Σ i V ) t i ( V ) −
1. The conclusion follows. (cid:3)
Corollary 3.7.
Let V be a C -module presented in finite degrees and i ∈ [ m ] , and suppose that hd ( D i V ) is finite. Then K i V is generated in finite degrees, V /K i V is presented in finite degrees, and t i ( V ) is finite.More precisely, gd( K i V ) , prd( V /K i V ) and t i ( V ) are all less than or equal to max { prd( V ) , hd ( D i V ) } .Proof. We may assume V is nonzero, for otherwise the statements are trivial. Since V is presented in finitedegrees, so are Σ i V and D i V by part (3) of Lemma 3.4, where we let S = { i } . Now break the exact sequence0 → K i V → V → Σ i V → D i V → → K i V → V → V /K i V → → V /K i V → Σ i V → D i V → . We have gd(
V /K i V ) gd( V ). From the long exact sequence of homology groups associated to the secondshort exact sequence, we deduce that:hd ( V /K i V ) max { hd (Σ i V ) , hd ( D i V ) } max { prd( V ) , hd ( D i V ) } , where we used Lemma 3.4 (3). Hence, V /K i V is presented in finite degrees.Similarly, from the long exact sequence associated to the first short exact sequence, we deduce that:gd( K i V ) max { gd( V ) , hd ( V /K i V ) } max { prd( V ) , hd ( D i V ) } . Hence, K i V is generated in finite degrees. By Lemma 3.4 (5), it follows that t i ( V ) max { prd( V ) , hd ( D i V ) } . (cid:3) WEE LIANG GAN AND LIPING LI A proof of the main theorem
Let V be a C -module. In this section we follow the algorithm described in [8, Section 3] to construct atree of quotient modules of V , and use it to prove the main theorem.An element i ∈ [ m ] is said to be a singular index of V if K i V = 0, and otherwise it is called a regularindex of V . Let S ( V ) and R ( V ) be, respectively, the sets of singular indices and regular indices of V . Forevery element i ∈ S ( V ), K i V is nonzero, so V /K i V is a proper quotient module of V . Definition 4.1.
Let V be any C -module. We call the collection of quotient modules V /K i V for all i ∈ S ( V )the children of V .We now construct a tree as follows. First, we put V on the zeroth level of the tree. Next, we put thechildren of V on level -1 of the tree, and draw an arrow from V to each of its children. We continue thisprocess for each child of V and keep going to get a tree T ( V ) of quotient modules of V . Each vertex in thistree (except V itself) is called a descendant of V . Clearly, for the s -th level on T ( V ), there are at most m − s vertices (note that s is a non-positive integer). But for an arbitrary C -module, T ( V ) might not be a finitetree since it may have infinitely many levels. Remark 4.2.
Subsets of R ( V ) are called nil subsets in [8], and R ( V ) is called the maximal nil subset.For any subset S ⊂ [ m ], define F S V = M i ∈ S D i V ! ⊕ M i ∈ [ m ] \ S Σ i V , that is, F S V = D S V ⊕ Σ [ m ] \ S V . In particular, F ∅ V = Σ [ m ] V and F [ m ] V = D [ m ] V . For each i ∈ [ m ], thereis an exact sequence 0 → V /K i V → Σ i V → D i V → . It follows that if S ⊂ T ⊂ [ m ], then there is a short exact sequence0 → M i ∈ T \ S V /K i V → F S V → F T V → . (4.1) Lemma 4.3.
Let V be a C -module. Then we have a short exact sequence → M i ∈S ( V ) V /K i V → F R ( V ) V → D [ m ] V → . That is, the kernel of the surjective map F R ( V ) V → D [ m ] V is the direct sum of the children of V .Proof. Immediate from (4.1) by taking S = R ( V ) and T = [ m ]. (cid:3) Remark 4.4.
We would like to point out the difference of notations between this paper and [8] to avoidpossible confusion. Specifically, for S ⊆ [ m ], the module D S V in this paper is the cokernel of the map V S → Σ S V , whereas D S V in [8] is the cokernel of the map V S → Σ [ m ] V . In other words, the module D S V in [8] is same as F S V in this paper; in particular, for S = R ( V ), the module D R ( V ) V defined in [8] isprecisely F R ( V ) V in this paper.The following lemma is from [8, Proposition 4.3] (keeping in mind the differences between our presentnotations and [8].) For the convenience of the reader, we give its proof since the assumptions in [8] aresomewhat different. Lemma 4.5.
Let V be a nonzero C -module. If S ⊂ R ( V ) , then reg( V ) reg( F S V ) + 1 . In particular, reg( V ) reg( F R ( V ) V ) + 1 .Proof. We shall prove by induction on s > V is any nonzero C -module and S ⊂ R ( V ), thenhd s ( V ) reg( F S V ) + s + 1 . This would imply that reg( V ) reg( F S V ) + 1.Let us first consider the case s = 0. We have hd ( V ) = gd( V ) = gd( D [ m ] V ) + 1, where we used Lemma3.4 (2). Since D [ m ] V is a quotient of F S V , we have gd( D [ m ] V ) gd( F S V ). Hence,hd ( V ) gd( F S V ) + 1 reg( F S V ) + 1 . ASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAIN PRODUCT CATEGORIES 9
Next, suppose s >
1. Take a short exact sequence 0 → W → P → V → P is a free C -modulewith gd( P ) = gd( V ). If W is zero, then V is a free module and we are done, so suppose W is nonzero. Since P is free, we have R ( P ) = [ m ], so R ( W ) = [ m ], and so S ⊂ R ( W ). Therefore,hd s ( V ) hd s − ( W ) reg( F S W ) + s, where the last inequality holds by induction hypothesis. It suffices to show that reg( F S W ) reg( F S V ) + 1.Since Σ S is an exact functor, we have the short exact sequence 0 → Σ S W → Σ S P → Σ S V → / / W S / / (cid:15) (cid:15) P S / / (cid:15) (cid:15) V S / / (cid:15) (cid:15) / / Σ [ m ] W / / Σ [ m ] P / / Σ [ m ] V / / . Hence we have an exact sequence 0 → F S W → F S P → F S V →
0. By Lemma 3.4 (1), F S P is projective andgd( F S P ) gd( P ) = gd( V ) = gd( D [ m ] V ) + 1 gd( F S V ) + 1 , so gd( F S W ) max { hd ( F S V ) , gd( F S P ) } max { hd ( F S V ) , hd ( F S V ) + 1 } . For each r > r ( F S W ) = hd r +1 ( F S V ). Hence, reg( F S W ) reg( F S V ) + 1. (cid:3) Now we are ready to prove the following main theorem.
Theorem 4.6.
Let V be a C -module. Then reg( V ) is finite if and only if V is presented in finite degrees.Proof. One direction is immediate since by the definition of regularity, if reg( V ) is finite, so are all homologicaldegrees of V . The proof of the other direction is based on an induction of gd( V ).Suppose that V is presented in finite degrees. If gd( V ) = −
1, that is, V = 0, the conclusion holds trivially.Assume that V is nonzero. Step 1:
We show that every vertex in the tree T ( V ) is presented in finite degrees. It suffices to showthat if a nonzero vertex W is presented in finite degrees, so are its children. Suppose i ∈ [ m ]. By statements(2) and (3) of Lemma 3.4, D i W is presented in finite degrees, andgd( D i W ) gd( D [ m ] W ) = gd( W ) − gd( V ) − W is a quotient module of V . Therefore, by the induction hypothesis, reg( D i W ) is finite, so in particularhd ( D i W ) is finite; one has: hd ( D i W ) reg( D i W ) + 2 . Consequently, all children of W are presented in finite degrees by Corollary 3.7. Step 2:
We show that T ( V ) is a finite tree. Let W be a nonzero vertex in T ( V ). We already proved that W is presented in finite degrees in the previous step, and reg( D [ m ] W ) is finite by the induction hypothesis.By Corollary 3.7, t ( W ) is a finite number; by Corollary 3.6, t ( W ′ ) < t ( W ) for any child W ′ of W . Since t ( U ) > − m for every vertex U in T ( V ), it can has only finitely many levels. As each level has only finitelymany vertices, the claim follows. Step 3:
We show that every vertex in the tree T ( V ) has finite regularity. Take an arbitrary vertex W on the lowest level of T ( V ). Of course we can assume that W is nonzero. Since W has no children, we have K [ m ] W = 0, so we have a short exact sequence0 → W ⊕ m → Σ [ m ] W → D [ m ] W → . The C -module D [ m ] W is presented in finite degrees, and has finite regularity by the induction hypothesis.Moreover, R ( W ) = [ m ] and F R ( W ) W = D [ m ] W . Thus by Lemma 4.5, reg( W ) reg( D [ m ] W ) + 1 is finite.Consequently, every vertex on the lowest level has finite regularity.Now we consider an arbitrary vertex U on the second lowest level. Note that all children of U have finiteregularity as they appear in the lowest level, and D [ m ] U has finite regularity by the induction hypothesis.Combining the conclusions of Lemma 4.3 and Lemma 4.5, conclude that U has finite regularity, too.Since the tree T ( V ) has only finitely many levels, recursively one can show that all vertices in it, including V , have finite regularity. (cid:3) Applying Theorem 4.6 to a skeleton of FI m , we immediate deduce the finiteness of regularity of FI m -modules presented in finite degrees, and hence partially answered a question proposed in [13, Subsection6.1]. Remark 4.7.
As mentioned before, the above proof relies on an inductive machinery originated in [7] andfurther developed in [8]. In the second paper, for a few other categories such as FI d and OI d and Noetheriancommutative coefficient rings, regularity of finitely generated representations is shown to be finite. But atthis moment we cannot establish a version of Theorem 4.6 for these categories because numerical invariantsfor their representations, sharing similar properties as t ( V ), are not available yet. Besides, for an FI m -module V presented in finite degrees, an explicit upper bound of reg( V ) is still missing for m >
1. In the special case m = 1, it is not hard to find an upper bound for reg( V T ), and use it to deduce an upper bound for reg( V ),as was done in [9]. However, for m >
1, we are not able to obtain a simple upper bound for the regularityof V T . 5. A few consequences
Category of modules presented in finite degrees.
A consequence Theorem 4.6 is:
Corollary 5.1.
The category of C -modules presented in finite degrees is abelian.Proof. Let α : U → V be a morphism such that both U and V are presented in finite degrees. We have toshow that the kernel, cokernel, and image of α are all presented in finite degrees.Consider the short exact sequence 0 → im α → V → coker α → . As a quotient module U , im α is generated in finite degrees. Applying the homology functor we deduce thatcoker α is presented in finite degrees. Therefore, both coker α and V have finite regularity by the previoustheorem, and hence their homological degrees are finite. Consequently, all homological degrees of im α arefinite as well.Now turn to the short exact sequence0 → ker α → U → im α → . By a similar argument, we deduce that all homological degrees of ker α are finite. (cid:3) Remark 5.2.
Let V be a C -module. Consider the following conditions:(1) V is presented in finite degrees;(2) all homological degrees of V are finite;(3) reg( V ) is finite;(4) the category of C -modules presented in finite degrees is abelian.Clearly, (3) implies (2), and (2) implies (1). Moreover, (1) and (2) are equivalent if and only if (4) holds.Indeed, if (1) implies (2), then the proof of the above corollary tells us that (4) holds. Conversely, supposethat (4) holds and V is presented in finite degrees. Let 0 → W → P → V → P is a free module generated in finite degrees (and so presented in finite degrees). Then W is presentedin finite degrees as well. In particular, hd ( V ) = hd ( W ) is finite. Replacing V by W and repeating theabove argument, one can eventually show that all homological degrees of V are finite.5.2. FI m -modules. By Corollary 5.1, the category of FI m -modules presented in finite degrees is an abeliancategory. Using this fact, many previously know results (for example in [13]) about finitely generatedFI m -modules over a commutative Noetherian coefficient ring, whose proofs only rely on the condition thatboth gd( V ) and prd( V ) are finite, can be extended to FI m -modules presented in finite degrees over anycommutative coefficient ring. For example, relative projective FI m -modules (also called semi-induced modulesor ♯ -filtered modules in literature) are defined in [13, Subsection 1.4] over any commutative coefficient ring.By [13, Theorem 1.3], these modules are presented in finite degrees. Accordingly, we can extend [13, Theorem1.5] to the setup of FI m -modules presented in finite degrees over any commutative ring. Theorem 5.3.
Let V be an FI m -module presented in finite degrees over a commutative ring k . Then thereexists a complex F • : 0 → V → F → F → . . . → F ℓ → such that the following statements hold:(1) each F j is a relative projective module with gd( F j ) gd( V ) − j ;(2) ℓ gd( V ) ,(3) all homology groups H j ( F • ) of this complex are torsion modules presented in finite degrees. ASTELNUOVO-MUMFORD REGULARITY OF REPRESENTATIONS OF CERTAIN PRODUCT CATEGORIES 11
Consequently, Σ n . . . Σ n m m V is a relative projective module if n i > t i ( H j ( F • )) + 1 for all j ℓ and i ∈ [ m ] .Proof. The proof of [13, Theorem 1.5] as well as its prerequisite results, including [13, Porposition 4.10,Lemma 4.8], only relies on the condition that both gd( V ) and t i ( V ) for all i ∈ [ m ] are finite. This conditionstill holds for FI m -modules presented in finite degrees over any commutative ring. (cid:3) Remark 5.4.
When m = 1, in [10, 16], Ramos and the second author proved that the cohomology groupsin the above finite complex are precisely the local cohomology groups of FI-modules. Furthermore, Nagpal,Sam, and Snowden showed that the regularity of an FI-module V can be described in terms of degrees ofthese cohomology groups; see [10, Conjecture 5.19] and [15, Theorem 1.1]. For m >
1, we expect theseresults still hold, though we could not establish them.
Remark 5.5.
A generalization of the category FI is the category FI G , where G is a (possibly infinite)group. This category encodes the wreath products of G and all symmetric groups, and share very similarrepresentation theoretic properties as FI; see [10]. The above results for FI m -modules can be extended toFI mG -modules using similar proofs. References [1] T. Church and J. Ellenberg, Homology of FI-modules, Geom. Topol. 21 (2017), 2373-2418, arXiv:1506.01022.[2] T. Church, J. Ellenberg, B. Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164(2015), 1833-1910, arXiv:1204.4533.[3] T. Church, J. Ellenberg, B. Farb, R. Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014), 2951-2984,arXiv:1210.1854.[4] N. Gadish, Representation stability for families of linear subspace arrangements, Adv. Math. 322 (2017), 341-377,arXiv:1603.08547.[5] N. Gadish, Categories of FI type: a unified approach to generalizing representation stability and character polynomials, J.Algebra 480 (2017), 450-486, arXiv:1608.02664.[6] W.L. Gan, L. Li, Noetherian property of infinite EI categories. New York J. Math. 21 (2015), 369-382, arXiv:1407.8235.[7] W. L. Gan, L. Li, An inductive machinery for representations of categories with shift functors, Trans. Amer. Math. Soc.371 (2019), no. 12, 8513-8534, arXiv:1610.09081.[8] W. L. Gan, L. Li, Asymptotic behavior of representations of graded categories with inductive functors, J. Pure Appl.Algebra 223 (2019), 188-217, arXiv:1705.00882.[9] L. Li, Upper bounds of homological invariants of FI G -modules, Arch. Math. (Basel) 107 (2016), 201-211, arXiv:1703.06832.[10] L. Li, E. Ramos, Depth and the local cohomology of FI G -modules, Adv. Math. 329 (2018), 704-741, arXiv:1602.04405.[11] L. Li, E. Ramos, Local cohomology and the multi-graded regularity of F I m -modules, to appear in J. Comm. Algebra,arXiv:1711.07964.[12] L. Li, N. Yu, Filtrations and homological degrees of FI-modules, J. Algebra 472 (2017) 369-398, arXiv:1511.02977.[13] L. Li, N. Yu, FI m -modules over Noetherian rings, J. Pure Appl. Algebra 223 (2019), no. 8, 3436-3460, arXiv:1705.00876.[14] B. Mitchell, Rings with several objects, Adv. Math. 8 (1972), 1-161.[15] R. Nagpal, S. Sam, and A. Snowden, Regularity of FI-modules and local cohomology, Proc. Amer. Math. Soc. 146 (2018),4117-4216.[16] E. Ramos, On the degreewise coherence of F I G -modules, New York J. Math. 23 (2017), 873-896, arXiv:1606.04514.[17] S. Sam, A. Snowden, Gr¨obner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017),159-203, arXiv:1409.1670.[18] C. Weibel, An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge UniversityPress, Cambridge, 1994. Department of Mathematics, University of California, Riverside, CA 92521, USA
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