Filtrations and homological degrees of FI-modules

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Filtrations and homological degrees of FI-modules

Liping Li^a,Nina Yu^b,∗

aKeyLaboratoryofPerformanceComputingandStochasticInformation Processing(MinistryofEducation),CollegeofMathematicsandComputer Science,HunanNormalUniversity,Changsha,Hunan410081,China

bSchoolofMathematicalSciences,XiamenUniversity,Xiamen,Fujian,361005, China

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received9February2016 Availableonline2December2016 CommunicatedbyChangchangXi

Keywords:

FI-modules Filtrations

Homologicaldegrees Projectivedimensions

Letk be a commutative Noetherianring. Inthis paper we consider -filtered modules of the category FI firstly intro- ducedin[12].Weshow that a finitelygeneratedFI-module V is-filteredifandonlyifitshigherhomologiesallvanish, andif andonly if a certain homology vanishes. Using this homologicalcharacterization, wecharacterize finitely gener- atedC-modulesV whoseprojectivedimensionpd(V) isfinite, anddescribeanupperboundforpd(V).Furthermore,wegive anew proofforthefact thatV inducesa finitecomplexof -filteredmodules,anduseitaswellasaresultofChurchand Ellenbergin[1]toobtainanotherupperboundforhomologi- caldegreesofV.

©2016ElsevierInc.Allrightsreserved.

✩ ThefirstauthorwassupportedbyChinaNSF11541002,theConstructProgramoftheKeyDiscipline inHunanProvince,andtheStart-UpFundsofHunanNormalUniversity830122-0037.Thesecondauthor wassupportedbyChinaNSF11601452.Bothauthorshighlyappreciatetheanonymousrefereeforcarefully checkingthemanuscriptandprovidingmanyvaluablecommentsandsuggestions.

* Correspondingauthor.

E-mailaddresses:[email protected](L. Li),[email protected](N. Yu).

http://dx.doi.org/10.1016/j.jalgebra.2016.11.019 0021-8693/©2016ElsevierInc.Allrightsreserved.

1. Introduction

1.1. Motivation

Thecategory FI,whose objectsarefinitesets andmorphismsareinjections between them,hasplayedacentralroleinrepresentationstabilitytheory introducedbyChurch and Farb in [3]. It has many interesting properties, which were used to prove quite a few stability phenomena observed in[1–3,5,14]. Among these properties, the existence ofashiftfunctor Σ isextremelyuseful.Forinstances,itwasappliedtoshowthelocally Noetherian property of FI over any Noetherian ring by Church, Ellenberg, Farb, and Nagpalin[4],andtheKoszulityofFIoverafieldofcharacteristic0byGanandthefirst author in[7].Recently,Nagpalprovedthatforanarbitraryfinitelygeneratedrepresen- tation V of FI,when N is largeenough,Σ_NV hasaspecialfiltration,where Σ_N is the N-thiterationofΣ;see[12,Theorem A].ChurchandEllenbergshowedthatFI-modules haveCastlenuovo–Mumfordregularity(foradefinitionincommutativealgebra,see[6]), and gaveanupperbound fortheregularity;see[1,Theorem A].

The main goal of this paper is to use the shift functor to investigate homological degreesandspecialfiltrationsofFI-modules.Specifically,wewantto:

(1) obtainahomologicalcharacterizationof-filteredmodulesfirstlyintroducedbyNag- palin[12];and

(2) usethese -filteredmodules,whichplayalmostthesameroleasprojective modules forhomologicalcalculations,to obtainupperbounds forprojectivedimensions and homologicaldegreesoffinitely generatedFI-modules.

In contrast to the combinatorial approach described in [1], our methods to realize these objectives aremostlyconceptualandhomological.Wedonotrelyonany specific combinatorial structure of the category FI. The main technical tools we use are the shiftfunctorΣ anditsinducedcokernelfunctorD introducedin[4,Subsection2.3]and [1,Section3].Therefore,itis hopefulthatourapproach,withadaptablemodifications, can be applied to other combinatorial categories recently appearing in representation stability theory[15].

1.2. Notation

Beforedescribingthemainresults,wefirstintroducenecessarynotation.Throughout this paper we let C be askeletal category of FI, whose objects are [n] ={1,2,. . . ,n} for n ∈ Z+, the set of nonnegative integers. By convention, [0] = ∅. By k we mean a commutative Noetherian ring with identity. Given a set S, S is the free k-module spanned byelements inS. LetC be thek-linearization ofC, which canbe regardedas both ak-linearcategory andak-algebrawithoutidentity.

Arepresentation ofC, oraC-module,isacovariantfunctor V fromC tok-Mod, the categoryof leftk-modules.Equivalently,aC-moduleis aC-module,whichbydefinition is a k-linear covariant functor from C to k-Mod. It is well known that C-Mod is an abelian category. Moreover, it has enough projectives. In particular, for i ∈ Z+, the k-linearizationC(i,−) oftherepresentablefunctor C(i,−) isprojective.

Arepresentation V of Cis saidto be finitely generated if thereexists afinite subset S ofV such thatany submodulecontaining S coincides withV; or equivalently, there existsasurjectivehomomorphism

i∈Z+

C(i,−)^⊕ai →V

such that

i∈Z+ai < ∞. It is said to be generated in degrees N if in the above surjectiononecanletai= 0 foralli> N.Obviously, V isfinitelygeneratedifandonly ifit is generated indegreesN for acertain N ∈ Z+ and the valuesof V on objects iN arefinitelygeneratedk-modules.SinceCislocallyNoetherianbythefundamental resultin[4],thecategoryC-mod offinitelygeneratedC-modulesisabelian.Inthispaper weonlyconsiderfinitely generatedC-modulesovercommutativeNoetherianrings.

Givenafinitely generatedC-module V andanobjecti∈Z+,wedenote itsvalueon ibyVi.For everyn∈Z+,onecandefine atruncation functor τn :C-mod→C-mod as follows:ForV ∈C-mod,

(τ_nV)_i:=

0, i < n Vi, in

AfinitelygeneratedC-moduleV iscalled torsionifthereexists someN ∈Z+ suchthat τ_NV = 0.Inother words,V_i = 0 foriN.

ThecategoryChasaself-embeddingfunctor ι:C→Cwhichisfaithfulandsendsan objecti∈Z+ toi+ 1.Foramorphismα∈C(i,j) (whichisaninjectionfrom[i] to[j]), ι(α) isaninjection from[i+ 1] to[j+ 1] defined asfollows:

(ι(α))(r) =

1, r= 1∈[i+ 1];

α(r−1) + 1, 1=r∈[i+ 1]. (1.1) Thefunctor ι induces apull-back ι^∗ : C-mod→C-mod.The shift functor Σ isdefined tobe ι^∗◦τ1.Fordetails,see[4,7,11].

Bythedirectedstructure,thek-linearcategoryChasatwo-sidedideal

J =

0i<j

C(i, j).

Therefore,

C0=

i∈Z+

C(i, i)

is aC-moduleviaidentifyingitwithC/J.

Given afinitely generatedC-moduleV,itstorsiondegree isdefinedtobe td(V) = sup{i∈Z+ |Hom_C(C(i, i), V)= 0}

or −∞iftd(V)=∅.InthelattercasewesaythatV istorsionless.Its0-thhomology is definedto be

H₀(V) =V /J V ∼=C0⊗CV.

Since C0⊗C−isrightexact,wedefine thes-th homology Hs(V) = Tor^C_s(C0, V)

for s 1. Note thatthis is a C-module since C0 is a(C,C)-bimodule. Moreover, it is finitely generatedand torsion.Fors∈Z+,thes-thhomological degree issettobe

hds(V) = td(Hs(V)).

Sometimeswecallthe0-thhomologicaldegreegeneratingdegree,anddenoteitbygd(V).

Remark1.1.Theabovedefinitionof torsiondegreesseemsmysterious, soletusgivean equivalent but more concrete definition. Let V be a finitely generated C-module and i ∈ Z+ be an object. If there exists a nonzero v ∈ V_i and α₀ ∈ C(i,i+ 1) such that α₀·v = 0, we claim that for all α ∈ C(i,i+ 1), one has α·v = 0. Indeed, since the symmetricgroupSi+1=C(i+ 1,i+ 1) actstransitivelyonC(i,i+ 1) fromtheleftside, for an arbitrary α ∈ C(i,i+ 1), we canfind an element g ∈ C(i+ 1,i+ 1) (which is unique)suchthatα=gα₀.Therefore,α·v=gα₀·v= 0.

Thisobservationtellsusthatα₀(andhenceallα∈C(i,i+1))sendstheC(i,i)-module C(i,i)·v to 0. Indeed, for every g ∈ C(i,i), since α0g ∈ C(i,i+ 1), one has (α0g)· v = 0 bythe argumentinthepreviousparagraph. ButtheC(i,i)-moduleC(i,i)·v can be regarded as a C-module in a natural way, so Hom_C(C(i,i),V) = 0. Conversely, if Hom_C(C(i,i),V)= 0, onecaneasilyfind anonzero elementv ∈V_i suchthatα·v = 0 forα∈C(i,i+ 1).

Theaboveobservationsimmediately imply

td(V) = sup{i∈Z+| ∃0=v∈Vi andα∈C(i, i+ 1) such thatα·v= 0}.

Remark1.2. HomologiesofFI-modulesweredefinedto behomologiesofaspecialcom- plex in[4, Section2.4]. Ganand the first author proved in[9] thathomologies ofthis special complex coincide with ones defined in the above way. Since Tor is a classical homologicalconstruction,inthispaperwetaketheabovedefinition.

Since each kS_i = C(i,i) is a subalgebra of C for i ∈ Z+, given a kS_i-module T, it induces aC-module C⊗kSiT.Wecall these modulesbasic -filtered modules. Afinitely generatedC-moduleV iscalled -filtered byNagpalifithasafiltration

0 =V⁻¹⊆V⁰⊆. . .⊆Vⁿ=V

such thatVⁱ⁺¹/Vⁱ is isomorphic to abasic -filtered module for −1 i n−1;see [12,Definition1.10].Thereaderwillseethat-filteredmoduleshavesimilarhomological behaviorsasprojective modules.

1.3. Main results

Now we are ready to state main results of this paper. The first result characterizes -filteredmodulesbyhomologicaldegrees.

Theorem 1.3 (Homological characterizations of -filtered modules). Let k be a commu- tative Noetherian ring and let V be a finitely generated C-module. Then the following statements areequivalent:

(1) V is-filtered;

(2) hds(V)=−∞foralls1;

(3) hd₁(V)=−∞;

(4) hd_s(V)=−∞forsome s1.

Remark1.4. This theorem was also independently provedalmost at thesame time by Ramosin[13,Theorem B]viaadifferent approach.

Using these homologicalcharacterizations, onecandeduce an upper bound forpro- jectivedimensionsoffinitelygenerated C-moduleswhoseprojectivedimensionisfinite.

Theorem1.5 (Upper boundsof projective dimensions).Letk bea commutative Noethe- rianring whose finitisticdimensionfindimk isfinite,¹ andlet V be afinitely generated C-module with gd(V)=n. Then theprojective dimension pd(V) is finite if andonly if

1 Bydefinition, thefinitistic dimensionis thesupremumofprojectivedimensionsof finitelygenerated k-moduleswhoseprojectivedimensionisfinite.Thefamousfinitisticdimensionconjectureassertsthatif kis a finitedimensionalalgebra, thenfindimk < ∞. However, thefinitistic dimension of anarbitrary commutativeNoetherianringmightbeinfinity.

for0in,onehas

Vⁱ/Vⁱ⁻¹∼=C⊗kSi(Vⁱ/Vⁱ⁻¹)_i and

pd_kSi((Vⁱ/Vⁱ⁻¹)_i)<∞, where Vⁱ isthesubmoduleof V generated by

jiVj. Moreover,in thatcase pd(V) = max{pd_kSi((Vⁱ/Vⁱ⁻¹)i)}ⁿi=0= max{pd_k((Vⁱ/Vⁱ⁻¹)i)}ⁿi=0 findimk.

Remark 1.6. This theorem asserts that finitely generated C-modules which are not -filtered have infinite projective dimension. Moreover, if the finitistic dimension of k is 0, or in particular the global dimension gldimk is 0, then the projective dimension of a C-module is either 0 or infinity. This special result has been pointed out in [8, Corollary 1.6]forfields ofcharacteristic0.

AnotherimportantapplicationofTheorem 1.3istoprovethefactthateveryfinitely generatedC-modulecanbeapproximatedbyafinitecomplexof-filteredmodules,which was firstly proved by Nagpal in [12, Theorem A]. We give a new proof based on the conclusionofTheorem 1.3aswell astheshiftfunctor.

Theorem 1.7(-Filtered complexes). Letkbe acommutative Noetherianring andletV be afinitely generatedC-module.Then thereexistsacomplex

F^•: 0→V →F⁰→F¹→. . .→Fⁿ →0 satisfying thefollowingconditions:

(1) each Fⁱ isa-filteredmodulewith gd(Fⁱ)gd(V)−i;

(2) ngd(V);

(3) the homology in each degree of the complex is a torsion module, including the ho- mologyatV.

In particular,ΣdV isa-filteredmoduleford0.

Remark1.8. Thiscomplexof-filteredmodulesgeneralizesthefiniteinjectiveresolution describedin[8],whereitwasusedbyGanandthefirstauthortogiveahomologicalproof for the uniform representation stability phenomenon observedand proved in[2,3]. For anarbitraryfield,italsoimpliesthepolynomialgrowthoffinitelygeneratedC-modules;

see [4,Theorem B]andRemark 3.14.

Homological characterizations of -filtered modules also play a very important role in estimating homological degrees of finitely generated C-modules V. Relying on an existingupperbound described in[1,Theorem A], weobtainanother upperbound for homologicaldegreesofC-modules,removingtheunnecessaryassumptionthatkisafield ofcharacteristic0in[11, Theorem 1.17].

Theorem1.9(Castlenuovo–Mumfordregularity).LetkbeacommutativeNoetherianring andletV be afinitely generatedC-module.Then fors1,

hd_s(V)max{2 gd(V)−1,td(V)}+s. (1.2) Remark1.10.Theorem Ain[1]assertsthat

hds(V)hd0(V) + hd1(V) +s−1

fors1.Theconclusionoftheabovetheoremrefinesthisboundfortorsionlessmodules since inthat casetd(V) = −∞and Corollary 3.4 tells us that bya certain reduction onecanalways assumethatgd(V)<hd1(V).Furthermore, itismorepracticalsinceit iseasiertofindtd(V) thanhd₁(V).Thereadermayreferto [11,Example5.20].

Remark1.11.GivenafinitelygeneratedC-moduleV,thereexistsashortexactsequence 0→VT →V →VF →0

suchthatV_T is torsionand V_F istorsionless.Note thatgd(V_F)gd(V) andtd(V_T)= td(V).Usingthelongexactsequenceinducedbythisshortexactsequence,oneintuitively seesthatthetorsionpartVT contributesto thetermtd(V) ininequality(1.2), andVF

contributes to the term 2gd(V)−1 in inequality (1.2). Indeed, if V is a torsionless C-module,thenwehavehd_s(V)2gd(V)+s−1 fors1.Ontheotherhand,ifV is atorsionmodule, thenhds(V)td(V)+sfors0.

1.4. Organization

The paper is organized as follows. In Section 2 we introduce some elementary but importantpropertiesoftheshiftfunctor Σ andits inducedcokernelfunctor D.Inpar- ticular,ifV isatorsionlessC-module,thenanadaptableprojectiveresolutiongivesriseto anadaptableprojectiveresolution ofDV;seeDefinition 2.8and Proposition 2.11.This observationprovidesus ausefultechniquetoestimate homologicaldegreesofV.Those -filtered modules are studied in details in Section 3. We characterize -filtered mod- ulesbyhomologicaldegrees,anduseitto provethateveryfinitely generatedC-module becomes-filteredafter applyingthe shiftfunctor enoughtimes. In thelast section we proveallmain resultsmentionedbefore.

2. Preliminaryresults

Throughout this section let k be a commutativeNoetherian ring, and let C be the skeletal subcategoryofFI withobjects[n],n∈Z+.

2.1. FunctorΣ

The shiftfunctor Σ:C-mod→C-mod has been definedinthe previoussection. We list certainproperties.

Proposition 2.1. LetV beaC-module.Then onehas:

(1) Σ(C(i,−))∼=C(i,−)⊕C(i−1,−)^⊕i.

(2) If gd(V)n,then gd(ΣV)n;conversely, ifgd(ΣV)n,thengd(V)n+ 1.

(3) The C-moduleV isfinitely generatedif andonlyif soisΣV. (4) If V istorsionless, soisΣV.

Proof. Statement (1) is well known, and it immediately implies the first half of (2) since Σ is an exact functor. For the second half of (2), onemay refer to the proof of [11, Lemma3.4].Statement (3)is immediately impliedby (2)as aC-moduleis finitely generated ifand onlyif itsgenerating degree isfinite and itsvalue oneach objectis a finitely generatedk-module.

Toprove(4),oneobservesthatV istorsionlessifandonlyifthefollowingconditions hold:fori∈Z+,0=v ∈V_i,and α∈C(i,i+ 1),onealwayshasα·v = 0.IfΣV is not torsionless, we can finda nonzero element v ∈ (ΣV)i and α∈ C(i,i+ 1) for acertain i∈Z+suchthatα·v= 0.But(ΣV)_i=V_i+1,andι(C(i,i+ 1))⊆C(i+ 1,i+ 2) whereιis theself-embeddingfunctor inducingΣ.Therefore,byregardingvas anelementinVi+1

one has ι(α)·v = 0. Consequently, V is not torsionless either. The conclusionfollows from thiscontradiction. 2

2.2. Homologicaldegreesundershift

Thefollowing lemmaisadirectapplicationof[11,Proposition4.5].

Lemma 2.2.LetV beafinitely generated C-module.Thenfors0, hd_s(V)max{hd₀(V) + 1, . . . ,hd_s−1(V) + 1,hd_s(ΣV) + 1}.

If V isatorsion module,thenΣ_dV = 0 foralargeenoughd.Thus theabovelemma canbeused toestimatehomologicaldegreesoftorsionmodules.

Proposition 2.3. [11, Theorem 1.5] If V is a finitely generated torsion C-module, then fors∈Z+,onehas

hd_s(V)td(V) +s.

2.3. FunctorD

Thefunctor D wasintroduced in[1,2]. Here we briefly mentionits definition. Since thefamilyof inclusions

{πi: [i]→[i+ 1], r→r+ 1|i0} (2.1) gives a natural transformation π from the identity functor Id_C to the self-embedding functor ι, we obtain a natural transformation π^∗ from the identity functor on C-mod to Σ,whichinduces anaturalmap π_V^∗ :V →ΣV for eachC-moduleV.Thefunctor D isdefinedtobe thecokernel ofthismap.Clearly,D isaright exactfunctor. Thatis, it preservessurjection.

ThefollowingpropertiesofDplayakey roleinourapproach.

Proposition2.4. LetD bethefunctor definedasabove.

(1) ThefunctorD preservesprojective C-modules.Moreover, DC(i,−)∼=C(i−1,−)^⊕i. (2) AC-moduleV istorsionlessif andonlyif thereisashort exactsequence

0→V →ΣV →DV →0.

(3) LetV beafinitely generated C-module.Then:

gd(DV) =

−∞ if gd(V) = 0or − ∞ gd(V)−1 if gd(V)1.

Proof. Statements(1)and(2) havebeen establishedin[1,Lemma3.6], soweonlygive aproofof(3). TakeasurjectionP →V →0 suchthatP isaprojectiveC-module and gd(P)= gd(V)=n.SinceDisarightexactfunctor,wegetasurjectionDP →DV →0.

When n= 0 or −∞, we know thatDP = 0, and hence DV = 0, so gd(DV)= −∞. If n>0,then DP is aprojective module with gd(DP)=n−1 by (1). Consequently, gd(DV)n−1.Wefinishtheproofbyshowingthatgd(DV)n−1 aswell.

LetV be thesubmoduleofV generated by

in−1V_i.This isapropersubmodule ofV sincegd(V)=n. Let V=V /V, whichis notzero.Moreover, V is aC-module generated indegreen. Applyingthe right exactfunctorD to V →V →0 onegetsa surjection DV → DV → 0. But oneeasily sees that (DV)i = 0 for i < n−1 and (DV)n−1= 0.Consequently,gd(DV)n−1.Thisforces gd(DV)n−1. 2

Remark2.5.IfV isnottorsionless,wehaveanexactsequence 0→VT →V →VF →0

suchthatVT = 0 is torsionandVF istorsionless.Itinducesashortexactsequence 0→ΣVT →ΣV →ΣVF →0.

Using snakelemma,oneobtainsexactsequences

0→K→V →ΣV →DV →0 and

0→K→VT →ΣVT →DVT →0.

Inparticular, Kis atorsionmodule.

Animmediateconsequenceis:

Corollary 2.6. A short exact sequence 0 → W → M → V → 0 of finitely generated torsionlessC-modulesgivesrisetothefollowingcommutativediagramsuchthatallrows and columnsareshortexact sequences

0 W ΣW DW 0

0 M ΣM DM 0

0 V ΣV DV 0.

Proof. Since W and M are torsionless. By [1,Lemma 3.6], W →ΣW and M →ΣM are injective,andonegetsacommutativediagram

0 W ΣW DW

α

0

0 M ΣM DM 0,

whichbythesnakelemmainduces thefollowingexactsequence 0→kerα→V →ΣV →DV →0.

But since V is torsionless, the map V → ΣV is injective. Therefore, kerα = 0. The conclusionfollows. 2

Thenextlemma assertsthatthefunctorD “almost”commuteswithΣ.

Lemma2.7. LetV be afinitelygenerated C-module.ThenΣDV ∼=DΣV.

Proof. It is sufficient to construct a natural isomorphism between ΣD and DΣ. This hasbeen donebyRamosin[13,Lemma3.5].Notethatinthesettingofthatpaper,the self-embeddingfunctor isdefinedinawaydifferent from ours;see[13, Definition2.20].

Therefore,wehavetoslightlymodifytheproofin[13,Lemma3.5].Inoursetting, (ΣDV)_n=V_n+2/π_n+1(V_n+1);

(DΣV)_n=V_n+2/(ι(π_n))(V_n+1)

whereι istheself-embedding functorand πn isdefinedin(2.1).By(1.1)and(2.1), we have

π_n+1: [n+ 1]→[n+ 2], i→i+ 1;

ι(π_n) : [n+ 1]→[n+ 2], i→

1, i= 1;

i+ 1, 2in+ 1.

Now thereadercansee that(DΣV)n and (ΣDV)n areisomorphic undertheaction of anbijectionα_n+2: [n+ 2]→[n+ 2] whichpermutes1 and2 andfixesallotherelements.

Moreover,thefamilyofsuchbijections{α_n|n0}givesanaturalisomorphismbetween ΣD andDΣ. 2

2.4. Adaptableprojectiveresolutions

A standard way to compute homologies and hence homological degrees is to use a suitableprojectiveresolution.

Definition2.8.LetV be afinitely generatedC-module.Aprojective resolution . . .→P^s→P^s−1→. . .→P⁰→V →0

of V is said to be adaptable iffor every s −1, gd(P^s+1)= gd(Z^s), where Z^s is the s-thcycleandbyconventionZ⁻¹=V.

Lemma 2.9.Let 0→W → P → V →0 be ashort exact sequence of finitely generated C-modulessuch thatP isprojective andgd(V)= gd(P).Then

gd(W)max{hd1(V),gd(V)}= max{gd(V), gd(W)}.

Proof. TheconclusionsholdforV = 0 byconvention,so weassumethatV isnonzero.

Thegivenshort exactsequencegivesriseto

0→H1(V)→H0(W)→H0(P)→H0(V)→0.

Clearly,

gd(W) = td(H₀(W))max{td(H₁(V)),td(H₀(P))}= max{hd₁(V),gd(V)}. Moreover, ifhd1(V) gd(V), then gd(W) gd(V) as well. If hd1(V) >gd(V), then hd1(V)= gd(W).Theequalityfollows fromthisobservation. 2

Given afinitelygenerated C-moduleV, thefollowingcorollaryrelatesgenerating de- grees of componentsinan adaptableprojective resolution of V to homologicaldegrees of V.Thatis:

Corollary 2.10.LetV be afinitely generatedC-module,andlet . . .→P^s→P^s−1→. . .→P⁰→V →0 be anadaptable projectiveresolutionof V.Letd_s= gd(P^s).Then

d_smax{hd₀(V), . . . ,hd_s−1(V), hd_s(V)} and

max{d0, . . . , ds−1, ds}= max{hd0(V), . . . , hds−1(V),hds(V)}.

Proof. Weuse inductionon s. Ifs= 0, nothingneeds to show. Supposethatthe con- clusion holdsfors=n0,and considers=n+ 1.

Considertheshortexactsequence

0→Zⁿ →Pⁿ →Zⁿ⁻¹→0.

ByLemma 2.9andthedefinitionof adaptableprojectiveresolutions,

dn+1= gd(Zⁿ)max{gd(Zⁿ⁻¹),hd1(Zⁿ⁻¹)}= max{dn, hdn+1(V)} sinceV =Z⁻¹ byconvention.However,byinduction,

d_nmax{hd₀(V), . . . ,hd_n(V)}.

Thelast twoinequalitiesimplytheconclusionforn+ 1.Thisestablishestheinequality.

Toshowtheequality,oneobserves thattheinequalitywejustprovedimpliesthat max{d₀, . . . , d_s−1, d_s}max{hd₀(V), . . . ,hd_s−1(V),hd_s(V)}.

However,one observes from the definitionof homologies thatdi hdi(V) fori ∈ Z+. Thuswealsohave

max{d0, . . . , ds−1, ds}max{hd0(V), . . . ,hds−1(V),hds(V)}. 2 Using thefunctor D,onemay relatethehomologicaldegrees ofafinitely generated C-moduleV tothoseof DV.

Proposition2.11.LetV beafinitely generated torsionlessC-module,andlet . . .→P^s→P^s−1→. . .→P⁰→V →0

beanadaptable projectiveresolutionofV.Thenitinducesanadaptableprojectivereso- lution

. . .→DP^s→DP^s−1→. . .→DP⁰→DV →0 suchthat fors∈Z+

gd(DP^s) =

−∞ if gd(P^s) = 0 or − ∞ gd(P^s)−1 if gd(P^s)1.

Proof. Let P^• → V → 0 be the resolution.Since V and all cycles are torsionless, by Corollary 2.6onegetsacommutativediagram

0 P^• ΣP^• DP^• 0

0 V ΣV DV 0.

Theconclusionthenfollows fromProposition 2.4. 2

Asanimmediateconsequenceoftheaboveresult,wehave:

Corollary2.12.LetV beafinitely generatedtorsionless C-module.Thenfors∈Z+

max{hd₀(V), . . . , hd_s(V)}max{hd₀(DV), . . . ,hd_s(DV)}+ 1.

Moreover,theequalityholds ifgd(V)1.

Proof. Theconclusion holds triviallyif gd(V)= 0 or −∞since inthatcase DV = 0.

So we assume gd(V)1, and DV = 0. Let P^• → V → 0 bean adaptable projective resolution.ByCorollary 2.10 andProposition 2.11,onehas

max{gd(V), . . . ,hds(V)}= max{gd(P⁰), . . . , gd(P^s)} and

max{gd(DV), . . . , hds(DV)}= max{gd(DP⁰), . . . , gd(DP^s)}.

Moreover, gd(Pⁱ)gd(DPⁱ)+ 1 fori∈Z+, andtheequalityholds ifgd(Pⁱ)1.The desiredinequalityandequalityfollow fromthese observations. 2

3. FiltrationsofFI-modules

Intheprevioussectionweuseadaptableprojectiveresolutionstoestimatehomological degrees of finitely generated C-modules. However, since finitely generated C-modules usually haveinfiniteprojectivedimension,theresolutionsare ofinfinitelength.Forthe purpose of estimating homological degrees, -filtered modules play a more subtle role since we will show that every finitely generated C-module V gives rise to a complex of -filtered modules which is of finite length. Moreover, we will see thatthese special modules, coinciding with projective moduleswhen k is afield ofcharacteristic 0, have similar homologicalpropertiesasprojective modules.

3.1. Ahomological characterizationof-filtered modules

Recall thatafinitelygeneratedC-moduleis -filtered ifthereexists achain 0 =V⁻¹⊆V⁰⊆. . .⊆Vⁿ =V

such thatVⁱ/Vⁱ⁻¹ isisomorphicto C⊗kSiT_i for0in, whereS_i isthesymmetric grouponiletters,andT_i isafinitelygeneratedkS_i-module.

AnimportantfactofFI,whichcanbe easilyobserved,is:

Lemma 3.1.Forn∈Z+,theC-moduleC1_n isaright free kS_n-module.

Proof. Notethatforn∈Z+ andmn,thegroupSn=C(n,n) actsfreelyonC(n,m) from therightside.Theconclusionfollows. 2

This elementary observation implies that higher homologies of -filtered modules vanish.

Lemma 3.2.If V isa-filteredmodule, thenHs(V)= 0foralls1.

Proof. Firstly we consider aspecial case: V is basic.That is, V =C⊗kSi Vi for some i∈Z+.Let

0→Wi→Pi→Vi→0

be ashort exact sequenceof kS_i-modulessuch thatP_i is projective. Since Cis aright projectivekSi-module,we getanexactsequence

0→W =C⊗kSiW_i→P =C⊗kSiP_i→V =C⊗kSiV_i→0.

Notethatthemiddletermis aprojective C-module.ByapplyingC0⊗C−onerecovers thefirstexactsequence, so H₁(V)= 0. ReplacingV byW, onededuces thatH₂(V)= H₁(W)= 0.Theconclusionfollowsbyrecursion.

For the general case, one may take a filtration for V, each component of which is abasic-filteredmodule. Theconclusionfollows from astandardhomologicalmethod:

shortexactsequencesinducelongexactsequencesonhomologies. 2

Thefollowing lemma wasproved in[12, Lemma2.2]. Here we givetwo proofs from thehomologicalviewpoint.

Lemma3.3.LetV beafinitely generatedC-modulegeneratedinonedegree.Ifhd1(V) gd(V),then V isa-filteredmodule.

Proof. Theconclusionholds triviallyforV = 0, soweassumegd(V)=n0.Consider theshortexactsequence

0→W →P →V →0

where P is aprojective C-modulewith gd(P) =n. Since hd1(V)n, oneknows that gd(W)nbyLemma 2.9. Ifgd(W)< n,then W = 0 sinceWi = 0 foralli< n. Thus V ∼=P isclearlya-filteredmodule.Nowweconsider thecasethatgd(W)=n.

Since0→W_n →P_n →V_n →0 isashort exactsequenceof kS_n-modulesand Cisa rightprojectivekS_n-module,weobtainashort exactsequence

0→C⊗kSnW_n →C⊗kSnP_n→C⊗kSnV_n→0.

NotethatW,P,andV areallgeneratedindegreen.Viathemultiplicationmapweget acommutativediagramsuchthatallverticalmapsaresurjective:

0 C⊗kSnWn C⊗kSnPn C⊗kSnVn 0

0 W P V 0.

Butthemiddleverticalmapisactuallyanisomorphism.Thisforcestheothertwovertical maps tobeisomorphismsbysnakelemma,and theconclusionfollows. 2

Proof. If V = 0, nothing needs to show. Otherwise, let gd(V) = n 0. Since V is generated indegreen,there isashort exactsequence

0→K→V˜ =C⊗kSnVn →V →0 whichinduces anexactsequence

0→H₁(V)→H₀(K)→H₀( ˜V)→H₀(V)→0

bythepreviouslemma. Notethatthemap H0(V)→H0(V) isanisomorphism.Conse- quently, H₀(K)∼=H₁(V). Inparticular, gd(K)= hd₁(V)gd(V)=n. Butit isclear thatK_i= 0 forallin.Therefore, theonlypossibility isthatK= 0. 2

A usefulresultis:

Corollary 3.4. Let V be a finitely generated C-module. Then there exists a short exact sequence

0→U →V →W →0

such thatHs(U)=Hs(V)fors1andW is-filtered.Moreover, ifV isnot-filtered, one alwayshas hd₁(U)>gd(U).

Proof. Suppose that V is nonzero. If hd1(V) > gd(V), then we can let U = V and W = 0,so theconclusionholds trivially.Otherwise,onehasashortexactsequence

0→V→V →V→0 where V is the submoduleof V generated by

igd(V)−1Vi, whichmight be 0.Then V= 0.Thelongexactsequence

. . .→H1(V)→H1(V)→H0(V)→H0(V)→H0(V)→0 tellsusthat

hd₁(V)max{gd(V),hd₁(V)}gd(V) = gd(V).

ButthepreviouslemmaassertsthatV is-filtered.

Ifgd(V)<hd₁(V),thentheaboveshortexactsequenceiswhatwewant.Otherwise, we cancontinue this process for V. Since gd(V) < gd(V), it must stopafter finitely manysteps.SinceV isnot-filtered,inthelaststepwemustgetasubmoduleUofV with

hd1(U)>gd(U).Moreover,sinceW is-filtered,oneeasilydeducesthatHs(U)∼=Hs(V) fors1. 2

Afinitelygenerated C-moduleis-filteredifandonlyifitshigherhomologiesvanish, andifandonlyifitsfirsthomologyvanishes.

Theorem3.5. Letk be acommutative Noetherianring and letV be a finitely generated C-module.Thenthefollowingstatements areequivalent:

(1) V isa-filteredmodule;

(2) Hi(V)= 0 foralli1;

(3) H1(V)= 0.

Proof. (1)⇒(2):ThisisLemma 3.2.

(2)⇒(3): Trivial.

(3) ⇒(1): Supposethat H₁(V) = 0. Thatis, hd₁(V) =−∞. If V is not -filtered, thenbyCorollary 3.4,there existsashortexactsequence

0→U →V →W →0

suchthatH1(U)∼=H1(V)= 0 andhd1(U)>gd(U).Thisisabsurd. 2

An extra bonus of this characterization is that: the category of finitely generated -filteredmodulesisclosed undertaking kernelsandextensions.

Corollary3.6. Let0→U →V →W →0beashortexact sequence.

(1) Ifboth V and W are -filtered,so isU. (2) Ifboth U andW are -filtered,sois V.

Proof. Usethe longexactsequence inducedbythe givenshort exactsequence andthe abovetheorem. 2

3.2. Properties of-filteredmodules

Inthissubsectionwe explorecertainimportantpropertiesof-filteredmodules.

Fora-filteredmoduleV, oneknowsthatithasafiltrationbybasic-filteredmod- ulesfrom the definition.Thefollowing resulttells usan explicitconstruction ofsuch a filtration.

Proposition3.7. LetV be a-filteredmodulewith gd(V)=n.Thenthereexistsachain ofC-modules

0 =V⁻¹⊆V⁰⊆V¹⊆. . .⊆Vⁿ=V

such that for −1 s n−1, V^s is the submodule of V generated by

isVi and V^s+1/V^s is0orabasic -filteredmodule.

Proof. Weprovethestatementbyinductionongd(V). Theconclusionholdsobviously forn= 0 orn=−∞.Supposethatn1.Wehaveashort exactsequence

0→V→V →V→0 such thatV is the submodule generated by

in−1V_i. Then V is generated inde- green.TheconclusionfollowsimmediatelyafterweshowthatV isa-filteredmodule.

However,wecanapplythesameargument asintheproof ofCorollary 3.4. 2

Remark3.8. Sofarwe donotuseany specialproperty ofthe category FIto provethe above resultsin this section. Therefore, all these results hold for k-linearcategories D satisfyingthefollowingconditions:

(1) objectsofDareparameterizedbynonnegativeintegers;

(2) there isnononzeromorphismsfrom biggerobjectstosmallerobjects;

(3) Dislocallyfinite;thatis,D(x,y) arefinitelygenerated k-modulesforx,y∈Z+; (4) D(x,y) isarightprojectiveD(x,x)-moduleforx,y∈Z+.

Now we begin to use somespecial propertiesof FIto deduce moreresults. Thefol- lowing propositiontellsusthatthepropertyof being-filteredis preservedbyfunctors Σ and D.

Proposition 3.9. Let V be a finitely generated C-module. If V is -filtered, then it is torsionless. Moreover, ΣV andDV arealso -filtered.

Proof. Clearly,wecansuppose thatV isnonzero.SinceV isa-filteredmodule, ithas afiltrationsuchthateachcomponentofwhichhastheformC⊗kSiTi,whereeachTiisa finitelygeneratedkSi-module.SinceV isatorsionlessmoduleifandonlyifHom_C(C0,V) is 0,to showthatV istorsionless,itsufficestoverifyHom_C(C0,C⊗kSiT_i)= 0;thatis, eachC⊗kSiT_i istorsionless.By[12,Lemma2.2],there isanaturalembedding

0→C⊗kSiMi→Σ(C⊗kSiMi).

ByProposition 2.4,this-filteredmodulemustbe torsionless.

Toprovethesecondstatement,weuseinductiononthegeneratingdegreen= gd(V).

For n= 0, theconclusionis implied by [12, Lemma2.2]. For n1,oneconsiders the short exactsequence

0→V→V →V→0

whereVisthesubmoduleofV generatedby

in−1Vi.ByProposition 3.7,alltermsin thisshort exactsequence are-filtered. Moreover,Visabasic-filteredmodule. Since we just proved thatthey are all torsionless, by Corollary 2.6, we have acommutative diagrameachroworcolumnofwhichisanshortexactsequence:

0 V ΣV DV 0

0 V ΣV DV 0

0 V ΣV DV 0.

Byinductionhypothesis,ΣVis-filtered.Moreover,ΣVis-filteredby[12,Lemma2.2], soΣV is-filteredaswell.ThesamereasoningtellsusthatDV arealso-filtered. 2 3.3. Finitely generatedC-modules become -filteredunder enoughshifts

ThemaintaskofthissubsectionistousefunctorD aswellas thehomologicalchar- acterizationof-filteredmodulestoshow thatfor everyfinitely generatedC-moduleV, ΣNV is-filteredforN 0.

Asthestartingpoint,weconsider C-modulesgeneratedindegree0.

Lemma3.10.EverytorsionlessC-modulegeneratedin degree 0is-filtered.

Proof. SinceV istorsionless,onegetsashort exactsequence 0→V →ΣV →DV →0.

ByProposition 2.4,DV = 0,soV ∼= ΣV. Inparticular, onehas V0∼=V1∼=V2∼=. . . .

ThusV ∼=C⊗C(0,0)V0. 2

Lemma3.11.LetV beafinitelygenerated torsionlessC-module.IfDV is-filtered,then gd(V)hd1(V).

Proof. One may assume that V is nonzero. If gd(V) = 0, then V is -filtered by Lemma 3.10andtheconclusionholds.Forgd(V)1,sincehd1(DV)=−∞,byCorol- lary 2.12,

max{gd(V),hd1(V)}= max{gd(DV) + 1,hd1(DV) + 1}= gd(DV) + 1 = gd(V), wherethelast equalityfollowsfromProposition 2.4.Thisimpliesthedesiredresult. 2

The following lemma, similar to Proposition 2.11, shows theconnections between a finitely generatedC-moduleV and DV.

Lemma 3.12. LetV be a finitely generated torsionless C-module.If DV is -filtered, so is V.

Proof. Weuseinductionongd(V).Theconclusionforgd(V)= 0 hasbeenestablishedin Lemma 3.10.Supposethatitholdsforallfinitelygenerated C-moduleswithgenerating degree at mostn, and letV be afinitely generated C-modulewith gd(V)=n+ 1. As before, considertheexactsequence0→V→V →V→0 whereV isthesubmodule generated by

inV_i.

Byconsideringthelongexactsequence

. . .→H1(V)→H1(V)→H0(V)→H0(V)→H0(V)→0

and using the fact gd(V) n and n+ 1 = gd(V) hd₁(V) which is proved in Lemma 3.11, one deduces that hd₁(V) n+ 1 = gd(V). Consequently, V is a -filteredmodule byLemma 3.3,andhenceis torsionlessbyProposition 3.9.Moreover, DV is-filteredaswell byProposition 3.9.

SinceVistorsionless,theaboveexactsequencegivesrisetoacommutativediagram

0 V ΣV DV 0

0 V ΣV DV 0

0 V ΣV DV 0.

Note thatboth DV andDV are -filtered.ByCorollary 3.6,DV is-filteredas well.

But clearly V is torsionless since it is a submodule of the torsionless module V. By inductionhypothesis,V is-filtered,soisV.Theconclusionfollowsfrominduction. 2

Now wearereadytoshow thefollowing result.

Theorem3.13. Letkbe acommutative NoetherianringandletV beafinitelygenerated C-module.If d0,thenΣdV is -filtered.

Proof. Again,assumethatV isnonzero.Sinced0,onemaysupposethattd(V)< d.

Therefore, Σ_dV is torsionless. Weuse induction on gd(V). If gd(V) = 0, then Σ_dV is generated indegree 0, so the conclusion holds by Lemma 3.10. Now suppose that the conclusionholds forall moduleswithgenerating degreesat mostn. Wethendealwith gd(V)=n+ 1.

Considertheexactsequence

0→Σ_dV →Σ_d+1V →Σ_dDV ∼=DΣ_dV →0.

If gd(Σ_dV) = 0, nothing needs to show. So we suppose thatgd(Σ_dV) 1. Note that gd(DV)=n.Therefore,byinductionhypothesis, ΣdDV is -filtered. Thatis, DΣdV is -filtered.ByLemma 3.12, Σ_dV is-filteredas well. 2

Remark3.14.Thistheoremgivesanotherproofforthepolynomialgrowth phenomenon observedin[4].SinceforasufficientlylargeN ∈Z+,Σ_NV hasafiltrationeachcompo- nentof whichis exactlyof theform M(W) asin[2, Definition2.2.2], andeach M(W) satisfies the polynomial growth property, so is Σ_NV. Consequently, τ_NV satisfies the polynomialgrowth property.

Because-filteredmodulescoincidewithprojectivemoduleswhenkisafieldofchar- acteristic0,onehas:

Corollary 3.15. Let k be a field of characteristic 0 and let V be a finitely generated C-module.Thenford0,Σ_dV isprojective.

4. Proofsofmainresults

Inthissectionweproveseveral mainresultsmentionedinSection1.

4.1. Proofof Theorem 1.3

Theorem 3.5 has established the equivalence of the first three statements inTheo- rem 1.3.Inthissubsectionweshowtheequivalencebetweenthefirststatementandthe lastone.

Lemma4.1.LetV beafinitely generatedC-module.IfH₂(V)= 0,thenV istorsionless.

Proof. Considerashort exactsequence

0→W →P →V →0.

SinceH1(W)=H2(V)= 0,W is-filtered.ApplyingΣ toitonegets 0→ΣW →ΣP →ΣV →0.

Theyinducethefollowingcommutativediagram:

0 W ΣW DW

α

0

0 P ΣP DP 0,

and henceanexactsequence

0→kerα→V →ΣV →DV →0.

Since W is -filtered,DW is -filteredas well,and is torsionlessby Proposition 3.9.

Therefore, kerαas asubmoduleofDW istorsionlessas well.However,as explainedin Remark 2.5, thekernelofV →ΣV is torsionsinceitisasubmoduleof VT,thetorsion partofV.This happensifandonlyifthekernelis0.Thatis, V istorsionless. 2

Theconclusionofthislemma canbestrengthened.

Lemma 4.2.LetV beafinitely generated C-module.IfH2(V)= 0,thenV is-filtered.

Proof. We already know that V is torsionless from the previous lemma. Now we use inductionongd(V).Ifgd(V)= 0,thentheconclusionfollowsfromLemma 3.10.Other- wise,wehaveacommutativediagram

0 W ΣW DW 0

0 P ΣP DP 0

0 V ΣV DV 0,

where P is projective andW is-filtered. ByProposition 3.9, DW is -filteredas well.

Moreover, gd(DV)<gd(V). Therefore, byinduction hypothesis, DV is -filtered. But byLemma 3.12,V mustbe-filteredas well. 2

TheconclusionofCorollary 3.6canbestrengthened asfollows:

Corollary 4.3.Let0→U →V →W →0be ashortexact sequenceoffinitely generated C-modules.Iftwotermsare -filtered,soisthethird one.

Proof. ItsufficestoshowthatifUandV are-filtered,soisW.Thelongexactsequence . . .→H2(U)→H2(V)→H2(W)→H1(U)→. . .

tells us H2(W) = 0 since H1(U) = H2(V) = 0. The conclusion follows from Lemma 4.2. 2

Nowwearereadyto provethefirstmain theoremmentionedinSection1.

Proposition4.4. LetV beafinitely generated C-module.ThenV is-filteredifandonly ifHs(V)= 0 forsomes1.

Proof. Onedirectionistrivial.Fortheotherdirection,wecanassumethats2.Take anadaptableprojectiveresolutionP^•→0 andletZⁱ bethei-thcycle. Thenwehave

0 =H_s(V) =H₁(Z^s−1).

Consequently,Z^s−1 is a-filteredmodule. Applyingthepreviouscorollary totheshort exactsequence

0→Z^s−1→P^s−2→Z^s−2→0

onededucesthatZ^s−2 is-filteredaswell.Theconclusionfollowsfrom recursion. 2 4.2. Upperbounds forprojectivedimensions

InthissubsectionweproveTheorem 1.5.Weneedawellknownresultonrepresenta- tiontheoryoffinitegroups.

Lemma 4.5.Let Gbe afinite group andk be a commutative Noetherianring. LetV be afinitely generatedkG-module.If pd_kG(M) isfinite,thenpd_kG(M)= pd_k(M).

Proof. TheproofusesEckmann–ShapiroLemma.Fordetails,see[10,Theorem 4.3]. 2 IfV isabasic-filteredmodule,thenits projectivedimensioncoincides withthatof thefinitelygeneratedgrouprepresentationinducingV.Thatis,

Lemma4.6. LetT beafinitely generated kSi-module.Then pd_kSi(T) = pd_C(C⊗kSiT).

Proof. Weclaimthat

pd_kSi(T)pd_C(C⊗kSiT).

Ifpd_kSi(T)=∞,theinequalityholds.Otherwise, thereisaprojective resolutionQ^•→ T →0 ofkSi-modulessuchthatZ^s isprojectiveforspd_kSi(T),whereZ^s isthes-th cycle.ApplyingtheexactfunctorC⊗kSi−onegetsaprojectiveresolutionofC-modules

C⊗kSiQ^•→C⊗kSiT →0,

whose s-th cycle C⊗kSi Z^s is a projective C-module for s pd_kSi(T). The claim is proved.

Now weshowthat

pd_kSi(T)pd_C(C⊗kSiT).

If pd_C(C⊗kSiT)=∞,theinequalityholds.Otherwise,there isafiniteprojectivereso- lution ofC-modules

P^•→C⊗kSiT →0

suchthateachtermofwhichisgenerated indegreei.Restricting thisresolutionto the objecti onegetsafiniteresolutionofkSi-modules

1_iP^•→1_i(C⊗kSiT) =T →0.

Theaboveinequalityfollows fromthisobservation. 2

Thefollowing lemmatellsusthattoconsider theprojectivedimensionofa-filtered module, itisenoughto consideritsbasicfiltrationcomponents.

Lemma4.7. LetV beafinitelygenerated-filteredC-moduleandsupposethatpd(V)<∞. Let

0 =V⁻¹⊆V⁰⊆V¹⊆. . .⊆Vⁿ=V

be thefiltrationgiven inProposition 3.7.Then for0in,one has pd(Vⁱ/Vⁱ⁻¹)pd(V).

Proof. Oneuses inductiononthe lengthn, which isprecisely gd(V). Ifn = 0 or −∞, nothing needsto show.Forn1,onehasashort exactsequence

0→U →V →W →0 whereUisthesubmodulegeneratedby

in−1Vi.Itsufficestoshowthatbothpd(U) pd(V) and pd(W)pd(V) since inthatcaseby inductionhypothesisoneknows that eachbasicfiltrationcomponentofU hasprojectivedimensionatmostpd(U)pd(V).

Take two surjectionsP⁰→U andQ⁰ →W suchthatgd(P⁰)= gd(U)n−1 and Q⁰ isgenerated indegreen.Wegetacommutativediagram