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Journal of Algebra
www.elsevier.com/locate/jalgebra
Filtrations and homological degrees of FI-modules
✩Liping Lia,Nina Yub,∗
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:lipingli@hunnu.edu.cn(L. Li),ninayu@xmu.edu.cn(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,ΣNV hasaspecialfiltration,where ΣN 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,
(τnV)i:=
0, i < n Vi, in
AfinitelygeneratedC-moduleV iscalled torsionifthereexists someN ∈Z+ suchthat τNV = 0.Inother words,Vi = 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+ |HomC(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) = TorCs(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 ∈ Vi 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 HomC(C(i,i),V) = 0. Conversely, if HomC(C(i,i),V)= 0, onecaneasilyfind anonzero elementv ∈Vi 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 kSi = C(i,i) is a subalgebra of C for i ∈ Z+, given a kSi-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) hds(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
pdkSi((Vi/Vi−1)i)<∞, where Vi isthesubmoduleof V generated by
jiVj. Moreover,in thatcase pd(V) = max{pdkSi((Vi/Vi−1)i)}ni=0= max{pdk((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,
hds(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
suchthatVT is torsionand VF istorsionless.Note thatgd(VF)gd(V) andtd(VT)= 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,thenwehavehds(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 ∈Vi,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=Vi+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, hds(V)max{hd0(V) + 1, . . . ,hds−1(V) + 1,hds(ΣV) + 1}.
If V isatorsion module,thenΣdV = 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
hds(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 IdC to the self-embedding functor ι, we obtain a natural transformation π∗ from the identity functor on C-mod to Σ,whichinduces anaturalmap πV∗ :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−1Vi.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=Vn+2/πn+1(Vn+1);
(DΣV)n=Vn+2/(ι(πn))(Vn+1)
whereι istheself-embedding functorand πn isdefinedin(2.1).By(1.1)and(2.1), we have
πn+1: [n+ 1]→[n+ 2], i→i+ 1;
ι(πn) : [n+ 1]→[n+ 2], i→
1, i= 1;
i+ 1, 2in+ 1.
Now thereadercansee that(DΣV)n and (ΣDV)n areisomorphic undertheaction of anbijectionαn+2: [n+ 2]→[n+ 2] whichpermutes1 and2 andfixesallotherelements.
Moreover,thefamilyofsuchbijections{αn|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 . . .→Ps→Ps−1→. . .→P0→V →0
of V is said to be adaptable iffor every s −1, gd(Ps+1)= gd(Zs), where Zs 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 . . .→Ps→Ps−1→. . .→P0→V →0 be anadaptable projectiveresolutionof V.Letds= gd(Ps).Then
dsmax{hd0(V), . . . ,hds−1(V), hds(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,
dnmax{hd0(V), . . . ,hdn(V)}.
Thelast twoinequalitiesimplytheconclusionforn+ 1.Thisestablishestheinequality.
Toshowtheequality,oneobserves thattheinequalitywejustprovedimpliesthat max{d0, . . . , ds−1, ds}max{hd0(V), . . . ,hds−1(V),hds(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 . . .→Ps→Ps−1→. . .→P0→V →0
beanadaptable projectiveresolutionofV.Thenitinducesanadaptableprojectivereso- lution
. . .→DPs→DPs−1→. . .→DP0→DV →0 suchthat fors∈Z+
gd(DPs) =
−∞ if gd(Ps) = 0 or − ∞ gd(Ps)−1 if gd(Ps)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), . . . , hds(V)}max{hd0(DV), . . . ,hds(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(Ps)} and
max{gd(DV), . . . , hds(DV)}= max{gd(DP0), . . . , gd(DPs)}.
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⊗kSiTi for0in, whereSi isthesymmetric grouponiletters,andTi isafinitelygeneratedkSi-module.
AnimportantfactofFI,whichcanbe easilyobserved,is:
Lemma 3.1.Forn∈Z+,theC-moduleC1n isaright free kSn-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 kSi-modulessuch thatPi is projective. Since Cis aright projectivekSi-module,we getanexactsequence
0→W =C⊗kSiWi→P =C⊗kSiPi→V =C⊗kSiVi→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→Wn →Pn →Vn →0 isashort exactsequenceof kSn-modulesand Cisa rightprojectivekSn-module,weobtainashort exactsequence
0→C⊗kSnWn →C⊗kSnPn→C⊗kSnVn→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 thatKi= 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, Vs is the submodule of V generated by
isVi and Vs+1/Vs 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−1Vi. 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 isatorsionlessmoduleifandonlyifHomC(C0,V) is 0,to showthatV istorsionless,itsufficestoverifyHomC(C0,C⊗kSiTi)= 0;thatis, eachC⊗kSiTi 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
inVi.
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, ΣdV is torsionless. Weuse induction on gd(V). If gd(V) = 0, then ΣdV 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→ΣdV →Σd+1V →ΣdDV ∼=DΣdV →0.
If gd(ΣdV) = 0, nothing needs to show. So we suppose thatgd(ΣdV) 1. Note that gd(DV)=n.Therefore,byinductionhypothesis, ΣdDV is -filtered. Thatis, DΣdV is -filtered.ByLemma 3.12, ΣdV is-filteredas well. 2
Remark3.14.Thistheoremgivesanotherproofforthepolynomialgrowth phenomenon observedin[4].SinceforasufficientlylargeN ∈Z+,ΣNV hasafiltrationeachcompo- nentof whichis exactlyof theform M(W) asin[2, Definition2.2.2], andeach M(W) satisfies the polynomial growth property, so is ΣNV. Consequently, τNV 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,ΣdV 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 =Hs(V) =H1(Zs−1).
Consequently,Zs−1 is a-filteredmodule. Applyingthepreviouscorollary totheshort exactsequence
0→Zs−1→Ps−2→Zs−2→0
onededucesthatZs−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 pdkG(M) isfinite,thenpdkG(M)= pdk(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 pdkSi(T) = pdC(C⊗kSiT).
Proof. Weclaimthat
pdkSi(T)pdC(C⊗kSiT).
IfpdkSi(T)=∞,theinequalityholds.Otherwise, thereisaprojective resolutionQ•→ T →0 ofkSi-modulessuchthatZs isprojectiveforspdkSi(T),whereZs isthes-th cycle.ApplyingtheexactfunctorC⊗kSi−onegetsaprojectiveresolutionofC-modules
C⊗kSiQ•→C⊗kSiT →0,
whose s-th cycle C⊗kSi Zs is a projective C-module for s pdkSi(T). The claim is proved.
Now weshowthat
pdkSi(T)pdC(C⊗kSiT).
If pdC(C⊗kSiT)=∞,theinequalityholds.Otherwise,there isafiniteprojectivereso- lution ofC-modules
P•→C⊗kSiT →0
suchthateachtermofwhichisgenerated indegreei.Restricting thisresolutionto the objecti onegetsafiniteresolutionofkSi-modules
1iP•→1i(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