On Cohen-Macaulayness and depth of ideals in invariant rings

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Journal

of

Pure

and

Applied

Algebra

www.elsevier.com/locate/jpaa

On

Cohen–Macaulayness

and

depth

of

ideals

in

invariant

rings

Martin Kohlsa, Müfit Sezerb,∗,1

a

TechnischeUniversitätMünchen,ZentrumMathematik-M11,Boltzmannstrasse3,85748Garching, Germany

b_{DepartmentofMathematics,BilkentUniversity,Ankara06800,Turkey}

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

Articlehistory:

Received7September2015 Receivedinrevisedform12October 2015

Availableonline26October2015 CommunicatedbyS.Iyengar

MSC:

13A50

WeinvestigatethepresenceofCohen–Macaulayidealsininvariantringsandshow that anidealof aninvariant ringcorrespondingto amodular representationofa p-groupisnotCohen–Macaulayunlesstheinvariantringitselfis.Asanintermediate result,weobtainthatnon-Cohen–MacaulayfactorialringscannotcontainCohen– Macaulay ideals. For modular cyclic groups of prime order, we show that the quotientoftheinvariantringmodulothetransferidealisalwaysCohen–Macaulay, extendingaresultofFleischmann.

© 2015ElsevierB.V.All rights reserved.

1. Introduction

ThedepthandCohen–Macaulaypropertyofinvariantringshavealwaysbeenamongthemajorinterests ofinvarianttheorists,seethereferencesbelow.Inthispaper,weconsideridealsofinvariantrings(asmodules overthelatter),and investigatetheirdepthandCohen–Macaulayness. Theoriginalgoalofthispaper was tofindfiltrationsoftheinvariantringswithCohen–Macaulayquotients(aweakeningofbeing“sequentially Cohen–Macaulay”asintroducedin[18,SectionIII.2]).However,theresultsofthispapershowthatinmany cases,invariantringsfailtocontainanyCohen–Macaulayideal,sothegoalismissedinthefirststepalready. Before wego into more details,we fix oursetup. Let V be afinite dimensionalrepresentation of agroup

G over a fieldK. The representation is called modular if thecharacteristic of K divides the order of G. Otherwise, it iscalled nonmodular.There is an inducedaction onthe symmetric algebraK[V ] := S(V∗) givenbyσ(f )= f◦ σ−1 forσ∈ G andf ∈ K[V ].Welet

K[V ]G:={f ∈ K[V ] | σ(f) = f for all σ ∈ G}

* Correspondingauthor.

E-mailaddresses:kohls@ma.tum.de(M. Kohls),sezer@fen.bilkent.edu.tr(M. Sezer).

1

SecondauthorissupportedbyagrantfromTÜBITAK:115F186.

http://dx.doi.org/10.1016/j.jpaa.2015.10.014 0022-4049/© 2015ElsevierB.V.All rights reserved.

denote the subalgebra of invariant polynomials in K[V ]. For any nonmodular representation, K[V ]G _is

a Cohen–Macaulay ring [12]. In the modular case, on the other hand, K[V ]G _{almost always fails to be}

Cohen–Macaulay,see [13].ThedepthofK[V ]G hasattractedmuchattentionandhasbeen determinedfor variousfamiliesofrepresentations,seeforexample[4,9,11,14,17].InthispaperweconsideridealsofK[V ]G

as modulesoverK[V ]G.Weshowthat,ifV isamodularrepresentationofap-group,thenK[V ]G doesnot contain aCohen–Macaulay idealunless K[V ]G _{is Cohen–Macaulay itself. Combining this witharesult of}

Broer allowsus to show the equivalenceof the transferidealbeing Cohen–Macaulay or principaland the invariantringbeing adirectsummandofthepolynomialringforthesegroups,seeCorollary 10.

However,ourresultsholdinabroadergenerality.WefirstshowthataCohen–Macaulayidealinanaffine domaincannothaveheightbiggerthanone.If,inaddition,theaffinedomainisfactorial,thenonlyprincipal ideals canbeCohen–Macaulay. Soweget thedesiredimplication forthe groupsandtheirrepresentations whose invariants are factorial. We also include an example that shows that the condition thatthe affine domain is factorialcan not be dropped. Wethen restrictto modularrepresentations of a cyclic groupof primeorder.OurmainresulthereisthatthequotientK[V ]G_/IG _{oftheinvariantringmodulothetransfer}

ideal IG _{is Cohen–Macaulay. Note thatthis extendsresults ofFleischmann [10] inthis case, namelythat}

K[V ]G_/√_IG _{is Cohen–Macaulay, and that}√_IG _{= I}G _{if V is projective. This also allows us to compute}

the depth of the transfer ideal. We end with a reduction result that reduces computing the depth of a

K[V ]G_{-moduletocomputingagradeofthetransferideal.}

Wereferthereaderto[1,5,6]formorebackgroundinmodularinvarianttheory. 2. Preliminaries

Inthissectionwesummarizeournotationaswellassomebasicresultsthatweuseinourcomputations. LetR =∞_d=0Rd beagradedaffineK-algebrasuchthatR0= K,andM =

∞

d=0Md afinitelygenerated

gradednonzeroR-module.WecallR+:=∞d=1RdthemaximalhomogeneousidealofR.Asequenceof

ho-mogeneouselementsa1,. . . ,ak∈ R+iscalledM -regularifeachaiisanonzerodivisoronM/(a1,. . . ,ai−1)M

fori= 1,. . . ,k. ForahomogeneousidealI⊆ R+,themaximallengthofanM -regularsequence lyinginI

is calledthegradeofI onM ,denotedbygrade(I,M ).Furthermore,onecallsdepth(M ):= grade(R+,M )

the depthofM . Recallthatwe havedepth(M )≤ dim(M) (wheredim(M ):= dim(R/AnnR(M )) denotes

theKrull dimension),and M iscalledCohen–Macaulay if equalityholds.

ByNoether-Normalization,R containsahomogeneoussystemofparameters(h.s.o.p.),i.e.,algebraically independent homogeneouselements a1,. . . ,an ∈ R such thatR isfinitely generated as amoduleover the

(polynomial)subalgebraA:= K[a1,. . . ,an].Notethatn= dim(R) isuniquelydetermined.Anysubsetofan

h.s.o.p.iscalledapartialh.s.o.p.(p.h.s.o.p.).IfR isalsoadomain,thenhomogeneouselementsa1,. . . ,ak ∈

R+ form ap.h.s.o.p. ifand onlyifheight(a1,. . . ,ak)= k,(see [13,Lemma1.5],[3, Theorem A.16]). Note

thatM isalsoanA-module,andfrom thegradedAuslander–Buchsbaumformula[7,Exercise19.8]weget that M isfree as anA-module if andonly ifits depthas an A-module isequal to dim(A).But sincethe depths of M as anA- andas anR-module are equal(see [6,Lemma3.7.2]or [3,Exercise 1.2.26]),this is also equivalenttotheconditionthatthedepthofM asanR-moduleisdim(R)= dim(A).Inother words,

M is freeas anA-moduleifandonlyifM is Cohen–Macaulayanddim(M )= dim(R),i.e., M is maximal Cohen–Macaulay.

Weincludethefollowingstandardfactsaboutdepthforthereader’sconvenience.

Lemma 1.(See[3,Proposition 1.2.9].) Assumethat I isahomogeneous nonzeroproperideal ofthegraded affine ring R.Thenwe havethefollowinginequalities.

(a) depth(R)≥ min{depth(I),depth(R/I)}. (b) depth(I)≥ min{depth(R),depth(R/I)+ 1}.

Thislemma impliesthatdepth(I) anddepth(R/I) areoftenstronglyrelated:

Lemma2.Assume thatI isa homogeneousnonzeroproperideal ofthegradedaffine ringR.

(a) Ifone ofthefollowingconditions

(i) depth(R)> depth(I),

(ii) depth(R)> depth(R/I),

(iii) R isaCohen–Macaulay domain holds,then

depth(I) = depth(R/I) + 1. (b) Ifdepth(I)> depth(R),then depth(R/I)= depth(R).

(c) Ifdepth(R/I)> depth(R),then depth(I)= depth(R).

Proof. (a)Assumefirstdepth(R)> depth(I).FromLemma 1(b)itfollowsthatdepth(I)≥ depth(R/I)+1, and from Lemma 1 (c) we get depth(R/I) ≥ depth(I)− 1, implying the desired equality. Secondly, as-sume depth(R)> depth(R/I). From Lemma 1 (b) it follows that depth(I) ≥ depth(R/I)+ 1, and from

Lemma 1 (c) we have depth(R/I) ≥ depth(I)− 1 so we obtain the result again. Finally assume R is a Cohen–Macaulay domain.As R is adomain and I = {0}, it follows thatdim(R/I)< dim(R). Hencewe havetheinequalitydepth(R/I)≤ dim(R/I)< dim(R)= depth(R),sotheassertionfollowsfrom (ii).

Statement (b) follows similarly from Lemma 1 (a) and (c). Statement (c) follows from Lemma 1 (a) and (b). 2

Forexample,ifR haspositivedepth,thenthehomogeneousmaximalidealalwayshasdepthonebythe abovelemma,asitsquotientiszero-dimensional.Nowwenotethatforanygivennumber1≤ k ≤ depth(R), thereexists anidealofdepthk:

Lemma3.Assumethatthehomogeneouselementsa1,. . . ,ak ofpositivedegreeformaregularsequenceof R.

Then

depth((a1, . . . , ak)R) = depth(R) + 1− k.

Proof. We have depth(R/(a1,. . . ,ak)R) = depth(R)− k < depth(R), and the result follows from the

previouslemma. 2

3. Cohen–Macaulay idealsinaffinedomains

ThemainresultofthissectionisTheorem 5whereitisshownthatonlyprincipalidealscanbeCohen– Macaulayinfactorialaffinedomains.Nevertheless,evenwhentheaffinedomainisnotfactorial,theheight ofaCohen–Macaulayidealcanbe atmostone.

Lemma 4.Assume that R isa graded affine domain, andI = R a homogeneous idealof heightat least 2.

ThenI isnot Cohen–Macaulayasan R-module.

Proof. AsI ishomogeneousofheightatleasttwo,itcontainsap.h.s.o.p.p,q ofR.Weextendthisp.h.s.o.p. toanh.s.o.p.h1,. . . ,hnwithh1= p,h2= q andconsidertheK-subalgebraA ofR generatedbythish.s.o.p.,

i.e., A = K[h1. . . ,hn]. Then A is isomorphic to a polynomialring over K in dim(R) variables. Assume

there exist elements g1,. . . ,gm∈ I suchthatI =

m

i=1Agi.As p,q areelementsof I,wecanfindunique

elementsai,bi ∈ A fori= 1,. . . ,m suchthatp=mi=1aigiandq =mi=1bigi.Multiplyingbothequations

with q andp respectively, wegetm_i=1(qai)gi = pq =mi=1(pbi)gi.Asqai,pbi∈ A andg1,. . . ,gmisafree

A-basisofI,wegetqai= pbi foralli= 1,. . . ,m.Asp,q aredifferentvariablesinthepolynomialringA,it

follows p|ai andq|bi foralli.Thereforethereexistbi ∈ A suchthatbi= qbi fori= 1,. . . ,m.Henceweget

q =m_i=1bigi=i=1m qbigi,andasweareinadomaindividingbyq yields1=mi=1bigi∈mi=1Agi= I.

This impliesI = R,contradictingthehypothesisI= R of thelemma. 2

Theorem 5. Assume that R is agraded factorial affine domain. If I = R is a homogeneous idealwhich is notprincipal,thenI isnotCohen–MacaulayasanR-module.Therefore,ifR isnotCohen–Macaulay,then R does not containany nonzerohomogeneousCohen–Macaulayideal (asanR-module).

Proof. AssumebywayofcontradictionthatI isCohen–Macaulayandnotprincipal.Leta1,. . . ,an denote

afiniteset ofgeneratorsofI.Asweareinafactorialring,we canconsider thegreatestcommon divisord

of thoseelements.Then wehaveI = (a1,. . . ,an) (d), wheretheinclusionis strictasI isnotaprincipal

idealbyassumption.AsR isadomainandd| ai foralli,theelements a_di ∈ R arewelldefined,andwecan

consider theideal J := (a1 d,. . . ,

an

d) =

1

dI. Note thatfrom I  (d) it follows thatJ  (1) = R,so J is a

properidealofR.Multiplicationbyd yieldsanR-moduleisomorphismfromJ toI,andthereforeJ isalso Cohen–Macaulay as anR-module.From Lemma 4 itfollows thattheheight ofJ is at most1.ButR isa domain and J = 0,so the height ofJ is 1.It follows thatthere exists aprimeideal ℘ of R ofheight one suchthatI⊆ ℘.AsR isfactorial,heightoneprimesareprincipal,andso℘ isgeneratedbyaprimeelement

p,so wehaveJ ⊆ ℘= (p),whichimpliesthatp isacommondivisorof a1 d,. . . ,

an

d .Thisisacontradiction,

as d isthegreatestcommondivisorofa1,. . . ,an.

NowthesecondassertionofthetheoremfollowsfromthefirstandthefactthatprincipalidealsofR are

isomorphicto R asR-modules. 2

We demonstrate two examplesof affine domains with non-principal Cohen–Macaulay ideals. First one is aCohen–Macaulay ring, the second one is not. Therefore, anon-Cohen–Macaulay ring may contain a Cohen–Macaulay ideal, and the hypothesis of R being factorial can not be dropped out in the previous theorem.

Example6.ConsiderthesubalgebraR = K[x2,y2,xy] ofthepolynomialringK[x,y] intwovariables.Note that R is not factorial as the equality x2_{· y}2 _{= (xy)· (xy) shows. We claim that the ideal I = (x}2_,xy)

of R is Cohen–Macaulay and not principal. Clearly I is not principal, because R is a graded ring that starts in degree2. Wenow consider the h.s.o.p.x2_{, y}2 _{of R andthe subalgebra A = K[x}2_,y2_{] generated}

by theh.s.o.p..We claimthatwehavethe directsumdecompositionsR = A⊕ Axy andI = Ax2⊕ Axy.

In bothcases, thesumis directbecauseinthe firstsummands allx degrees areeven, whileinthe second summands all x degrees are odd. As both sums contain the respective ideal generators, it only remains to show thatboth sums are invariantunder multiplication with xy.For the sumfor R,this follows from

xy· xy = x2y2∈ A.ForthesumforI,wehavexy· x2= x2· xy ∈ Axy andxy· xy = y2· x2∈ Ax2.Therefore

R andI arefreeA-modules,soR andI areCohen–Macaulayas R-modules.AlsonotethatI⊆(x2_),as

(xy)2_{= y}2_{· x}2_{∈ (x}2_{).Thus,height(I)≤ height((x}2_{))= 1,as predictedbyLemma 4.}

Westatethefollowingexampleasaproposition.

Proposition7.ConsiderthesubalgebraR := K[x4_,x3_y,xy3_,y4_{] ofthepolynomialringK[x,y].Thentheideal}

I := (x4_,x3_{y) ofR isofheightoneandCohen–MacaulayasanR-module.(WhileR isnotCohen–Macaulay}

Proof. FirstnotethatR iswellknowntobenon-Cohen–Macaulay,see[6,Example2.5.4],andasx4_{· y}4₌

(x3y)· (xy3),R isalsonotfactorial.Alsosince(x3y)4= y4x8· x4∈ (x4) wehaveI⊆(x4_),whichshows

thattheheightof I is1.Weconsiderthesubalgebra A:= K[x4,y4] of R generatedbyanh.s.o.p.,andwe willshow thatI isafreeA-module, whichimpliesthatI isCohen–MacaulayasanR-module.Weset

a := x4, p := x3y, q := xy3, b := y4

andclaimthat

I = (a, p) = Aa⊕ Aaq ⊕ Ap ⊕ Ap2.

Theinclusion“⊇”isclear.Wefirstshowthatthesumontherighthandsideisindeeddirect.Let denote

themapfromthesetofmonomialsofK[x,y] toN2

0givenby(xiyj):= (i,j).Wecomputetheepsilonvalues

oftheA-modulegeneratorsmodulo4:

(a) = (4, 0), (aq) = (5, 3)≡ (1, 3), (p) = (3, 1), (p2) = (6, 2)≡ (2, 2).

As(m)≡ (0,0) forany monomialm inA, itfollows thatthe-valuesofmonomialsinAa, Aaq,Ap,Ap2

fallinto differentcongruenceclassesmodulo4,sothesumisindeeddirect.

We now verify the inclusion “⊆”. Clearly, a,p ∈ S := Aa⊕ Aaq ⊕ Ap⊕ Ap2_{, and S is closed under}

multiplication witha andb.It remainsto show thatS is closed undermultiplication withp and q, which followsfrom

p(Aa) = Aap⊆ Ap, q(Aa) = Aaq,

p(Aaq) = Aax4_y4_{⊆ Aa, q(Aaq) = Ax}6_y6_{= Abp}2_{⊆ Ap}2_,

p(Ap) = Ap2_{, q(Ap) = Apq = Ax}4_y4_{⊆ Aa,}

p(Ap2) = Ax9y3= Aa2q⊆ Aaq, q(Ap2) = Ax7y5= Ax4y4p⊆ Ap. 2

Remark8.WelearnedfromRogerWiegandthattherearetheoremsthatsaythat,forsomespecialclasses ofrings,non-freemaximalCohen–Macaulaymoduleshavehighranks.Sincetherankofanidealinadomain isone,and non-principalidealsare non-free,Lemma 4 andTheorem 5 readilyfollow forsuchringswhose non-free maximal Cohen–Macaulay modules areknown to havea high rank. Butwe can notexpectthat a non-free maximal Cohen–Macaulay module will always have rank > 1 as the previous two examples demonstrate.

Wenotetwo applicationsofTheorem 5tomodularinvariantrings.

Corollary9. Assumethat K isof positivecharacteristic p andG isa finitep-group. Forany finite dimen-sionallinear representationV of G overK suchthat theinvariantringK[V ]G _{isnotCohen–Macaulay,no}

nonzerohomogeneousideal ofK[V ]G _{isCohen–Macaulay(as aK[V ]}G_-module).

Proof. ItiswellknownthatK[V ]G _{isfactorial,see forinstance[5,Theorem 3.8.1].Theclaimnowfollows}

from Theorem 5. 2

ThetransferidealIG _{isdefinedastheimageof thetransfermapTr,i.e.}

IG= Tr(K[V ]), with Tr : K[V ]→ K[V ]G, f → Tr(f) = 

σ∈G

Corollary 10. Assume that K is of positive characteristic p and G is a finite p-group. Then thefollowing are equivalent.

(1) K[V ]G isadirectsummand of K[V ] as agradedK[V ]G-module.

(2) IG isaprincipal idealof K[V ]G.

(3) IG _{isCohen–Macaulay.}

Proof. Theequivalenceofthefirsttwostatementsisestablishedin[2,Corollary4].Assumenowthatoneand hencebothofthemhold.Itiswellknownthatfrom(1)itfollowsthatK[V ]G_{isCohen–Macaulay.Theideal}

IG _{isprincipalby(2),henceI}G _{is alsoCohen–Macaulay.Conversely, assumethatI}G _{isCohen–Macaulay.}

Then sinceK[V ]G _{isfactorial,Theorem 5appliesandsoI}G _{isprincipal. 2}

4. Depthofidealsandquotientofthetransferininvariantringsforacyclicgroupofprime order

InthissectionwespecializetoacyclicgroupG ofprimeorderp equaltothecharacteristicofthefield K, which we assume to be algebraically closed. Fixa generatorσ of G. There are exactly p indecomposable G-modulesV1,. . . ,VpoverK andeachindecomposablemoduleViisaffordedbyaJordanblockofdimension

i with 1’sonthediagonal. LetV beanarbitrary G-moduleoverK. AssumethatV has l summands and so wecanwriteV =_1≤j≤lVnj.Noticethatl = dim V

G_{.Wealsoassumethatnoneofthesesummandsis}

trivial,i.e.,nj > 1 for1≤ j ≤ l.WesetK[V ]= K[xi,j | 1≤ i≤ nj, 1≤ j ≤ l] andtheactionofσ isgiven

byσ(xi,j)= xi,j+ xi−1,j for1< i≤ nj andσ(x1,j)= x1,j.Wedefinethenorm

N (f ) := 

τ∈G

τ (f ) for all f ∈ K[V ].

Notice that for 1 ≤ i ≤ nj, N (xi,j) is monic of degree p as a polynomial in xi,j. For simplicity we set

Nj := N (xnj,j) for 1 ≤ j ≤ l. By a famous theorem of Ellingsrud and Skjelbred [8], depth(K[V ]

G_{) =}

min{dimK(VG)+ 2,dimK(V )}= min{l + 2,dimK(V )}.In[4],thisresultisextendedto someotherclasses

ofgroups,andtheproofisalsomademoreelementaryandexplicit.Restrictingtheresultsof[4]toourcase, wegetthefollowingdescriptionofamaximalK[V ]G_{-regularsequence,whichallowstoexplicitlyconstruct}

anidealofagivendepthat mostthatoftheinvariantring. Proposition 11.A maximal K[V ]G_{-regularsequenceisgivenby}

x1,1, x1,2, N1, . . . , Nl if l > 1;

x1,1, N (x2,1), N1 if l = 1, n1> 2;

x1,1, N1 if l = 1, n1= 2.

LetIk denotetheidealofK[V ]Ggeneratedbythefirstk elementsofthesequence.Thenwehavedepth Ik=

depth(K[V ])G+ 1− k for 1≤ k ≤ depth(K[V ]G).

Proof. Let b denote the second element of the sequence, i.e., x1,2 or N (x2,1) = xp2,1 − x2,1xp−11,1 . As x1,1

and b are coprime in K[V ], and both are invariant, they form aregular sequence in K[V ]G_{. Proceeding}

by induction, we assume thatthe elements x1,1,b,N1,. . . ,Nk−1 form aregular sequence for somek < l.

Consider the standard basis vector enk,k ∈ V corresponding to the variable xnk,k. Then enk,k is a fixed

point, and U := Kenk,k is a 1-dimensional submodule of V . Since no element of the regular sequence x1,1,b,N1,. . . ,Nk−1 contains the variable xnk,k, [4, Corollary 17] applies to U and xnk,k, so the regular

sequence canbe extendedbytheelement Nk.Since thelengthofthegiven sequenceequalsdepth(K[V ]G)

The transfer ideal often plays an important role in computing the invariant ring and its various as-pects have been subject to research. The vanishing set of IG _{equals the fixed point space V}G _{(see [5,}

Theorem 9.0.10]), inparticularwe havedim(K[V ]G/IG)= dim(VG)= l. Wewill showthatK[V ]G/IG is Cohen–Macaulay,whichalsoallowsustocomputethedepthofthetransferideal.Todothisweprovethat

N1,. . . ,Nl is aK[V ]G/IG-regular sequence. Let f ∈ K[V ] and 1≤ j1 < j2 <· · · < jt ≤ l be arbitrary.

We denote thedegreeof f as apolynomialinxnj,j by degjf . Since Nj1 is amonic polynomialof degree

p in xn_j1,j1, we can write f = q1Nj1 + r1, where degj1r1 < p. Next we divide r1 by Nj2 and we get a

decompositionf = q1Nj1+ q2Nj2+ r2,where degj1r2,degj2r2 < p anddegj1q2< p.Inthis waywe geta

decomposition

f = q1Nj1+· · · + qtNjt+ r,

where deg_jir < p for1≤ i≤ t and deg_jiqi < p for i< i.This iscalled thenormdecomposition andr is

calledtheremainderoff withrespecttoNj1,. . . ,Njt.Noticethatr isunique.Iff ∈ K[V ]

G _{isaninvariant,}

then the quotients qi for 1≤ i ≤ t and the remainder r are also invariant, see [16, Proposition 2.1]. We

denotethecosetofanelementf ∈ K[V ]G inK[V ]G/IG byf .

Theorem12.ThealgebraK[V ]G_/IG_{isCohen–Macaulay,andanh.s.o.p.isgivenbytheset{N}

j | 1≤ j ≤ l}.

In particular,wehave depth(IG)= l + 1.

Proof. WeshowthatN1,. . . ,NlformsaregularsequenceforK[V ]G/IG.Asitslengthl equalsthedimension

of K[V ]G_/IG_{, it followsthatthis ring isCohen–Macaulay. First,we show thatN}

i is aK[V ]G/IG-regular

element for 1 ≤ i ≤ l. Assume f Ni ∈ IG for some invariant f . Then f Ni = Tr(g) for someg ∈ K[V ].

Considerthenormdecompositiong = qNi+ r ofg withrespectto Ni.Wehave

f Ni= Tr(qNi+ r) = Tr(q)Ni+ Tr(r).

Hence,

0 = (f− Tr(q))Ni+ Tr(r).

Notethatthegroupactionpreserves thexni,i-degree,sowehave

deg_iTr(r)≤ deg_ir < p = deg_iNi.

So, we getthat f− Tr(q) = 0 andTr(r) = 0.Therefore f ∈ IG_{, and N}

i is aK[V ]G/IG-regular element.

AssumenowbyinductionthatN1,. . . ,Nj−1 isaK[V ]G/IG-regularsequence,andwe have

f Nj= f1N1+· · · + fj−1Nj−1+ Tr(t), (1)

where f,fi ∈ K[V ]G for1≤ i ≤ j − 1 andt ∈ K[V ].Considerthe norm decompositionsof f andt with

respect to N1,. . . ,Nj−1. Since the quotients and the remainderin thedecomposition of f areinvariants,

we can replace f byits remainder. As for Tr(t),notice that Tr(t) and the transferof the remainder of t

differbyaK[V ]G_{-linearcombination ofN}

1,. . . ,Nj−1.Therefore,we canreplaceTr(t) withthetransferof

the remainder of t. Moreover, byconsidering the norm decomposition of fi with respect to N1,. . . ,Ni−1

for 1 ≤ i ≤ j − 1, we can replace fi with its corresponding remainder. Therefore, we may assume that

deg_ifi < p for1≤ i < i and 1≤ i ≤ j − 1. Notice alsothat thedegree of f and Tr(t) with respect to

any variable xni,i is < p for 1≤ i

 _{≤ j − 1. Now, considering Equation(1) as a polynomialequation in}

way we get f1 = f2 = · · · = fj−1 = 0. So, Equation (1) becomes f Nj = Tr(t). However, since Nj is a

K[V ]G_/IG_{-regular element,we havef ∈ I}G _{as desired.This shows thatN}

1,. . . ,Nl is aregular sequence.

From depth(K[V ]G/IG)= l < depth(K[V ]G)= min{l + 2,dimK(V )} (weassumeanon-trivialaction)and

Lemma 2,itnowfollowsthatdepth(IG)= l + 1. 2

Wealsoproveareductionresultforthedepthofamoduleovertheinvariantring,whichisbasedonthe following lemma. Thestatementis probablyfolklore,butfortheconvenienceof thereaderandthelackof areference,weprovide aproof.

Lemma 13. Assumethat R isagraded affinering andM is a finitely generatedgraded nonzeroR-module. If h1,. . . ,hr ∈ R+ form ahomogeneous M -regular sequence andI is ahomogeneous ideal of R such that



I + (h1, . . . , hr)R = R+,then

depth(M ) = grade(I, M/(h1, . . . , hr)M ) + r.

Proof. As the homogeneous elements h1,. . . ,hr ∈ R+ form an M -regular sequence, we have that

depth(M )= depth(M/(h1,. . . ,hr)M )+ r. Weshow that

grade(I, M/(h1, . . . , hr)M )≥ grade(R+, M/(h1, . . . , hr)M ),

asthereverseinequalityisobvious.Letf1,. . . ,fd∈ R+beamaximalhomogeneousM/(h1,. . . ,hr)M -regular

sequence. Since taking powersdoes not change the property of being aregular sequence, we canassume thatallelements inthesequenceare containedinI + (h1,. . . ,hr)R.Thereforefor 1≤ i≤ d wecanwrite

fi= gi+ bi withhomogeneouselementsgi∈ I andbi∈ (h1,. . . ,hr)R.Since bi isintheannihilatorof

M/(h1, . . . , hr, f1, . . . , fi−1)M = M/(h1, . . . , hr, g1, . . . , gi−1)M

itfollowsthatgiisregularonM/(h1,. . . ,hr,g1,. . . ,gi−1)M aswell.Hencetheelementsg1,. . . ,gdofI form

anM/(h1,. . . ,hr)M -regularsequence. 2

WerecallthatforanyidealI oftheinvariantringK[V ]G,wehave√I =IK[V ]∩ K[V ]G.This holds generallywhenG isareductivegroup[15, Lemma3.4.2],and anelementaryproofforfinite groupscanbe found in[5,Lemma12.1.1].

Proposition 14. Let M be a finitely generated gradedK[V ]G-module on which thenorms N1,. . . ,Nl form

an M -regularsequence. Then

depth(M ) = grade(IG, M/(N1, . . . , Nl)M ) + l.

Proof. WehavealreadymentionedthatthezerosetofIGisgivenbyVG =l_i=1Keni,i.Asforanelement v = l_i=1λieni,i ∈ V

G _{with λ}

i ∈ K, we have Ni(v) = λpi, the common zero set of IG+ (N1,. . . ,Nl) is

zero, hence by the Nullstellensatz (IG_{+ (N}

1, . . . , Nl))K[V ] = K[V ]+. From the paragraph before the

proposition we obtain that the radical ideal of IG _{+ (N}

1,. . . ,Nl) equals K[V ]G+, and the lemma above

applies. 2

Examples wherethepropositionappliesincludethecasel = 1 andM = I anonzerohomogeneous ideal ofK[V ]G_{.Thecorollaryalsoappliesforarbitraryl andM = K[V ]}G_{byProposition 11.Inthe“non-trivial”}

caseswheredepth(K[V ]G_{)= l + 2,itfollowsfromdepth(K[V ]}G_{)= grade(I}G_{,K[V ]}G_/(N

1,. . . ,Nl))+ l,that

Therefore, there is a maximal K[V ]G_{-regular sequence consisting of the l norms and two transfers. Also}

comparewiththeknownfactthatgrade(IG_{,K[V ]}G_{)= 2 inthese cases,see[4,Propositions20and22].As}

depth(K[V ]G)= l + 2,this alsoshowsthatdepth(M )= grade(IG,M ) ingeneral. Acknowledgements

Wethank Tübitakfor fundingavisit ofthe firstauthor to Bilkent University,and Gregor Kemper for inviting thesecond author to TUMünchen. We alsothank theanonymous referee ofthis paper formany remarksthatimprovedtheexpositionofthepaper,inparticularforsuggestingtoincludeCorollary 10inthe paper.SpecialthanksgotoRogerWiegandforpointingouttousthatnon-freemaximalCohen–Macaulay moduleshavehighranksforsomerings.Finally,wethankFabianReimersformanyinspiringconversations, andparticularlyfordrawingourattentiontosomerelationsbetweenthedepthofanidealanditsquotient. References

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