ORDINAL LENGTH AND THE CANONICAL TOPOLOGY HANSSCHOUTENS 3 1 0 ABSTRACT. Weextendtheclassicallengthfunctiontoanordinal-valuedinvariantonthe 2 classofallfinite-dimensionalNoetherianmodules.Weshowhowtocalculatethiscombi- n natorialinvariantbymeansofthefundamentalcycleofthemodule,thuslinkingthelattice a ofsubmodulestohomologicalpropertiesofthemodule.Usingthis,wedefineonamodule J itscanonicaltopology,inwhicheverymorphismiscontinuous. 8 2 ] 1. INTRODUCTION C A Thepurposeofthispaperistolaythefoundationsofanew,ordinal-valuedinvariantin CommutativeAlgebra:the(ordinal)lengthlen(M)ofaNoetherianmoduleM,measuring . h (by meansof an ordinal)the longestdescendingchain of submodulesin M. Recall that t a anordinalisanisomorphismclassofatotalwell-order(=admittingthedescendingchain m condition),andtheclassofordinals,orderedbytheinitialsegmentrelation,isagainawell- [ order;eachnaturalnumberisanordinalbyidentifyingitwithachainofthatlength;theor- derof(N,≤)isdenotedωandisequaltothesupremumofalln∈N;theorder-typeofthe 1 lexicographicalorderon Nd isdenotedωd. Apartfromtheusual(non-commutative)sum v 7 α+β,wealsoneedthe(commutative)shuffleornaturalsumα⊕β givenbycoefficient- 5 wiseadditionintheCantornormalform(fordetails,see §2). Sincedescendingchainsin 4 a Noetherian module M are well-ordered with respect to the reverse inclusion, they are 6 ordinals,andwedefinelen(M)asthesupremumofallordinalsobtainedthisway. . 1 In[8]weshowedthat,whereaslengthcannolongerbeadditiveonexactsequences,it 0 isstillsemi-additive: 3 1 Theorem(Semi-additivity). If0→N →M →Q→0isanexactsequenceofNoether- : ianR-modules,then v i (1) len(Q)+len(N)≤len(M)≤len(Q)⊕len(N). X r Moreover,ifthesequenceissplit,thenthelastinequalityisanequality. a As an application, we obtain that a Noetherian ring has length ωd if and only if it is a domain of Krull dimension d. In this paper, we prove the remarkable fact that this combinatorial invariant can be described in terms of homological invariants: recall that the local multiplicity e (M) at a prime ideal p is defined as the length of the p-torsion p submoduleofM (whenceisnon-zeroifandonlyifpisanassociatedprimeofM). p MainTheorem. ThelengthofM isequalto e (M)ωdim(R/p). Lp p Combiningthiswithsemi-additivity,leadstomanyinterestingapplications([8,6,7]). Inthispaper,weapplythetheorytodefineacanonicaltopologyoneverymodule:theopen submodulesinthistopologyarepreciselythosethathavethesamelengthasM. Weshow Date:January29,2013. Key words and phrases. Commutative Algebra; ordinal length; local cohomology; Chow group; local multiplicity. 1 2 HANSSCHOUTENS thatanymorphismbetweenmodulesiscontinuousinthistopology. Anyopensubmodule is essential, and in [7], we will characterize those modules for which the converse also holds. In the last section, as an applicationof this material, we discuss the phenomenon of degradation: how source and target of a module may force the morphism to become (almost)zero.Forinstance,weshow Theorem. If an endomorphism on M factors through a module that has no associated primesincommonwithM,thenitmustbenilpotent. 2. ORDINALS AND ORDINAL LENGTH Apartialorderingiscalleda (partial)well-order ifithasthedescendingchaincondi- tion, that is to say, any descendingchain must eventuallybe constant. A total orderis a well-orderifandonlyifeverynon-emptysubsethasaminimalelement.Anordinalisthen anequivalenceclass,uptoanorder-preservingisomorphism,ofatotalwell-order.Theset of ordinals is a transfinite extension of the set of natural numbers N, in which the usual inductionisreplacedbytransfiniteinduction.Wesaythatα≤β ifαcanbeembeddedas atotalorderinβ. Anyboundedsubsetofordinalshasthenaninfimumandasupremum. The(Cantor)sumα+β istheordinalcorrespondingtothewell-orderon α⊔β obtained bylettinganyelementofβ belargerthananyelementofα. Thus,1+ωisthesameasω, whenceinparticulardifferentfrom ω+1. Wewillnotneedarbitraryordinalmultiplica- tion,butonlyproductsoftheformnαwithn ∈N,simplydefinedasthesumofncopies of α (be aware that logicianswould use the more awkwardnotation α·n forthis). The supremumofthenωisdenotedω2andistheorder-typeofthelexicographicalorderingon N2. Theωdaresimilarlydefined,andtheirsupremumisdenotedωω. LetObethecollectionofordinalsstrictlybelowωω. Anyα ∈ OhasauniqueCantor normalform (2) α=a ωd+···+a ω+a d 1 0 with a ∈ N, called its Cantor coefficients. The support of α, denoted Supp(α) ⊆ N, n consistsof all i forwhich a 6= 0. The maximumandminimumof the supportof α are i called respectivelyits degree and order. An ordinalis a successor ordinal(=of the form α+1forsomeordinalα) ifandonlyif itsorderis zero. Thesumofall a iscalledthe i valenceofα. Theordering≤onordinalscorrespondstothelexicographicalorderingon thetuplesofCantorcoefficients(a ,...,a ). Letussaythatαisweakerthanβ,denoted d 0 α (cid:22) β, if a ≤ b foralli, where,likewise, theb aretheCantorcoefficientsof β. Note i i i that≤extendsthepartialorder(cid:22)toatotalorder. Apart from the usual ordinal sum, we make use of the natural or shuffle sum α⊕β giveninCantornormalformas (a +b )ωd +···+a +b . Notethattheshufflesum d d 0 0 iscommutative,andα+β willin generalbesmallerthan α⊕β. Infact, weshowedin [8, Theorem 7.1] that the shuffle sum is the largest possible ordinal sum one can obtain fromwritingbothordinalsasasumofsmallerordinalsandthenrearrangingtheirterms. Inparticular,Oisclosedunderbothadditions.Moreover,sinceα(cid:22)β ifandonlyifthere existsγ suchthatα⊕γ =β,wemayview(O,⊕,(cid:22))asapartiallyorderedcommutative semi-group. 3. LENGTHAND SEMI-ADDITIVITY Allringswillbe commutative,Noetherian,of finiteKrulldimension,andallmodules will be finitely generated. Throughout, if not specified otherwise, R denotes a (finite- dimensional, Noetherian) ring and M denotes some (finitely generated) R-module. By ORDINALLENGTHANDTHECANONICALTOPOLOGY 3 Noetherianity,thecollectionofsubmodulesofM orderedbyinverseinclusionisapartial well-order,calledtheGrassmanianofM anddenotedGr (M)(orjustGr(M)). Inpar- R ticular,anychaininGr (M)is(equivalentto)anordinal. Thesupremumofallpossible R ordinalsarisingasachaininthiswayiscalledthelengthofM andisdenotedlen (M),or R whenthebaseringisclear,simplybylen(M). NotethatthelengthofM asanR-module isthesameasthatofanR/Ann (M)-module,andsowemayassume,ifnecessary,that R M isfaithful. ItfollowsfromtheJordan-Ho¨ldertheorythat thisordinallengthcoincides withtheusuallengthformodulesoffinitelength. Thelengthofaringisthatofamodule overitself. Inotherwords,len(R)isthelongestdescendingchainofidealsinR. Itisusefultohavealsoatransfinitedefinitionoflength:wedefineaheightrankl(·)on Gr (M),asfollows. Putl(M) := 0. GivenasubmoduleN ⊆ M,atasuccessorstage, R wesaythatl(N) ≥ α+1, if thereexistsa submoduleN′ ⊆ M containingN suchthat l(N′) ≥ α. Ifλisalimitordinal(thatistosay,notasuccessorordinal),thenwesaythat l(N)≥λ,providedforeachα<λ,thereexistsasubmoduleN ⊆M containingN with α l(N )≥α. Finally,wesaythatl(N)=αifl(N)≥αbutnotl(N)≥α+1.Weprovein α [8,Theorem3.10]thattheheightrankofthezeromoduleisthelengthofM.Infact,more generally,foranysubmoduleN ⊆M,itsheightrankequalsitsco-length,thatistosay, (3) l(N)=len(M/N). Notethatheightranksatisfiesthefollowingcontinuityproperty:l(N)islessthanorequal tothesupremumofalll(W)+1,forallW strictlycontainingN. Usingsemi-additivity (seeintroduction)weshowedin[8]: 3.1.Theorem(Dimension). LetM beafinitelygeneratedmoduleoverafinite-dimensional NoetherianringR. Thenthedegreeoflen(M)isequaltothedimensionofM. Inpartic- ular,Risad-dimensionaldomainifandonlyiflen(R)=ωd. (cid:3) The order of a module is by definition the order of its length, and will be denoted ord(M); the valence val(M) is the valence of its length. We will calculate these two invariantsinCorollary4.5below. Letuscallanexactsequencestronglyequilateralifwe haveequalityatbothsidesof(1);ifweonlyhaveequalityattheright,wecallthesequence equilateral. Beingstronglyequilateralisreallyapropertyofordinals: α+β = α⊕β if andonlyifthedegreeofβ isatmosttheorderofα,andhence 3.2.Corollary. AnexactsequenceoffinitelygeneratedR-modules 0→N →M →Q→0, isstronglyequilateralifandonlyifdimN ≤ord(Q). (cid:3) 4. LENGTH ASA COHOMOLOGICAL RANK Ascustomary,wedefinethedimensionofaprimeidealp⊆R,denoteddim(p),asthe KrulldimensionoftheresidueringR/p.Wedenotethecollectionofallassociatedprimes (= prime ideals of the form Ann(a) with a ∈ M) by Ass(M); it is always a finite set. We willmakefrequentuse, fora shortexactsequence 0 → N → M → Q → 0, ofthe followingtwoinclusions(see,forinstance,[2,Lemma3.6]) (4) Ass(N)⊆Ass(M); (5) Ass(M)⊆Ass(N)∪Ass(Q) 4 HANSSCHOUTENS We define the the (zero-th) local cohomology1of M at p, denoted H0(M), as the p- p torsionofM ,thatistosay,thesubmoduleofM consistingofallelementsthatarekilled p p bysomepowerofp. AsH0(M)isamoduleoffinitelengthoverR ,wedenotethislength p p bye (M) and call itthe localmultiplicity ofM atp (see, forinstance, [2, p. 102]). An p alternativeformulationisthroughthenotionofthefinitisticlengthofamoduleM,defined asthesupremumlenfin(M)ofalllen(N)withN ⊆Mandlen(N)<ω.ByNoetherianity, M hasalargestsubmoduleH offinitelength,andhencelenfin(M) = len(H). Withthis notation,wehave (6) e (M)=lenfin(M ), p p foranyprimeidealp,andthisisnon-zeroifandonlyifpisanassociatedprimeofM. We nowdefinethecohomologicalrankofamoduleM as coh(M):=Mep(M)ωdim(p). p It is instructiveto view this fromthe pointof view ofChow cycles. Let A(R) be the ChowringofR, definedasthefreeAbeliangroupon Spec(R). AnelementD ofA(R) will be called a cycle, and will be represented as a finite sum a [p ], where [p] is the P i i symboldenotingthefreegeneratorcorrespondingtotheprimeidealp. Thesumofalla i iscalledthedegreedeg(D)ofthecycleD. WedefineapartialorderonA(R)bytherule thatD (cid:22)E,ifa ≤b ,foralli,whereE = b [p ].Inparticular,denotingthezerocycle i i P i i simplyby0,wecallacycleD effective,if0 (cid:22) D,andweletA+(R)bethesemi-group of effective cycles. This allows us to define a map from effective cycles to ordinals by sendingtheeffectivecycleD = a [p ]totheordinal Pi i i o(D):=Maiωdim(pi). i Clearly,ifD andE areeffective,theno(D+E) = o(D)⊕o(E). Moreover,ifD (cid:22) E, theno(D)(cid:22)o(E),sothatwegetamap(A+(R),+,(cid:22))→(O,⊕,(cid:22))ofpartiallyordered semi-groups. ToanyR-moduleM,wecanassignitsfundamentalcycle,bytherule (7) cycR(M):=Xep(M)[p] p Immediately from (6) we get o(cyc(M)) = coh(M). Our main result now links this cohomologicalinvarianttoourcombinatoriallengthinvariant: 4.1.Theorem. ForanyfinitelygeneratedmoduleM overafinite-dimensionalNoetherian ringR,wehavelen(M)=o(cyc(M))=coh(M). Beforewegivetheproof,wederivetwolemmas. Itisimportanttonoticethatthefirst oftheseisnottrueatthelevelofcycles. 4.2.Lemma. IfM →QisapropersurjectivemorphismofR-modules,then coh(Q) d, showing thatcoh(Q) and coh(M) can only start differingat a coefficientof ωi for i ≤ d. So let 1Thisdeviatesslightlyfromthepracticein[2,App.4]aswealsolocalize. ORDINALLENGTHANDTHECANONICALTOPOLOGY 5 p∈Ass(M)∩Supp(N)havedimensiond.Ingeneral,localcohomologyisonlyleftexact, but by Lemma 4.3 below, we have in fact an exact sequence (8). Since e (N) 6= 0, we p mustthereforehavee (Q)

Ordinal length and the canonical topology

Similar Ordinal length and the canonical topology