The Steinberg group of a monoid ring, nilpotence,

arXiv:math/0601400v3[math.KT]7Mar2006THESTEINBERGGROUPOFAMONOIDRING,NILPOTENCE,ANDALGORITHMSJOSEPHGUBELADZEAbstract.ForaregularringRandanaffinemonoidMthehomothetiesofMactnilpotentlyontheMilnorunstablegroupsofR[M].ThisstrengthenstheK2partofthemainresultof[G5]intwoways:thecoefficientfieldofcharacteristic0isextendedtoanyregularringandthestableK2-groupissubstitutedbytheunstableones.Theproofisbasedonapolyhedral/combinatorialtechnique,computationsinSteinberggroups,andasubstantiallycorrectedversionofanoldresultonelementarymatricesbyMushkudiani[Mu].AsimilarstrongernilpotenceresultforK1andalgorithmicconsequencesforfactorizationofhighFrobeniuspowersofinvertiblematricesarealsoderived.1.Introduction1.1.Mainresult.Intherecentwork[G5]weprovedthefollowingresult.Letkbeafieldofcharacteristic0,MbeanadditivesubmonoidofZnwithoutnontrivialunits,andibeanonnegativeinteger.Thenforanyelementx∈Ki(k[M])andanynaturalnumberc≥2thereexistsanintegerjx≥0suchthat(cj)∗(x)∈Ki(k)forallj≥jx.HereforanaturalnumbercthegroupendomorphismofKi(k[M]),inducedbythemonoidendomorphismM→M,m7→mc,isdenotedbyc∗.Themotivationforthisresultisthatitisanaturalhigherversionofthetrivialityofalgebraicvectorbundlesonaffinetoricvarieties[G1],containsQuillen’sfunda-mentalresultonhomotopyinvariance,andeasilyextendstoglobaltoricvarieties.Seetheintroductionof[G5]forthedetails.Thisresultconfirmsthenilpotenceconjectureforaspecialclassofcoefficientsrings.TheconjectureassertsthesimilarnilpotencepropertyofhigherK-groupsofmonoidalgebrasoverany(commutative)regularcoefficientring.Themainresultinthispaperisastrongerunstableversionofthenilpotenceprop-ertyforthefunctorsK1,randK2,rforanyregularcoefficientring.Moreover,whenthecoefficientringisafieldtheargumentleadstoanalgorithmforfactorizationofhigh‘Frobeniuspowers’ofinvertiblematricesintoelementaryones.Inthespecialcaseofthepolynomialringsk[Zn+]=k[t1,...,tn]thealgorithmicstudyoffactorizationsofinvertiblematriceshasapplicationsinsignalprocessing[LiXW,PW].ThestartingpointhereisSuslin’swellknownpaper[Su].Inthisspe-cialcasethereisnoneedtotakeFrobeniuspowersofinvertiblematrices.However,aK-theoreticalobstructionshowsthatthisisnolongerpossibleonceweleavethe2000MathematicsSubjectClassification.14M25,19B14,19C09,20G35,52B20.12JOSEPHGUBELADZEclassoffreemonoids,seeRemark2.5.Therefore,ouralgorithmicfactorizationisanoptimal‘sparseversion’oftheexistingalgorithmforpolynomialrings.Hereisthemainresult:Theorem1.1.LetMbeacommutativecancellativetorsionfreemonoidwithoutnontrivialunits,c≥2anaturalnumber,Racommutativeregularringandkafield.Then:(a)Foranyelementz∈K2,r(R[M]),r≥max(5,dimR+3),thereexistsanintegerjz≥0suchthat(cj)∗(z)∈K2,r(R)=K2(R),j≥jz.(b)ForanymatrixA∈GLr(R[M]),r≥max(3,dimR+2),thereexistsanintegernumberjA≥0suchthat(cj)∗(A)∈Er(R[M])GLr(R),j≥jA.(c)ThereisanalgorithmwhichforanymatrixA=SLr(k[M]),r≥3,findsanintegernumberjA≥0andafactorizationoftheform:(cjA)∗(A)=Ykepkqk(λk),epkqk(λk)∈Er(k[M]).Here:foracommutativeringΛitsKrulldimensionisdenotedbydimΛ,K2,r(−)referstotheMilnor’srthunstableK2,foranaturalnumbercthegroupendomorphismsGLr(R[M])→GLr(R[M])andK2,r(R[M])→K2,r(R[M]),inducedbythemonoidendomorphismM→M,m7→mc,arebothdenotedbyc∗.fortwosubgroupH1andH2ofagroupGweusethenotationH1H2={h1h2|h1∈H1,h2∈H2}.Remark1.2.Wedonotgiveadetaileddescriptionoftheactualalgorithm,men-tionedinTheorem1.1(c).Rather,throughoutthetext,wehighlighttheexplicitnatureoftheproofofTheorem1.1(b)whichimpliesthepossibilityofconvertingtheargumentintoanimplementedalgorithmwhenthecoefficientsareinafield.Remark1.3.ItisnotdifficulttoshowthattheproofofTheorem1.1,givenbelow,worksforamoregeneralclassofringsofcoefficients.Infact,alloneneedsfromtheringRisthevalidityoftheclaims(a),(b)and(c)forthepolynomialextensionR[t1,...,tn],n=rankM–aclassicallyknownfactwhenRisaregularring,seeSection2.Remark1.4.Wedonotknowwhetherthereisauniformboundji,dependingonM,Randi,butnotonx∈Ki(R[M]),suchthat(cji)∗(x)∈Ki(R).Nontrivialexamplesin[G3]indicatethatsuchboundsmayinfactexist,atleastforK1.AwordisinorderonthepreviousresultsandtheproofofTheorem1.1.TheproofofthenilpotenceofKi(k[M])asgivenin[G5]–eveninthecaseofMilnor’sK2–usesaseriesofdeepfactsinhigherK-theoryofrings,obtainedfromSTEINBERGGROUPOFAMONOIDRING3theearly1990son(themostrecentofwhichis[Cor]).TheproofofTheorem1.1,givenbelow,makesnouseofanyoftheseresults.ItisbasedoncomputationsinEr(R[M]),essentiallyduetoMushkudiani[Mu],andsimilarcomputationsinStr(R[M]).Theexplicitnatureofthesecomputationsisalsothesourceofthealgo-rithmicconsequencesforSLr(k[M]).Obviously,nosuchapurealgebraicapproachispossibleforhigherK-groups.Actually,theweakerstableversionofTheorem1.1(a)forK2isclaimedin[Mu]andthepresentworkgrewupfromourattemptstounderstandMushkudiani’sargument.Eventually,whatsurvivedfrom[Mu]ishispreliminarycomputationsinthegroupofelementarymatrices–animportanttechnicalfactwhosecorrectedandstrongerunstableversionisgiveninthelastSection8;seeRemarks5.4,6.3and6.5.1TherestofthepaperisdevotedtothereductionofTheorem1.1tothistechnicalfact.Inthecourseoftheproofwealsodevelopaneffective/algorithmicexcisiontech-niqueforth