AlgebraicGroups
TheTheoryofGroupSchemesofFinite TypeoveraField
J.S.MILNE
UniversityofMichigan,AnnArbor
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bHopfalgebras ...........................65
cHopfalgebrasandalgebraicgroups ...............66
dHopfsubalgebras .........................67
eHopfsubalgebrasof O .G/ versussubgroupsof G ........68
fSubgroupsof G.k/ versusalgebraicsubgroupsof G .......68
gAffinealgebraicgroupsincharacteristiczeroaresmooth ....70
hSmoothnessincharacteristic p ¤ 0 ................72
iFaithfulflatnessforHopfalgebras ................73
jThehomomorphismtheoremforaffinealgebraicgroups ....74
kFormsofalgebraicgroups ....................76 Exercises ................................81
4LinearRepresentationsofAlgebraicGroups83 aRepresentationsandcomodules ..................83
bStabilizers .............................85
cRepresentationsareunionsoffinite-dimensionalrepresentations86 dAffinealgebraicgroupsarelinear .................86
eConstructingallfinite-dimensionalrepresentations .......88
hChevalley’stheorem ........................94
5GroupTheory;theIsomorphismTheorems98
´ etaleexactsequence
kTheisomorphismtheoremsforfunctorstogroups ........118
lTheisomorphismtheoremsforsheavesofgroups ........118 mTheisomorphismtheoremsforalgebraicgroups .........119 nSomecategorytheory .......................121 Exercises ................................122
6SubnormalSeries;SolvableandNilpotentAlgebraicGroups124
dThederivedgroupsandcommutatorgroups
7AlgebraicGroupsActingonSchemes138
8TheStructureofGeneralAlgebraicGroups148
9TannakaDuality;JordanDecompositions163
10TheLieAlgebraofanAlgebraicGroup186 aDefinition
eDescriptionoftheLiealgebraintermsofderivations
iAnexampleofChevalley
jTheuniversalenvelopingalgebra
kTheuniversalenveloping p -algebra
lThealgebraofdistributions(hyperalgebra)ofanalgebraicgroup207
11FiniteGroupSchemes209 aGeneralities
fStructureoftheunderlyingschemeofafinitegroupscheme..218 gFinitegroupschemesoforder n arekilledby
hFinitegroupschemesofheightatmostone
iTheVerschiebungmorphism
jTheWittschemes
12GroupsofMultiplicativeType;LinearlyReductiveGroups230
gClassificationofgroupsofmultiplicativetype
13ToriActingonSchemes254 aThesmoothnessofthefixedsubscheme
14UnipotentAlgebraicGroups279
15CohomologyandExtensions302
16TheStructureofSolvableAlgebraicGroups324
17BorelSubgroupsandApplications352
gChevalley’sdescriptionoftheunipotentradical
hProofofChevalley’stheorem ...................373 iBorelandparabolicsubgroupsoveranarbitrarybasefield ....375 jMaximaltoriandCartansubgroupsoveranarbitrarybasefield.376 kAlgebraicgroupsoverfinitefields ................382 lSplitalgebraicgroups
Exercises ................................385
18TheGeometryofAlgebraicGroups387 aCentralandmultiplicativeisogenies
bTheuniversalcovering ......................388 cLinebundlesandcharacters ....................389 dExistenceofauniversalcovering .................392 eApplications ............................393 fProofoftheorem18.15 ......................395 Exercises ................................396
19SemisimpleandReductiveGroups397 aSemisimplegroups ........................397 bReductivegroups
cTherankofagroupvariety
20AlgebraicGroupsofSemisimpleRankOne407 aGroupvarietiesofsemisimplerank0 ...............407 bHomogeneouscurves .......................408 cTheautomorphismgroupoftheprojectiveline ..........409 dAfixedpointtheoremforactionsoftori .............410 eGroupvarietiesofsemisimplerank1. ..............412 fSplitreductivegroupsofsemisimplerank1. ...........414 gPropertiesofSL2 .........................415
hClassificationofthesplitreductivegroupsofsemisimplerank1418 iTheformsofSL2 ,GL2 ,andPGL2 ................419 jClassificationofreductivegroupsofsemisimplerankone ....421 kReviewofSL2 ..........................422
Exercises ................................423
21SplitReductiveGroups424 aSplitreductivegroupsandtheirroots ...............424 bCentresofreductivegroups ....................427 cTherootdatumofasplitreductivegroup .............428 dBorelsubgroups;Weylgroups;Titssystems ...........433 eComplementsonsemisimplegroups ...............439 fComplementsonreductivegroups ................442
gUnipotentsubgroupsnormalizedby T
jTherootdataoftheclassicalsemisimplegroups
22RepresentationsofReductiveGroups463
aThesemisimplerepresentationsofasplitreductivegroup
bCharactersandGrothendieckgroups
cSemisimplicityincharacteristiczero
dWeyl’scharacterformula
23TheIsogenyandExistenceTheorems
gStatementoftheexistencetheorem;applications
hProofoftheexistencetheorem
24ConstructionoftheSemisimpleGroups
gThegeometricallyalmost-simplegroupsoftype C
jThegeometricallyalmost-simplegroupoftypes B and D
mThetrialitariangroups(groupsofsubtype 3D4 and 6D4 )....542 Exercises ................................542
25AdditionalTopics544
aParabolicsubgroupsofreductivegroups
cTheSatake–Titsclassification
AppendixAReviewofAlgebraicGeometry566
AppendixBExistenceofQuotientsofAlgebraicGroups586
AppendixCRootData607
Preface
Foronewhoattemptstounravelthestory,theproblems areasperplexingasamassofhempwithathousand looseends.
DreamoftheRedChamber,TsaoHsueh-Chin.
Thisbookrepresentsmyattempttowriteamodernsuccessortothethree standardworks,alltitled LinearAlgebraicGroups,byBorel,Humphreys,and Springer.Morespecifically,itisanexpositionofthetheoryofgroupschemesof finitetypeoverafield,basedonmodernalgebraicgeometry,butwithminimal prerequisites.
Ithasbeenclearforfiftyyearsthatsuchaworkhasbeenneeded.1 When Borel,Chevalley,andothersintroducedalgebraicgeometryintothetheoryof algebraicgroups,thefoundationstheyusedwerethoseoftheperiod(e.g.,Weil 1946),andmostsubsequentwritersonalgebraicgroupshavefollowedthem. Specifically,nilpotentsarenotallowed,andtheterminologyusedconflictswith thatofmodernalgebraicgeometry.Forexample,algebraicgroupsareusually identifiedwiththeirpointsinsomelargealgebraicallyclosedfield K ,andan algebraicgroupoverasubfield k of K isanalgebraicgroupover K equipped witha k -structure.Thekernelofa k -homomorphismofalgebraic k -groupsisan objectover K (not k ) whichneednotbedefinedover k . Inthemodernapproach,nilpotentsareallowed,2 analgebraic k -groupis intrinsicallydefinedover k ,andthekernelofahomomorphismofalgebraic groupsover k is(ofcourse)definedover k .Insteadofidentifyinganalgebraic groupwithitspointsinsome“universal”field,itismoreconvenienttoidentifyit withthefunctorof k -algebrasitdefines.
Theadvantagesofthemodernapproacharemanifold.Forexample,the infinitesimaltheoryisbuiltintoitfromthestartinsteadofenteringonlyinanad hocfashionthroughtheLiealgebra.TheNoetherisomorphismtheoremsholdfor
1 “Anotherremorseconcernsthelanguageadoptedforthealgebrogeometricalfoundationofthe theory...twosuchlanguagesarebrieflyintroduced...thelanguageofalgebraicsets...andthe Grothendiecklanguageofschemes.Lateron,thepreferenceisgiventothelanguageofalgebraicsets ...Ifthingsweretobedoneagain,Iwouldprobablyratherchoosetheschemeviewpoint...whichis notonlymoregeneralbutalso,inmanyrespects,moresatisfactory.”Tits1968,p.2.
2 Toanyonewhoaskedwhyweneedtoallownilpotents,Grothendieckwouldsaythattheyare alreadythereinnature;neglectingthemobscuresourvision.
algebraicgroupschemes,andsotheintuitionfromabstractgrouptheoryapplies. Thekernelsofinfinitesimalhomomorphismsbecomevisibleasalgebraicgroup schemes.
ThefirstsystematicexpositionofthetheoryofgroupschemeswasinSGA3. Aswasnaturalforitsauthors(Demazure,Grothendieck,...),theyworkedover anarbitrarybaseschemeandtheyusedthefulltheoryofschemes(EGAand SGA).Mostsubsequentauthorsongroupschemeshavefollowedthem.The onlybooksIknowofthatgiveanelementarytreatmentofgroupschemesare Waterhouse1979andDemazureandGabriel1970.Inwritingthisbook,Ihave reliedheavilyonboth,butneithergoesveryfar.Forexample,neithertreatsthe structuretheoryofreductivegroups,whichisacentralpartofthetheory.
Asnoted,themoderntheoryismoregeneralthantheoldtheory.Theextra generalitygivesaricherandmoreattractivetheory,butitdoesnotcomeforfree: someproofsaremoredifficult(becausetheyprovestrongerstatements).Inthis work,Ihaveavoidedanyappealtoadvancedschemetheory.UnpleasantlytechnicalargumentsthatIhavenotbeenabletoavoidhavebeenplacedinseparate sectionswheretheycanbeignoredbyallbutthemostseriousstudents.Byconsideringonlyschemesalgebraicoverafield,weavoidmanyofthetechnicalities thatplaguethegeneraltheory.Also,thetheoryoverafieldhasmanyspecial featuresthatdonotgeneralizetoarbitrarybases.
Acknowledgements: Theexpositionincorporatessimplificationstothegeneral theoryfromIversen1976,Luna1999,Steinberg1999,Springer1998,andother sources.Inwritingthisbook,thefollowingworkshavebeenespeciallyuseful tome:DemazureandGabriel1966;DemazureandGabriel1970;Waterhouse 1979;theexpositorywritingsofSpringer,especiallySpringer1994,1998;online notesofCasselman,Ngo,Perrin,andPink,aswellasthediscussions,often anonymous,on https://mathoverflow.net/.AlsoIwishtothankallthose whohavecommentedonthevariousnotespostedonmywebsite.
Introduction
Thebookcanbedividedroughlyintofiveparts.
A.Basictheoryofgeneralalgebraicgroups(Chapters1–8)
Thefirsteightchapterscoverthegeneraltheoryofalgebraicgroupschemes(not necessarilyaffine)overafield.Afterdefiningthemandgivingsomeexamples, weshowthatmostofthebasictheoryofabstractgroups(subgroups,normal subgroups,normalizers,centralizers,Noetherisomorphismtheorems,subnormal series,etc.)carriesoverwithlittlechangetoalgebraicgroupschemes.We relateaffinealgebraicgroupschemestoHopfalgebras,andweprovethatall algebraicgroupschemesincharacteristiczeroaresmooth.Westudythelinear representationsofalgebraicgroupschemesandtheiractionsonalgebraicschemes. Weshowthateveryalgebraicgroupschemeisanextensionofan ´ etalegroup schemebyaconnectedalgebraicgroupscheme,andthateverysmoothconnected groupschemeoveraperfectfieldisanextensionofanabelianvarietybyanaffine groupscheme(Barsotti–Chevalleytheorem).
BeginningwithChapter9,allgroupschemesareaffine.
B.Preliminariesonaffinealgebraicgroups(Chapters9–11)
Thenextthreechaptersarepreliminarytothemoredetailedstudyofaffine algebraicgroupschemesinthelaterchapters.TheycoverbasicTannakian theory,inwhichthecategoryofrepresentationsofanalgebraicgroupscheme playstheroleofthetopologicaldualofalocallycompactabeliangroup,Jordan decompositions,theLiealgebraofanalgebraicgroup,andthestructureoffinite groupschemes.ThroughoutthisworkweemphasizetheTannakianpointofview inwhichthegroupanditscategoryofrepresentationsareplacedonanequal footing.
C.Solvableaffinealgebraicgroups(Chapters12–16)
Thenextfivechaptersstudysolvablealgebraicgroupschemes.Amongtheseare thediagonalizablegroups,theunipotentgroups,andthetrigonalizablegroups.
Analgebraicgroup G isdiagonalizableifeverylinearrepresentationof G isadirectsumofone-dimensionalrepresentations;inotherwordsif,relativeto somebasis,theimageof G liesinthealgebraicsubgroupofdiagonalmatricesin GLn .Analgebraicgroupthatbecomesdiagonalizableoveranextensionofthe basefieldissaidtobeofmultiplicativetype.
Analgebraicgroup G isunipotentifeverynonzerorepresentationof G containsanonzerofixedvector.Thisimpliesthateveryrepresentationhasa basisforwhichtheimageof G liesinthealgebraicsubgroupofstrictlyupper triangularmatricesinGLn .
Analgebraicgroup G istrigonalizableifeverysimplerepresentationhas dimensionone.Thisimpliesthateveryrepresentationhasabasisforwhich theimageof G liesinthealgebraicsubgroupofuppertriangularmatricesin GLn .Thetrigonalizablegroupsareexactlytheextensionsofdiagonalizable groupsbyunipotentgroups.Trigonalizablegroupsaresolvable,andtheLie–Kolchintheoremsaysthatallsmoothconnectedsolvablealgebraicgroupsbecome trigonalizableoverafiniteextensionofthebasefield.
D.Reductivealgebraicgroups(Chapters17–25)
Thisistheheartofthebook,Thefirstsevenchaptersdevelopindetailthestructure theoryofsplitreductivegroupsandtheirrepresentationsintermsoftheirroot data.Chapter24exhibitsallthealmost-simplealgebraicgroups,andChapter25 explainshowthetheoryofsplitgroupsextendstothenonsplitcase.
E.Appendices
Thefirstappendixreviewsthedefinitionsandstatementsfromalgebraicgeometry neededinthebook.Expertsneedonlynotethat,aswealwaysworkwithschemes offinitetypeoverabasefield k ,itisnaturaltoignorethenonclosedpoints (whichwedo).
Thesecondappendixprovestheexistenceofaquotientofanalgebraicgroup byanalgebraicsubgroup.Thisisanimportantresult,buttheexistenceof nilpotentsmakestheproofdifficult,andsomostreadersshouldsimplyaccept thestatement.
Thethirdappendixreviewsthecombinatorialobjects,rootsystemsandroot data,onwhichthetheoryofsplitreductivegroupsisbased.
History
Apartfromoccasionalbriefremarks,weignorethehistoryofthesubject,which isquitecomplex.Manymajorresultswerediscoveredinonesituation,andthen extendedtoothermoregeneralsituations,sometimeseasilyandsometimesonly withdifficulty.Withouttoomuchexaggeration,onecansaythatallthetheoryof algebraicgroupschemesdoesisshowthatthetheoryofKillingandCartanfor
“local”objectsover C extendsinanaturalwayto“global”objectsoverarbitrary fields.
Conventionsandnotation
Throughout, k isafieldand R isafinitelygenerated k -algebra.1 All k -algebras and R -algebrasarerequiredtobecommutativeandfinitelygeneratedunlessitis specifiedotherwise.Noncommutativealgebrasarereferredtoas“algebrasover k ”ratherthan“k -algebras”.Unadornedtensorproductsareover k .Anextension of k isafieldcontaining k ,andaseparableextensionisaseparablealgebraic extension.When V isavectorspaceover k ,weoftenwrite VR for V ˝ R ;for v 2 V ,welet vR D v ˝ 1 2 VR .Thesymbol k a denotesanalgebraicclosureof k and k s (resp. k i )denotestheseparable(resp.perfect)closureof k in k a .The characteristicexponentof k is p or 1 accordingasitscharacteristicis p or 0. Thegroupofinvertibleelementsofaring R isdenotedby R .Thesymbol AlgR denotesthecategoryoffinitelygenerated R -algebras.
Analgebraicschemeover k (oralgebraic k -scheme)isaschemeoffinite typeover k .Analgebraicschemeisanalgebraicvarietyifitisgeometrically reducedandseparated.Bya“point”ofanalgebraicschemeorvarietyover k wealwaysmeanaclosedpoint.Foranalgebraicscheme .X; OX / over k ,we oftenlet X denotetheschemeand jX j theunderlyingtopologicalspaceofclosed points.Foralocallyclosedsubset Z of jX j (resp.subscheme Z of X ),the reducedsubschemeof X withunderlyingspace Z (resp. jZ j)isdenotedby Zred . Theresiduefieldatapoint x of X isdenotedby .x/.Whenthebasefield k is understood,weomitit,andwrite“algebraicscheme”for“algebraicschemeover k ”.Unadornedproductsofalgebraic k -schemesareover k .SeeAppendixAfor moredetails.
Welet Z denotetheringofintegers, R thefieldofrealnumbers, C thefield ofcomplexnumbers,and Fp thefieldof p elements(p prime).
Afunctorissaidtobeanequivalenceofcategoriesifitisfullyfaithfuland essentiallysurjective.Asufficientlystrongversionoftheaxiomofglobalchoice thenimpliesthatthereexistsaquasi-inversetothefunctor.Wesometimesloosely refertoanaturaltransformationoffunctorsasamapoffunctors.
Allcategoriesarelocallysmall(i.e.,themorphismsfromoneobjecttoa secondarerequiredtoformaset).Whentheobjectsformaset,thecategory issaidtobesmall.Acategoryisessentiallysmallifitisequivalenttoasmall subcategory.
Let P beapartiallyorderedset.Agreatestelementof P isa g 2 P such that a g forall a 2 P .Anelement m in P ismaximalif m a implies a D m. Agreatestelementisauniquemaximalelement.Leastandminimalelements aredefinedsimilarly.Whenthepartialorderisinclusion,wereplaceleastand greatestwithsmallestandlargest.Wesometimesuse Œx todenotetheclassof x underanequivalencerelation.
1 ExceptinAppendixC,where R isasetofroots.
FollowingBourbaki,welet N Df0;1;2;:::g.Anintegerispositiveifitlies in N.Asetwithanassociativebinaryoperationisasemigroup.Amonoidisa semigroupwithaneutralelement.
By A ' B wemeanthat A and B arecanonicallyisomorphic(orthatthereis agivenoruniqueisomorphism),andby A B wemeansimplythat A and B areisomorphic(thereexistsanisomorphism).Thenotation A B meansthat A isasubsetof B (notnecessarilyproper).Adiagram A ! B C isexactifthe firstarrowistheequalizerofthepairofarrows.
Supposethat p and q arestatementsdependingonafield k andwewish toprovethat p.k/ implies q.k/.If p.k/ implies p.k a / and q.k a / implies q.k/, thenitsufficestoprovethat p.k a / implies q.k a /.Insuchasituation,wesimply saythat“wemaysupposethat k isalgebraicallyclosed”.
Weoftenomit“algebraic”fromsuchexpressionsas“algebraicsubgroup”, “unipotentalgebraicgroup”,and“semisimplealgebraicgroup”.Afterp.162,all algebraicgroupsareaffine.
Weusetheterminologyofmodern(post1960)algebraicgeometry;for example,foralgebraicgroupsoverafield k; ahomomorphismisautomatically definedover k ,notoversomelargealgebraicallyclosedfield.2
Throughout,“algebraicgroupscheme”isshortenedto“algebraicgroup”.A statementheremaybestrongerthanastatementinBorel1991orSpringer1998 evenwhenthetwoarewordforwordthesame.3
Allconstructionsaretobeunderstoodasbeinginthesenseofschemes.For example,fibresofmapsofalgebraicvarietiesneednotbereduced,andthekernel ofahomomorphismofsmoothalgebraicgroupsneednotbesmooth.
Numbering
Areference“17.56”istoitem56ofChapter17.Areference“(112)”isto the112thnumberedequationinthebook(weincludethepagenumberwhere necessary).Section17cisSectioncofChapter17andSectionAcisSectioncof AppendixA.TheexercisesinChapter17arenumbered17-1,17-2,...
Foundations
WeusethevonNeumann–Bernays–Godel(NBG)settheorywiththeaxiomof choice,whichisaconservativeextensionofZermelo–Fraenkelsettheorywith theaxiomofchoice(ZFC).Thismeansthatasentencethatdoesnotquantify overaproperclassisatheoremofNBGifandonlyifitisatheoremofZFC.The advantageofNBGisthatitallowsustospeakofclasses.
Itisnotpossibletodefinean“unlimitedcategorytheory”thatincludesthe categoryof all sets,thecategoryof all groups,etc.,andalsothecategoriesof
2 Asmuchaspossible,ourstatementsmakesenseinaworldwithoutchoice,wherealgebraic closuresneednotexist.
3 AnexampleisChevalley’stheoremonrepresentations;see4.30.
functorsfromoneofthesecategoriestoanother.Instead,onemustconsider onlycategoriesoffunctorsfromcategoriesthataresmallinsomesense.To thisend,wefixafamilyofsymbols .Ti /i 2N indexedby N, 4 andlet Alg0 k denote thecategoryof k -algebrasoftheform kŒT0 ;:::;Tn =a forsome n 2 N andideal a in kŒT0 ;:::;Tn .Thustheobjectsof Alg0 k areindexedbytheidealsinsome subring kŒT0 ;:::;Tn of kŒT0 ;::: –inparticular,theyformaset,andso Alg0 k is small.Wecalltheobjectsof Alg0 k small k -algebras.If R isasmall k -algebra, thenthecategory Alg0 R ofsmall R -algebrashasasobjectspairsconsistingofa small k -algebra A andahomomorphism R ! A of k -algebras.Notethattensor productsexistin Alg0 k –infact,ifwefixabijection N $ N N,then ˝ becomes awell-definedbi-functor.
Theinclusionfunctor Alg0 k ,! Algk isanequivalenceofcategories.Choosing aquasi-inverseamountstochoosinganorderedsetofgeneratorsforeachfinitely generated k -algebra.Onceaquasi-inversehasbeenchosen,everyfunctoron Alg0 k hasawell-definedextensionto Algk .
Alternatively,readerswillingtoassumeadditionalaxiomsinsettheorymay useMacLane’s“one-universe”solutiontodefiningfunctorcategories(MacLane 1969)orGrothendieck’s“multi-universe”solution,anddefineasmall k -algebra tobeonethatissmallrelativetothechosenuniverse.
Prerequisites
Afirstcourseinalgebraicgeometry(includingbasiccommutativealgebra).Since thesevarygreatly,wereviewthedefinitionsandstatementsthatweneedfrom algebraicgeometryinAppendixA.Inafewproofs,whichcanbeskipped,we assumesomewhatmore.
References
Thecitationsareauthor–year,exceptforthefollowingabbreviations:
CA =Milne2017(APrimerofCommutativeAlgebra).
DG =DemazureandGabriel1970(Groupesalg ´ ebriques).
EGA =Grothendieck1967(El ´ ementsdeg ´ eometriealg ´ ebrique).
SGA3 =DemazureandGrothendieck2011(Sch ´ emasengroupes,SGA3).
SHS =DemazureandGabriel1966(S ´ eminaireHeidelberg–Strasbourg1965–66 ).
4 Better,use N itselfasthesetofsymbols.
C
HAPTER
DefinitionsandBasicProperties
Recallthat k isafield,andthatanalgebraic k -schemeisaschemeoffinitetype over k .Welet D Spm.k/
a.Definition
Analgebraicgroupover k isagroupobjectinthecategoryofalgebraicschemes over k .Indetail,thismeansthefollowing.
D EFINITION 1.1. Let G beanalgebraicschemeover k andlet mW G G ! G bearegularmap.Thepair .G;m/ isan algebraicgroup over k ifthereexist regularmaps
e W ! G; invW G ! G; suchthatthefollowingdiagramscommute:
When G isavariety,wecall .G;m/ a groupvariety,andwhen G isanaffine scheme,wecall .G;m/ an affinealgebraicgroup. Forexample, SLn def D Spm kŒT11 ;T12 ;:::;Tnn =.det.Tij / 1/
becomesanaffinegroupvarietywiththeusualmatrixmultiplicationonpoints. Formanymoreexamples,seeChapter2.
Similarly,an algebraicmonoid over k isanalgebraicscheme M over k togetherwithregularmaps mW M M ! M and e W ! M suchthatthediagrams (1)commute.
D EFINITION 1.2. A homomorphism ' W .G;m/ ! .G 0 ;m0 / ofalgebraicgroups isaregularmap ' W G ! G 0 suchthat ' ı m D m0 ı .' ' /.
Analgebraicgroup G is trivial if e W ! G isanisomorphism,andahomomorphism G ! G 0 is trivial ifitfactorsthrough e 0 W ! G 0 .Weoftenwrite e for thetrivialalgebraicgroup.
D EFINITION 1.3. An algebraicsubgroup ofanalgebraicgroup .G;mG / over k isanalgebraicgroup .H;mH / over k suchthat H isa k -subschemeof G andthe inclusionmapisahomomorphismofalgebraicgroups.Analgebraicsubgroupis calleda subgroupvariety ifitsunderlyingschemeisavariety.
Let .G;mG / beanalgebraicgroupand H anonemptysubschemeof G .If mG jH H and invG jH factorthrough H ,then .H;mG jH H/ isanalgebraic subgroupof G
Let .G;m/ beanalgebraicgroupover k .Foranyfield k 0 containing k ,the pair .Gk 0 ;mk 0 / isanalgebraicgroupover k 0 ,saidtohavebeenobtainedfrom .G;m/ by extensionofscalars or extensionofthebasefield.
Algebraicgroupsasfunctors
The K -pointsofanalgebraicscheme X with K afielddonotseethenilpotentsin thestructuresheaf.Thus,weareledtoconsiderthe R -pointswith R a k -algebra. Oncewedothat,thepointscapture all informationabout X .
1.4.An algebraicscheme X over k definesafunctor
X W Alg0 k ! Set;R X.R/:
Forexample,if X isaffine,say, X D Spm.A/,then
X.R/ D Homk -algebra .A;R/:
Thefunctor X X isfullyfaithful(Yonedalemma,A.33);inparticular, X determines X uniquelyuptoauniqueisomorphism.Wesaythatafunctorfrom small k -algebrastosetsisrepresentableifitisoftheform X foranalgebraic scheme X over k
If .G;m/ isanalgebraicgroupover k ,then R .G.R/;m.R// isafunctor fromsmall k -algebrastogroups.
Let X beanalgebraicschemeover k ,andsupposethatwearegivena factorizationof Q X throughthecategoryofgroups.Thenthemaps
x;y 7! xy W X.R/ X.R/ ! X.R/; 7! e W ! X.R/;x 7! x 1 W X.R/ ! X.R/
givenbythegroupstructuresonthesets X.R/ define,bytheYonedalemma, morphisms
mW X X ! X; ! X; invW X ! X makingthediagrams(1)and(2)commute.Therefore, .X;m/ isanalgebraic groupover k
Combiningthesetwostatements,weseethattogiveanalgebraicgroupover k amountstogivingafunctor Alg0 k ! Grp whoseunderlyingfunctortosetsis representablebyanalgebraicscheme.Wewrite G for G regardedasafunctorto groups.
Fromthisperspective, SLn canbedescribedasthealgebraicgroupover k sending R tothegroup SLn .R/ of n n matriceswithentriesin R anddeterminant 1.
Thefunctor R.R; C/ isrepresentedby Spm.kŒT /,andhenceisan algebraicgroup Ga .Similarly,thefunctor R .R ; / isrepresentedby Spm.kŒT;T 1 /,andhenceisanalgebraicgroup Gm .See2.1and2.2below. Weoftendescribeahomomorphismofalgebraicgroupsbygivingitsaction on R -points.Forexample,whenwesaythat invW G ! G isthemap x 7! x 1 , wemeanthat,forallsmall k -algebras R andall x 2 G.R/,inv.x/ D x 1 .
1.5. If .H;mH / isanalgebraicsubgroupof .G;mG /,then H.R/ isasubgroup of G.R/ forall k -algebras R .Conversely,if H isanalgebraicsubschemeof G suchthat H.R/ isasubgroupof G.R/ forallsmall k -algebras R ,thenthe Yonedalemma(A.33)showsthatthemaps
.h;h0 / 7! hh0 W H.R/ H.R/ ! H.R/
arisefromamorphism mH W H H ! H andthat .H;mH / isanalgebraic subgroupof .G;mG /.
1.6. Considerthefunctorof k -algebras 3 W R fa 2 R j a 3 D 1g.Thisis representedby Spm.kŒT =.T 3 1//,andsoitisanalgebraicgroup.Weconsider threecases.
(a)Thefield k isalgebraicallyclosedofcharacteristic ¤ 3.Then kŒT =.T 3 1/ ' kŒT =.T 1/ kŒT =.T / kŒT =.T 2 / where 1; ; 2 arethecuberootsof 1 in k .Thus, 3 isadisjointunionofthree copiesofSpm.k/ indexedbythecuberootsof 1 in k .
(b)Thefield k isofcharacteristic ¤ 3 butdoesnotcontainaprimitivecube rootof 1.Then
kŒT =.T 3 1/ ' kŒT =.T 1/ kŒT =.T 2 C T C 1/; andso 3 isadisjointunionof Spm.k/ and Spm.kŒ / where isaprimitive cuberootof 1 in k s .
(c)Thefield k isofcharacteristic 3.Then T 3 1 D .T 1/3 ,andso 3 is notreduced.Although 3 .K/ D 1 forallfields K containing k ,thealgebraic group 3 isnottrivial.Certainly, 3 .R/ maybenonzeroif R hasnilpotents.
Homogeneity
Recallthat,foranalgebraicscheme X over k ,wewrite jX j fortheunderlying topologicalspaceof X ,and .x/ fortheresiduefieldatapoint x of jX j (it isafiniteextensionof k ).Wecanidentify X.k/ withthesetofpoints x of jX j suchthat .x/ D k (CA13.4).Analgebraicscheme X over k issaidto be homogeneous ifthegroupofautomorphismsof X (asa k -scheme)acts transitivelyon jX j.Weshallseethatanalgebraicgroupishomogeneouswhen k isalgebraicallyclosed
1.7. Let .G;m/ beanalgebraicgroupover k .Themap m.k/W G.k/ G.k/ ! G.k/ makes G.k/ intoagroupwithneutralelement e. / andinversemap inv.k ). When k isalgebraicallyclosed, G.k/ D jG j,andso mW G G ! G makes jG j intoagroup.Themaps x 7! x 1 and x 7! ax (a 2 G.k/)areautomorphisms of jG j asatopologicalspace.
Ingeneral,when k isnotalgebraicallyclosed, m doesnotmake jG j into agroup,andevenwhen k isalgebraicallyclosed,itdoesnotmake jG j intoa topological group.
1.8. Let .G;m/ beanalgebraicgroupover k .Foreach a 2 G.k/,thereisa translationmap la W G 'fa g G m ! G;x 7! ax . For a;b 2 G.k/, la ı l
and le D id.Therefore
,andso la isanisomorphism sending e to a .Hence G ishomogeneouswhen k isalgebraicallyclosed(butnot ingeneralotherwise;see1.6(b)).
Densityofpoints
Becauseweallownilpotentsinthestructuresheaf,amorphism X ! Y of algebraicschemesisnotingeneraldeterminedbyitseffecton X.k/,evenwhen k isalgebraicallyclosed.Weintroducesometerminologytohandlethis.
D EFINITION 1.9. Let X beanalgebraicschemeover k and S asubsetof X.k/ Wesaythat S is schematicallydense (orjust dense)in X iftheonlyclosed subscheme Z of X suchthat S Z.k/ is X itself.
Let X D Spm.A/,andlet S beasubsetof X.k/.Let Z D Spm.A=a/ be aclosedsubschemeof X .Then S Z.k/ ifandonlyif a m forall m 2 S . Therefore, S isschematicallydensein X ifandonlyif T fm j m 2 S g D 0
P ROPOSITION 1.10. Let X beanalgebraicschemeover k and S asubsetof X.k/ jX j.Thefollowingconditionsareequivalent: (a) S isschematicallydensein X I
(b) X isreducedand S isdensein jX j;
(c) thefamilyofhomomorphisms
f 7! f.s/W OX ! .s/ D k;s 2 S; isinjective.
P ROOF. (a))(b).Let S denotetheclosureof S in jX j.Thereisaunique reducedsubscheme Z of X withunderlyingspace S .As S jZ j,thescheme Z D X ,andso X isreducedwithunderlyingspace S . (b))(c).Let U beanopenaffinesubschemeof X ,andlet A D OX .U/.Let f 2 A besuchthat f.s/ D 0 forall s 2 S \jU j.Then f.u/ D 0 forall u 2jU j because S \jU j isdensein jU j.Thismeansthat f liesinallmaximalidealsof A,andthereforeliesintheradicalof A,whichiszerobecause X isreduced(CA 13.11).
(c))(a).Let Z beaclosedsubschemeof X suchthat S Z.k/.Because Z isclosedin X ,thehomomorphism OX ! O Z issurjective.Because S Z.k/, themaps f 7! f.s/W OX ! .s/, s 2 S ,factorthrough O Z ,andsothemap OX ! O Z isinjective.Hencethemap OX ! O Z isanisomorphism,which impliesthat Z D X . ✷
P ROPOSITION 1.11. Aschematicallydensesubsetremainsschematicallydense underextensionofthebasefield.
P ROOF Let k 0 beafieldcontaining k ,andlet S X.k/ beschematicallydense. Wemaysupposethat X isaffine,say, X D Spm.A/.Let s 0 W Ak 0 ! k 0 bethemap obtainedfrom s W A ! .s/ D k byextensionofscalars.Thefamily s 0 , s 2 S ,is injectivebecausethefamily s , s 2 S ,isinjectiveand k 0 isflatover k . ✷
C OROLLARY 1.12. If X admitsaschematicallydensesubset S X.k/,thenit isgeometricallyreduced.
P ROOF. Theset S remainsschematicallydensein X.k a /,andso Xk a isreduced. ✷
P ROPOSITION 1.13. If S isschematicallydensein X and u;v W X Y are regularmapsfrom X toaseparatedalgebraicscheme Y suchthat u.s/ D v.s/ forall s 2 S ,then u D v .
P ROOF. Because Y isseparated,theequalizerofthepairofmapsisclosedin X . Asitsunderlyingspacecontains S ,itequals X . ✷
R EMARK 1.14. Someoftheabovediscussionextendstobaserings.Forexample,let X beanalgebraicschemeoverafield k andlet S beaschematically densesubsetof X.k/.Let R bea k -algebraand,for s 2 S ,let
s 0 D s Spm.k/ Spm.R/ X 0 D X Spm.k/ .R/:
Asintheproofof(1.11),thefamilyofmaps OX 0 ! O s 0 .s 0 / D R isinjective. Itfollows,asintheproofof(1.10),thattheonlyclosed R -subschemeof X 0 containingall s 0 is X 0 itself.
D EFINITION 1.15. Let X beanalgebraicschemeoverafield k ,andlet k 0 bea fieldcontaining k .Wesaythat X.k 0 / is dense in X iftheonlyclosedsubscheme Z of X suchthat Z.k 0 / D X.k 0 / is X itself.
P ROPOSITION 1.16. If X.k 0 / isdensein X ,then X isreduced.Conversely,if X.k 0 / isdenseinthetopologicalspace jXk 0 j and X isgeometricallyreduced, then X.k 0 / isdensein X .
P ROOF. Recallthat Xred isthe(unique)reducedsubschemeof X withunderlying space jX j.Moreover Xred .k 0 / D X.k 0 / because k 0 isreduced,andso Xred D X if X.k 0 / isdensein X .
Conversely,supposethat X isgeometricallyreducedand X.k 0 / isdense in jXk 0 j.Let Z beaclosedsubschemeof X suchthat Z.k 0 / D X.k 0 /.Then jZk 0 jDjXk 0 j bythedensitycondition.Thisimpliesthat Zk 0 D Xk 0 because Xk 0 isreduced,whichinturnimpliesthat Z D X (seeA.65). ✷
C OROLLARY 1.17. If X isgeometricallyreduced,then X.k 0 / isdensein X for everyseparablyclosedfield k 0 containing k .
P ROOF.Byastandardresult(A.48), X.k 0 / isdensein jXk 0 j. ✷
C OROLLARY 1.18. Let Z and Z 0 beclosedsubvarietiesofanalgebraicscheme X over k .If Z.k 0 / D Z 0 .k 0 / forsomeseparablyclosedfield k 0 containing k , then Z D Z 0 .
P ROOF. Theclosedsubscheme Z \ Z 0 of Z hasthepropertythat .Z \ Z 0 /.k 0 / D Z.k 0 /,andso Z \ Z 0 D Z .Similarly, Z \ Z 0 D Z 0 ✷
Thus,aclosedsubvariety Z of X isdeterminedbythesubset Z.k s / of X.k s /. Moreexplicitly,if X D Spm.A/ and Z D Spm.A=a/,then a isthesetof f 2 A suchthat f.P/ D 0 forall P 2 Z.k s /:
Algebraicgroupsoverrings
Althoughweareonlyinterestedinalgebraicgroupsoverfields,occasionallywe shallneedtoconsiderthemovermoregeneralbaserings.
1.19. Let R bea(finitelygenerated) k -algebra.Analgebraicschemeover R is ascheme X equippedwithamorphism X ! Spm.R/ offinitetype.Equivalently, X isanalgebraicschemeover k suchthat OX isequippedwithan R -algebra structurecompatiblewithits k -algebrastructure.Forexample,affinealgebraic schemesover R arethemax-spectraoffinitelygenerated R -algebras.Amorphism ofalgebraic R -schemes ' W X ! Y isamorphismof k -schemescompatiblewith the R -algebrastructures,i.e.,suchthat O Y ! ' OX isahomomorphismof sheavesof R -algebras.Let G beanalgebraicschemeover R andlet mW G G ! G beamorphismof R -schemes.Thepair .G;m/ isan algebraicgroupover R if thereexist R -morphisms e W Spm.R/ ! G and invW G ! G suchthatthediagrams (1)and(2)commute.Forexample,analgebraicgroup .G;m/ over k givesrise toanalgebraicgroup .GR ;mR / over R byextensionofscalars.
A SIDE 1.20. Bydefinitionanalgebraicgroupanditsmultiplicationmaparedescribed bypolynomials,butwerarelyneedtoknowwhatthepolynomialsare.Nevertheless,itis ofsomeinterestthatitisoftenpossibletorealizethecoordinateringofanaffinealgebraic groupasaquotientofapolynomialringinaconcretenaturalway(Popov2015).
N OTES Asnotedelsewhere,inmostoftheliterature,analgebraicgroupoverafield k is definedtobeagroupvarietyoversomealgebraicallyclosedfield K containing k together witha k -structure(see,forexample,Springer19981.6.14,2.1.1).Inparticular,nilpotents arenotallowed.Analgebraicgroupoverafield k inoursenseisagroupschemeof finitetypeover k inthelanguageofSGA3.Ournotionofanalgebraicgroupover k is essentiallythesameasthatinDG.
b.Basicpropertiesofalgebraicgroups
P ROPOSITION 1.21. If ' W .G;mG / ! .H;mH / isahomomorphismofalgebraic groups,then ' ı eG D eH and ' ı invG D invH ı' .Inparticular,themaps e and inv in(1.1)areuniquelydeterminedby .G;m/
P ROOF. Forevery k -algebra R ,themap '.R/ isahomomorphismofabstract groups .G.R/;mG .R// ! .H.R/;mH .R//,andsoitmapstheneutralelement of G.R/ tothatof H.R/ andtheinversionmapon G.R/ tothaton H.R/.The Yonedalemma(A.33)nowshowsthatthesameistruefor ' . ✷
Weoftenwrite e fortheimageof e W ! G in G.k/ or jG j.Recall(A.41) thatanalgebraicscheme X isseparatedifitsdiagonal X isclosedin X X .
P ROPOSITION 1.22. Algebraicgroupsareseparated(asalgebraicschemes).
P ROOF Let .G;m/ beanalgebraicgroup.Thediagonalin G G istheinverse imageoftheclosedpoint e 2 G.k/ underthemap m ı .id inv/W G G ! G sending .g1 ;g2 / to g1 g 1 2 ,andsoitisclosed. ✷
Therefore“groupvariety”=“geometricallyreducedalgebraicgroup”.
C OROLLARY 1.23. Let G beanalgebraicgroupover k andlet k 0 beanextension of k .If G.k 0 / isdensein G ,thenahomomorphism G ! H ofalgebraicgroups isdeterminedbyitsactionon G.k 0 /.
P ROOF Let '1 and '2 behomomorphisms G ! H suchthat '1 .a/ D '2 .a/ for all a 2 G.k 0 /.Because H isseparated,theequalizer Z of '1 and '2 isaclosed subschemeof G .As Z.k 0 / D G.k 0 /,wehave Z D G ✷
D EFINITION 1.24. Analgebraicgroup .G;m/ is commutative if m ı t D m, where t isthetranspositionmap .x;y/ 7! .y;x/W G G ! G G .
P ROPOSITION 1.25. Analgebraicgroup G iscommutativeifandonlyif G.R/ iscommutativeforall k -algebras R .Agroupvariety G iscommutativeif G.k s / iscommutative.
P ROOF. AccordingtotheYonedalemma(A.33), m ı t D m ifandonlyif m.R/ ı t.R/ D m.R/ forall k -algebras R ,i.e.,ifandonlyif G.R/ iscommutativefor all R .Thisprovesthefirststatement.Let G beagroupvariety.If G.k s / is commutative,then m ı t and m agreeon .G G/.k s /,whichisdensein G G (see1.17). ✷
Smoothness
Let X beanalgebraicschemeover k .For x 2jX j,wehave
dim.OX;x / dim.mx =m2 x /.
Here mx isthemaximalidealinthelocalring OX;x ,the“dim”atleftistheKrull dimension,andthe“dim”atrightisthedimensionasa .x/-vectorspace(see CA, 22).Whenequalityholds,thepoint x issaidtoberegular.Ascheme X is saidtoberegularif x isregularforall x 2jX j.Itispossiblefor X toberegular without Xk a beingregular.Toremedythis,weneedanothernotion.
Let kŒ" bethe k -algebrageneratedbyanelement " with "2 D 0.Fromthe homomorphism " 7! 0,wegetamap X.kŒ" / ! X.k/,andwedefinethe tangent space Tgtx .X/ atapoint x 2 X.k/ tobethefibreover x .Thus
Tgtx .X/ ' Homk -linear .mx =m2 x ;k/;
andso dimTgtx .X/ dim.OX;x /.Whenequalityholds,thepointissaidtobe smooth.Theformationofthetangentspacecommuteswithextensionofthe basefield,andsoapoint x 2 X.k/ issmoothon X ifandonlyifitissmoothon Xk a .Analgebraicscheme X overanalgebraicallyclosedfield k issaidtobe smoothifall x 2jX j aresmooth,andanalgebraicscheme X overanarbitrary field k issaidtobesmoothif Xk a issmooth.Smoothschemesareregular,and theconverseistruewhen k isalgebraicallyclosed.SeeSectionAh.
P ROPOSITION 1.26. Let G beanalgebraicgroupover k .
(a) If G isreducedand k isperfect,then G isgeometricallyreduced.
(b) If G isgeometricallyreduced,thenitissmooth(andconversely).
P ROOF.(a)Thisistrueforallalgebraicschemes(A.43).
(b)Wehavetoshowthat Gk a issmooth.But Gk a isanalgebraicvariety,and sosomepointonitissmooth(A.55),whichimpliesthateverypointissmoothby homogeneity(1.8). ✷
Therefore “groupvariety”=“smoothalgebraicgroup”.
Incharacteristiczero,allalgebraicgroupsaresmooth(3.23,8.39below).
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trade itself, which is thus continually recruited from the inexperienced. There is a flux among the workers, the union officials, and the employers themselves. Among women, the more or less ephemeral character of much of their work, their frequent change of occupation, and marriage, all operate against permanency. The girl who knocked at our door that night, to invite us to our first trades union meeting, is now in a profession.
Later, when we moved to Henry Street, Minnie, who lived in the next block, enlisted our sympathy in her efforts to organize the girls in her trade. She based her arguments for shorter hours on their need of time to acquire knowledge of housekeeping and homemaking before marriage and motherhood came to them, touching instinctively a fundamental argument against excessive hours for women.
We invited Minnie to a conference of philanthropists on methods for improving the condition of working girls, in order that she might give her conception of what would be advantageous. Representatives of the various societies reported on their work:
vacations provided, seats in stores, religious instruction, and so on. “We are the hands of the boss,” said Minnie when her turn came. “What does he care for us? I say, Let our hands be for him and our heads for ourselves. We must work for bread now, but we must think of our future homes. What time has a working girl to make ready for this? We never see a meal prepared. For all we know, soup grows on trees.”
Minnie, who was headlined by the press during a strike as a Joan of Arc leading militant hosts to battle, had no educational preparation for leadership; no equipment beyond her sound good sense and her woman’s subtlety. Speaking once of the difficulty of earning a living without training, she told me that her mother could do nothing but sell potatoes from a push-cart in the street, “among those rough people.” Then, repenting of her harshness, “Of course, some of those people must be nice, too, but it is hard to find a diamond in the mud.”
Frequent and prolonged conferences at the settlement with Minnie and Lottie, her equally intelligent companion, and with many others, inevitably led to some action on our part; and long anticipating the Women’s Trades Union League, we took the initiative in organizing a union at the time of a strike in the cloak trade. The eloquence of the girl leaders, the charm of our back yard as a meeting-place, and possibly our own conviction that only through organization could wages be raised and shop conditions improved, finally prevailed, and the union was organized. One of our residents and a brilliant young Yiddish-speaking neighbor took upon themselves some of the duties of the walking delegate. When the strike was settled, and agreements for the season were about to be signed by the contractors (or middlemen) and the leader of the men’s organization, I was invited into a smoke-filled room in Walhalla Hall long after
midnight, to be told that the girls were included in the terms of the contract.
Though its immediate object was accomplished, this union also proved to be an ephemeral organization. For years I held the funds, amounting to sixteen dollars, because the members had scattered and we could never assemble a quorum to dispose of the money.
When, in 1903, I was asked to participate in the formation of the National Women’s Trades Union League, I recognized the importance of the movement in enlisting sympathy and support for organizations among working women. To my regret I cannot claim to have rendered services of any value in the development of the League. It was inevitable that its purpose, as epitomized in its motto—“The Eight-hour Day; A Living Wage; To Guard the Home”—should draw to it effective participants and develop strong leaders among working women themselves. Those who are familiar with factory and shop conditions are convinced that through organization and not through the appeal to pity can permanent reforms be assured. It is undoubtedly true that the enforcement of existing laws is in large measure dependent upon watchful trades unions. The women’s trades union leagues, national and state, are not only valuable because of support given to the workers, but because they make it possible for women other than wage-earners to identify themselves with working people, and thus give practical expression to their belief that with them and through them the realization of the ideals of democracy can be advanced.
The imagination of New Yorkers has been fired from time to time by young working women who have had no little influence in helping to rouse public interest in labor conditions. My associates and I, in the early years of the settlement, owed much to a mother and daughter of singularly lofty mind and character, both working women, who for a time joined the settlement family. They had been affiliated with labor organizations almost all their lives. The ardor of the daughter continually prodded us to action, and the clear-minded, intellectual mother helped us to a completer realization of the deeplying causes that had inspired Mazzini and other great leaders, whose works we were re-reading.
More recently a young capmaker has stimulated recognition of the public’s responsibility for the well-being of the young worker. Despite her long hours, she found time to organize a union in her trade, not in a spurt of enthusiasm, but as a result of a sober realization that women workers must stand together for themselves and for those who come after them.
The inquiry that followed the disastrous fire in the factory of the Triangle Waist Company in March, 1911, when one hundred and forty-three girls were burned, or leaped from windows to their death, disclosed the fact that the owners of this factory, like many others, kept the doors of the lofts locked. Hundreds of girls, many stories above the streets, were thus cut off from access to stairs or fireescapes because of the fear of small thefts of material. The girls in this factory had tried, a short time before the fire, to organize a union to protest against bad shop conditions and petty tyrannies.
After the tragedy, at a meeting in the Metropolitan Opera House called together by horrified men and women of the city, this young capmaker stood at the edge of the great opera-house stage and in a voice hardly raised, though it reached every person in that vast audience, arraigned society for regarding human life so cheaply. No one could have been insensitive to her cry for justice, her anguish over the youth so ruthlessly destroyed; and there must have been many in that audience for whom ever after the little, brown-clad figure with the tragic voice symbolized the factory girl in the lofts high above the streets of an indifferent metropolis.
Before the fire the “shirt-waist strike” had brought out a wave of popular sympathy. This was due in part to the youth of a majority of the workers, to a realization of the heroic sacrifices some of them were making (an inkling of which got to the public), and in part also to disapproval of the methods used to break the strike. Fashionable women’s clubs held meetings to hear the story from the lips of girl strikers themselves, and women gave voice to their disapproval of judges who sentenced the young strikers to prison, where they were associated—often sharing the same cells—with criminals and prostitutes. Little wonder that women who had never known the bitterness of poverty or oppression found satisfaction in picketing side by side with the working girls who were paying the great cost of the strike. Many, among them settlement residents, readily went bail or paid fines for the girls who were arrested.
Cruel and dramatic exploitation of workers is in the main a thing of the past, but the more subtle injuries of modern industry, due to overstrain, speeding-up, and a minimum of leisure, have only recently attracted attention. It is barely three years (1912) since the New York Factory Law was amended to prohibit the employment of girls over sixteen for more than ten hours in one day or fifty-four hours a week. The legislation reflected the new compunction of the community concerning these workers, though unlimited hours are still permitted in stores during the Christmas season.
Few people realize what even a ten-hour day means, especially when the worker lives at a distance from the shop or factory and additional hours must be spent in going to and from the place of employment. And in New York travel during the rush hours may mean standing the entire distance.
Working girls, in their own vernacular, have “two jobs.” Those who have long hours and poor pay must live at the cheapest rate. Often they are not able to pay for more than part use of a bed, and however generous may be the provision of working girls’ hotels, the low-paid workers are not able to avail themselves of these. The girl
who receives the least wage must live down to the bone, cook her own meals, wash and iron her own shirtwaists, attend to all the necessary details for her home and person, and this after the long day. The cheapest worker is also likely to be the overtime worker, a fact that is most obvious to the public at Christmas time.
The Factory Investigating Commission, appointed after the Triangle fire to recommend measures for safety, was continued for the purpose of inquiry into the wages of labor throughout the state and also into the advisability of establishing a minimum wage rate. The reports of the commission, the public hearings, and the invaluable contributions to current periodicals are enlightening the community on the social perils due to giving a wage less than the necessary cost of decent living; and as the great majority of employees concerning whom this information has been gathered are young girls, the appeal to the public is bound to bring recommendations for safety in this respect. The dullness of life when pettiest economies must be forever practiced has also been well pictured in the testimony brought out by the commission.
In these chapters I have sought to portray the youth of our neighborhood at its more conscious and responsible period, when the age of greatest incorrigibility (said to be between thirteen and sixteen) has been passed. Labor discussions and solemn
conferences on social problems may seem an incongruous background for a picture of youth. Happily, its gayety is not easily suppressed, and comforting reassurance lies in the fact that recreation has ever for the young its strong and legitimate appeal; that art and music carry their message, and that the public conscience which recognizes the requirements of youth is reflected in the increasing provision for its pleasures. “Wider use of school buildings,” “recreation directors,” “social centers,” “municipal dances,” are new terms that have crept into our vocabularies.
Though the Italians have brought charming festas into our city streets, it was not until I admired the decorations that enhance the picturesque streets of Japan, and enjoyed the sight of the gay dancers on the boulevards of Paris on the day in July when the French celebrate, that it occurred to me that we might bring color
and gayety to the streets—even the ugly streets—of New York. For years Henry Street has had its dance on the Fourth of July, and the city and citizens share in the preparation and expense. The asphalt is put in good condition (once, for the very special occasion of the settlement’s twentieth birthday, the city officials hastened a contemplated renewal of the asphalt); the street-cleaning department gives an extra late-afternoon cleaning and keeps a white uniformed sweeper on duty during the festivity; the police department loans the stanchions and the park department the rope; the Edison Company illuminates with generosity; from the tenements and the settlement houses hang the flags and the bunting streamers, and the neighbors —all of us together—pay for the band. Asphalt, when swept and cleaned, makes an admirable dancing floor, and to this street dance come all the neighbors and their friends. The children play games to the music in their roped-off section, the young people dance, and all are merry. The first year of the experiment the friendly captain of the precinct asked what protection was needed. We had courage and faith to request that no officer should be added to the regular man on the beat, and the good conduct of the five or six thousand who danced or were spectators entirely justified the faith and the courage.
A I H P H S , C T A S
The protective legislation, the new terms in our vocabulary, and the dance on the street are but symbols of the acceptance by the community of its responsibility for protecting and nurturing its precious possession,—the youth of the city.
