PROPERTIESOFALGEBRAICSTACKS04X8Contents1.Introduction12.Conventionsand - PDF document

PROPERTIESOFALGEBRAICSTACKS04X8Contents1.Introduction12.Conventionsandabuseoflanguage13.Propertiesofmorphismsrepresentablebyalgebraicspaces34.Pointsofalgebraicstacks85.Surjectivemorphisms126.Quasi-compactalgebraicstacks127.Propertiesofalgebraicstacksdenedbypropertiesofschemes138.Monomorphismsofalgebraicstacks159.Immersionsofalgebraicstacks1610.Reducedalgebraicstacks2211.Residualgerbes2412.Dimensionofastack2913.Localirreducibility3014.Finitenessconditionsandpoints3115.Otherchapters31References321.Introduction04X9PleaseseeAlgebraicStacks,Section1forabriefintroductiontoalgebraicstacks,andpleasereadsomeofthatchapterforourfoundationsofalgebraicstacks.Thein-tentisthatinthatchapterwearecarefultodistinguishbetweenschemes,algebraicspaces,algebraicstacks,andstartingwiththischapterweemploythecustomaryabuseoflanguagewherealloftheseconceptsareusedinterchangeably.Thegoalofthischapteristointroducesomebasicnotionsandpropertiesofal-gebraicstacks.Afundamentalreferenceforthecaseofquasi-separatedalgebraicstackswithrepresentablediagonalis[LMB00].2.Conventionsandabuseoflanguage04XAWechooseabigfppfsiteSchfppf.AllschemesarecontainedinSchfppf.AndallringsAconsideredhavethepropertythatSpec(A)is(isomorphic)toanobjectofthisbigsite.WealsoxabaseschemeS,bytheconventionsaboveanelementofSchfppf.ThereaderwhoisonlyinterestedintheabsolutecasecantakeS=Spec(Z).Hereareourconventionsregardingalgebraicstacks: ThisisachapteroftheStacksProject,versiona577f147,compiledonSep05,2020.1 PROPERTIESOFALGEBRAICSTACKS2(1)WhenwesayalgebraicstackwewillmeananalgebraicstacksoverS,i.e.,acategorybredingroupoidsp:X!(Sch=S)fppfwhichsatisestheconditionsofAlgebraicStacks,Denition12.1.(2)Wewillsayf:X!Yisamorphismofalgebraicstackstoindicatea1-morphismofalgebraicstacksoverS,i.e.,a1-morphismofcategoriesbredingroupoidsover(Sch=S)fppf,seeAlgebraicStacks,Denition12.3.(3)A2-morphism:f!gwillindicatea2-morphisminthe2-categoryofalgebraicstacksoverS,seeAlgebraicStacks,Denition12.3.(4)GivenmorphismsX!ZandY!Zofalgebraicstacksweabusivelycallthe2-breproductXZYthebreproduct.(5)WewillwriteXSYfortheproductofthealgebraicstacksX,Y.(6)WewilloftenabusenotationandsaytwoalgebraicstacksXandYareisomorphiciftheyareequivalentinthis2-category.Hereareourconventionsregardingalgebraicspaces.(1)IfwesayXisanalgebraicspacethenwemeanthatXisanalgebraicspaceoverS,i.e.,Xisapresheafon(Sch=S)fppfwhichsatisestheconditionsofSpaces,Denition6.1.(2)Amorphismofalgebraicspacesf:X!YisamorphismofalgebraicspacesoverSasdenedinSpaces,Denition6.3.(3)WewillnotdistinguishbetweenanalgebraicspaceXandthealgebraicstackSX!(Sch=S)fppfitgivesriseto,seeAlgebraicStacks,Lemma13.1.(4)Inparticular,amorphismf:X!YfromXtoanalgebraicstackYmeansamorphismf:SX!Yofalgebraicstacks.SimilarlyformorphismsY!X.(5)Moreover,givenanalgebraicstackXwesayXisanalgebraicspacetoindicatethatXisrepresentablebyanalgebraicspace,seeAlgebraicStacks,Denition8.1.(6)Wewillusethefollowingnotationalconvention:Ifweindicateanalgebraicstackbyaromancapital(suchasX;Y;Z;A;B;:::)thenitwillbethecasethatitsinertiastackistrivial,andhenceitisanalgebraicspace,seeAlgebraicStacks,Proposition13.3.Hereareourconventionsregardingschemes.(1)IfwesayXisaschemethenwemeanthatXisaschemeoverS,i.e.,Xisanobjectof(Sch=S)fppf.(2)ByamorphismofschemeswemeanamorphismofschemesoverS.(3)WewillnotdistinguishbetweenaschemeXandthealgebraicstackSX!(Sch=S)fppfitgivesriseto,seeAlgebraicStacks,Lemma13.1.(4)Inparticular,amorphismf:X!YfromaschemeXtoanalgebraicstackYmeansamorphismf:SX!Yofalgebraicstacks.SimilarlyformorphismsY!X.(5)Moreover,givenanalgebraicstackXwesayXisaschemetoindicatethatXisrepresentable,seeAlgebraicStacks,Section4.Hereareourconventionsregardingmorphismsofalgebraicstacks:(1)Amorphismf:X!Yofalgebraicstacksisrepresentable,orrepresentablebyschemesifforeveryschemeTandmorphismT!YthebreproductTYXisascheme.SeeAlgebraicStacks,Section6. PROPERTIESOFALGEBRAICSTACKS3(2)Amorphismf:X!YofalgebraicstacksisrepresentablebyalgebraicspacesifforeveryschemeTandmorphismT!YthebreproductTYXisanalgebraicspace.SeeAlgebraicStacks,Denition9.1.InthiscaseZYXisanalgebraicspacewheneverZ!Yisamorphismwhosesourceisanalgebraicspace,seeAlgebraicStacks,Lemma9.8.NotethateverymorphismX!Yfromanalgebraicspacetoanalgebraicstackisrepresentablebyalgebraicspaces,seeAlgebraicStacks,Lemma10.11.Wewillusethisbasicresultwithoutfurthermention.3.Propertiesofmorphismsrepresentablebyalgebraicspaces04XBWewillstudypropertiesof(arbitrary)morphismsofalgebraicstacksinitsownchapter.Formorphismsrepresentablebyalgebraicspacesweknowwhatitmeanstobesurjective,smooth,orétale,etc.ThisappliesinparticulartomorphismsX!Yfromalgebraicspacestoalgebraicstacks.Inthissection,werecallhowthisworks,welistthepropertiestowhichthisapplies,andweproveafeweasylemmas.Ourrstlemmasaysamorphismisrepresentablebyalgebraicspacesifitissoafterabasechangebyaat,locallynitelypresented,surjectivemorphism.Lemma3.1.04ZPLetf:X!Ybeamorphismofalgebraicstacks.LetWbeanalgebraicspaceandletW!Ybesurjective,locallyofnitepresentation,andat.Thefollowingareequivalent(1)fisrepresentablebyalgebraicspaces,and(2)WYXisanalgebraicspace.Proof.Theimplication(1))(2)isAlgebraicStacks,Lemma9.8.Conversely,letW!Ybeasin(2).Toprove(1)itsucestoshowthatfisfaithfulonbrecategories,seeAlgebraicStacks,Lemma15.2.Assumption(2)impliesinparticularthatWYX!Wisfaithful.HencethefaithfulnessofffollowsfromStacks,Lemma6.9.LetPbeapropertyofmorphismsofalgebraicspaceswhichisfppflocalonthetargetandpreservedbyarbitrarybasechange.Letf:X!Ybeamorphismofalgebraicstacksrepresentablebyalgebraicspaces.ThenwesayfhaspropertyPifandonlyifforeveryschemeTandmorphismT!YthemorphismofalgebraicspacesTYX!ThaspropertyP,seeAlgebraicStacks,Denition10.1.Itturnsoutthatiff:X!YisrepresentablebyalgebraicspacesandhaspropertyP,thenforanymorphismofalgebraicstacksY0!YthebasechangeY0YX!Y0haspropertyP,seeAlgebraicStacks,Lemmas9.7and10.6.IfthepropertyPispreservedundercompositions,thenthisholdsalsointhesettingofmorphismsofalgebraicstacksrepresentablebyalgebraicspaces,seeAlgebraicStacks,Lemmas9.9and10.5.Moreover,inthiscaseproductsX1X2!Y1Y2ofmorphismsrepresentablebyalgebraicspaceshavingpropertyPhavepropertyP,seeAlgebraicStacks,Lemma10.8.Finally,ifwehavetwopropertiesP;P0ofmorphismsofalgebraicspaceswhicharefppflocalonthetargetandpreservedbyarbitrarybasechangeandifP(f))P0(f)foreverymorphismf,thenthesameimplicationholdsforthecorrespondingpropertyofmorphismsofalgebraicstacksrepresentablebyalgebraicspaces,see PROPERTIESOFALGEBRAICSTACKS4AlgebraicStacks,Lemma10.9.Wewillusethiswithoutfurthermentioninthefollowingandinthefollowingchapters.Thediscussionaboveappliestoeachofthefollowingpropertiesofmorphismsofalgebraicspaces(1)quasi-compact,seeMorphismsofSpaces,Lemma8.4andDescentonSpaces,Lemma10.1,(2)quasi-separated,seeMorphismsofSpaces,Lemma4.4andDescentonSpaces,Lemma10.2,(3)universallyclosed,seeMorphismsofSpaces,Lemma9.3andDescentonSpaces,Lemma10.3,(4)universallyopen,seeMorphismsofSpaces,Lemma6.3andDescentonSpaces,Lemma10.4,(5)universallysubmersive,seeMorphismsofSpaces,Lemma7.3andDescentonSpaces,Lemma10.5,(6)universalhomeomorphism,seeMorphismsofSpaces,Lemma53.4andDe-scentonSpaces,Lemma10.8,(7)surjective,seeMorphismsofSpaces,Lemma5.5andDescentonSpaces,Lemma10.6,(8)universallyinjective,seeMorphismsofSpaces,Lemma19.5andDescentonSpaces,Lemma10.7,(9)locallyofnitetype,seeMorphismsofSpaces,Lemma23.3andDescentonSpaces,Lemma10.9,(10)locallyofnitepresentation,seeMorphismsofSpaces,Lemma28.3andDescentonSpaces,Lemma10.10,(11)nitetype,seeMorphismsofSpaces,Lemma23.3andDescentonSpaces,Lemma10.11,(12)nitepresentation,seeMorphismsofSpaces,Lemma28.3andDescentonSpaces,Lemma10.12,(13)at,seeMorphismsofSpaces,Lemma30.4andDescentonSpaces,Lemma10.13,(14)openimmersion,seeMorphismsofSpaces,Section12andDescentonSpaces,Lemma10.14,(15)isomorphism,seeDescentonSpaces,Lemma10.15,(16)ane,seeMorphismsofSpaces,Lemma20.5andDescentonSpaces,Lemma10.16,(17)closedimmersion,seeMorphismsofSpaces,Section12andDescentonSpaces,Lemma10.17,(18)separated,seeMorphismsofSpaces,Lemma4.4andDescentonSpaces,Lemma10.18,(19)proper,seeMorphismsofSpaces,Lemma40.3andDescentonSpaces,Lemma10.19,(20)quasi-ane,seeMorphismsofSpaces,Lemma21.5andDescentonSpaces,Lemma10.20,(21)integral,seeMorphismsofSpaces,Lemma45.5andDescentonSpaces,Lemma10.22,(22)nite,seeMorphismsofSpaces,Lemma45.5andDescentonSpaces,Lemma10.23, PROPERTIESOFALGEBRAICSTACKS5(23)(locally)quasi-nite,seeMorphismsofSpaces,Lemma27.4andDescentonSpaces,Lemma10.24,(24)syntomic,seeMorphismsofSpaces,Lemma36.3andDescentonSpaces,Lemma10.25,(25)smooth,seeMorphismsofSpaces,Lemma37.3andDescentonSpaces,Lemma10.26,(26)unramied,seeMorphismsofSpaces,Lemma38.4andDescentonSpaces,Lemma10.27,(27)étale,seeMorphismsofSpaces,Lemma39.4andDescentonSpaces,Lemma10.28,(28)nitelocallyfree,seeMorphismsofSpaces,Lemma46.5andDescentonSpaces,Lemma10.29,(29)monomorphism,seeMorphismsofSpaces,Lemma10.5andDescentonSpaces,Lemma10.30,(30)immersion,seeMorphismsofSpaces,Section12andDescentonSpaces,Lemma11.1,(31)locallyseparated,seeMorphismsofSpaces,Lemma4.4andDescentonSpaces,Lemma11.2,Lemma3.2.04XCLetPbeapropertyofmorphismsofalgebraicspacesasabove.Letf:X!Ybeamorphismofalgebraicstacksrepresentablebyalgebraicspaces.Thefollowingareequivalent:(1)fhasP,(2)foreveryalgebraicspaceZandmorphismZ!YthemorphismZYX!ZhasP.Proof.Theimplication(2))(1)isimmediate.Assume(1).LetZ!Ybeasin(2).ChooseaschemeUandasurjectiveétalemorphismU!Z.ByassumptionthemorphismUYX!UhasP.ButthediagramUYX // ZYX U// Ziscartesian,hencetherightverticalarrowhasPasfU!Zgisanfppfcovering.ThefollowinglemmatellsusitsucestocheckPafterabasechangebyasurjective,at,locallynitelypresentedmorphism.Lemma3.3.04XDLetPbeapropertyofmorphismsofalgebraicspacesasabove.Letf:X!Ybeamorphismofalgebraicstacksrepresentablebyalgebraicspaces.LetWbeanalgebraicspaceandletW!Ybesurjective,locallyofnitepresentation,andat.SetV=WYX.Then(fhasP),(theprojectionV!WhasP):Proof.TheimplicationfromlefttorightfollowsfromLemma3.2.AssumeV!WhasP.LetTbeascheme,andletT!Ybeamorphism.Considerthe PROPERTIESOFALGEBRAICSTACKS6commutativediagramTYX TYW oo // W TTYVoo // Vofalgebraicspaces.Thesquaresarecartesian.Thebottomleftmorphismisasurjective,atmorphismwhichislocallyofnitepresentation,hencefTYV!Tgisanfppfcovering.HencethefactthattherightverticalarrowhaspropertyPimpliesthattheleftverticalarrowhaspropertyP.Lemma3.4.06TYLetPbeapropertyofmorphismsofalgebraicspacesasabove.Letf:X!Ybeamorphismofalgebraicstacksrepresentablebyalgebraicspaces.LetZ!Ybeamorphismofalgebraicstackswhichisrepresentablebyalgebraicspaces,surjective,at,andlocallyofnitepresentation.SetW=ZYX.Then(fhasP),(theprojectionW!ZhasP):Proof.ChooseanalgebraicspaceWandamorphismW!Zwhichissurjective,at,andlocallyofnitepresentation.BythediscussionabovethecompositionW!Yisalsosurjective,at,andlocallyofnitepresentation.DenoteV=WZW=VYX.ByLemma3.3weseethatfhasPifandonlyifV!WdoesandthatW!ZhasPifandonlyifV!Wdoes.Thelemmafollows.Lemma3.5.06M2LetPbeapropertyofmorphismsofalgebraicspacesasabove.Let2fetale;smooth;syntomic;fppfg.LetX!YandY!Zbemorphismsofalgebraicstacksrepresentablebyalgebraicspaces.Assume(1)X!Yissurjectiveandétale,smooth,syntomic,oratandlocallyofnitepresentation,(2)thecompositionhasP,and(3)Pislocalonthesourceinthetopology.ThenY!ZhaspropertyP.Proof.LetZbeaschemeandletZ!Zbeamorphism.SetX=XZZ,Y=YZZ.By(1)fX!Ygisacoveringofalgebraicspacesandby(2)X!ZhaspropertyP.By(3)thisimpliesthatY!ZhaspropertyPandwewin.Lemma3.6.04Y6Letg:X0!Xbeamorphismofalgebraicstackswhichisrepre-sentablebyalgebraicspaces.Let[U=R]!Xbeapresentation.SetU0=UXX0,andR0=RXX0.Thenthereexistsagroupoidinalgebraicspacesoftheform(U0;R0;s0;t0;c0),apresentation[U0=R0]!X0,andthediagram[U0=R0][pr] // X0g [U=R]// Xis2-commutativewherethemorphism[pr]comesfromamorphismofgroupoidspr:(U0;R0;s0;t0;c0)!(U;R;s;t;c). PROPERTIESOFALGEBRAICSTACKS7Proof.SinceU!Yissurjectiveandsmooth,seeAlgebraicStacks,Lemma17.2thebasechangeU0!X0isalsosurjectiveandsmooth.Hence,byAlgebraicStacks,Lemma16.2itsucestoshowthatR0=U0X0U0inordertogetasmoothgroupoid(U0;R0;s0;t0;c0)andapresentation[U0=R0]!X0.UsingthatR=VYV(seeGroupoidsinSpaces,Lemma21.2)thisfollowsfromR0=UXUXX0=(UXX0)X0(UXX0)seeCategories,Lemmas31.8and31.10.ClearlytheprojectionmorphismsU0!UandR0!Rgivethedesiredmorphismofgroupoidspr:(U0;R0;s0;t0;c0)!(U;R;s;t;c).Hencethemorphism[pr]ofquotientstacksbyGroupoidsinSpaces,Lemma20.1.Westillhavetoshowthatthediagram2-commutes.ItisclearthatthediagramU0prU f0// X0g Uf// X2-commuteswhereprU:U0!Uistheprojection.Thereisacanonical2-arrow:ft!fsinMor(R;X)comingfromR=UXU,t=pr0,ands=pr1.UsingtheisomorphismR0!U0X0U0wegetsimilarlyanisomorphism0:f0t0!f0s0.Notethatgf0t0=ftprRandgf0s0=fsprR,whereprR:R0!Ristheprojection.Thusitmakessensetoaskif(3.6.1)04Y7?idprR=idg?0:Nowwemaketwoclaims:(1)ifEquation(3.6.1)holds,thenthediagram2-commutes,and(2)Equation(3.6.1)holds.Weomittheproofofbothclaims.Hints:part(1)followsfromtheconstructionoff=fcanandf0=f0caninAl-gebraicStacks,Lemma16.1.Part(2)followsbycarefulyworkingthroughthedenitions.Remark3.7.04ZQLetYbeanalgebraicstack.Considerthefollowing2-category:(1)Anobjectisamorphismf:X!Ywhichisrepresentablebyalgebraicspaces,(2)a1-morphism(g;):(f1:X1!Y)!(f2:X2!Y)consistsofamorphismg:X1!X2anda2-morphism:f1!f2g,and(3)a2-morphismbetween(g;);(g0;0):(f1:X1!Y)!(f2:X2!Y)isa2-morphism:g!g0suchthat(idf2?)=0.Letusdenotethis2-categorySpaces=YbyanalogywiththenotationofTopologiesonSpaces,Section2.Nowweclaimthatinthis2-categorythemorphismcategoriesMorSpaces=Y((f1:X1!Y);(f2:X2!Y))areallsetoids.Namely,a2-morphismisarulewhichtoeachobjectx1ofX1assignsanisomorphismx1:g(x1)�!g0(x1)intherelevantbrecategoryofX2suchthatthediagramf2(x1) x1yy 0x1%% f2(g(x1))f2(x1)// f2(g0(x1)) PROPERTIESOFALGEBRAICSTACKS8commutes.Butsincef2isfaithful(seeAlgebraicStacks,Lemma15.2)thismeansthatifx1exists,thenitisunique!Inotherwordsthe2-categorySpaces=Yisveryclosetobeingacategory.Namely,ifwereplace1-morphismsbyisomorphismclassesof1-morphismsweobtainacategory.Wewilloftenperformthisreplacementwithoutfurthermention.4.Pointsofalgebraicstacks04XELetXbeanalgebraicstack.LetK;Lbetwoeldsandletp:Spec(K)!Xandq:Spec(L)!Xbemorphisms.Wesaythatpandqareequivalentifthereexistsaeld anda2-commutativediagramSpec( )//  Spec(L)q Spec(K)p// X:Lemma4.1.04XFThenotionabovedoesindeeddeneanequivalencerelationonmor-phismsfromspectraofeldsintothealgebraicstackX.Proof.Itisclearthattherelationisreexiveandsymmetric.Hencewehavetoprovethatitistransitive.Thiscomesdowntothefollowing:GivenadiagramSpec( )b// a Spec(L)q Spec( 0)b0oo a0 Spec(K)p// XSpec(K0)p0oo withbothsquares2-commutativewehavetoshowthatpisequivalenttop0.Bythe2-Yonedalemma(seeAlgebraicStacks,Section5)themorphismsp,p0,andqaregivenbyobjectsx,x0,andyinthebrecategoriesofXoverSpec(K),Spec(K0),andSpec(L).The2-commutativityofthesquaresmeansthatthereareisomorphisms:ax!byand0:(a0)x0!(b0)yinthebrecategoriesofXoverSpec( )andSpec( 0).Chooseanyeld 00andembeddings ! 00and 0! 00agreeingonL.ThenwecanextendthediagramabovetoSpec( 00) cxx q0 c0&& Spec( )b// a Spec(L)q Spec( 0)b0oo a0 Spec(K)p// XSpec(K0)p0oo withcommutativetrianglesand(q0)(0)�1(q0):(ac)x�!(a0c0)x0isanisomorphisminthebrecategoryoverSpec( 00).Hencepisequivalenttop0asdesired.Denition4.2.04XGLetXbeanalgebraicstack.ApointofXisanequivalenceclassofmorphismsfromspectraofeldsintoX.ThesetofpointsofXisdenotedjXj. PROPERTIESOFALGEBRAICSTACKS9Thisagreeswithourdenitionofpointsofalgebraicspaces,seePropertiesofSpaces,Denition4.1.Moreover,foraschemewerecovertheusualnotionofpoints,seePropertiesofSpaces,Lemma4.2.Iff:X!Yisamorphismofalgebraicstacksthenthereisaninducedmapjfj:jXj!jYjwhichmapsarepre-sentativex:Spec(K)!Xtotherepresentativefx:Spec(K)!Y.Thisiswelldened:namely2-isomorphic1-morphismsremain2-isomorphicafterpre-orpost-composingbya1-morphismbecauseyoucanhorizontallypre-orpost-composebytheidentityofthegiven1-morphism.Thisholdsinany(strict)(2;1)-category.IfX // Y W// Zisa2-commutativediagramofalgebraicstacks,thenthediagramofsetsjXj // jYj jWj// jZjiscommutative.Inparticular,ifX!YisanequivalencethenjXj!jYjisabijection.Lemma4.3.04XHLetZYX//  X Z// Ybeabreproductofalgebraicstacks.ThenthemapofsetsofpointsjZYXj�!jZjjYjjXjissurjective.Proof.Namely,supposegiveneldsK,LandmorphismsSpec(K)!X,Spec(L)!Z,thentheassumptionthattheyagreeaselementsofjYjmeansthatthereisacommonextensionKMandLMsuchthatSpec(M)!Spec(K)!X!YandSpec(M)!Spec(L)!Z!Yare2-isomorphic.AndthisisexactlytheconditionwhichsaysyougetamorphismSpec(M)!ZYX.Lemma4.4.04XILetf:X!Ybeamorphismofalgebraicstackswhichisrepre-sentablebyalgebraicspaces.Thefollowingareequivalent:(1)jfj:jXj!jYjissurjective,and(2)fissurjective(inthesenseofSection3).Proof.Assume(1).LetT!Ybeamorphismwhosesourceisascheme.Toprove(2)wehavetoshowthatthemorphismofalgebraicspacesTYX!Tissurjective.ByMorphismsofSpaces,Denition5.2thismeanswehavetoshowthatjTYXj!jTjissurjective.ApplyingLemma4.3weseethatthisfollowsfrom(1).Conversely,assume(2).Lety:Spec(K)!YbeamorphismfromthespectrumofaeldintoY.ByassumptionthemorphismSpec(K)y;YX!Spec(K)of PROPERTIESOFALGEBRAICSTACKS10algebraicspacesissurjective.ByMorphismsofSpaces,Denition5.2thismeansthereexistsaeldextensionKK0andamorphismSpec(K0)!Spec(K)y;YXsuchthattheleftsquareofthediagramSpec(K0)//  Spec(K)y;YX // X Spec(K) Spec(K)y// Yiscommutative.ThisshowsthatjXj!jYjissurjective.Hereisalemmaexplaininghowtocomputethesetofpointsintermsofapresen-tation.Lemma4.5.04XJLetXbeanalgebraicstack.LetX=[U=R]beapresentationofX,seeAlgebraicStacks,Denition16.5.ThentheimageofjRj!jUjjUjisanequivalencerelationandjXjisthequotientofjUjbythisequivalencerelation.Proof.Theassumptionmeansthatwehaveasmoothgroupoid(U;R;s;t;c)inalgebraicspaces,andanequivalencef:[U=R]!X.WemayassumeX=[U=R].Theinducedmorphismp:U!Xissmoothandsurjective,seeAlgebraicStacks,Lemma17.2.HencejUj!jXjissurjectivebyLemma4.4.NotethatR=UXU,seeGroupoidsinSpaces,Lemma21.2.HenceLemma4.3impliesthemapjRj�!jUjjXjjUjissurjective.HencetheimageofjRj!jUjjUjisexactlythesetofpairs(u1;u2)2jUjjUjsuchthatu1andu2havethesameimageinjXj.Combiningthesetwostatementswegettheresultofthelemma.Remark4.6.04XKTheresultofLemma4.5canbegeneralizedasfollows.LetXbeanalgebraicstack.LetUbeanalgebraicspaceandletf:U!Xbeasurjectivemorphism(whichmakessensebySection3).LetR=UXU,let(U;R;s;t;c)bethegroupoidinalgebraicspaces,andletfcan:[U=R]!XbethecanonicalmorphismasconstructedinAlgebraicStacks,Lemma16.1.ThentheimageofjRj!jUjjUjisanequivalencerelationandjXj=jUj=jRj.TheproofofLemma4.5workswithoutchange.(Ofcourseingeneral[U=R]isnotanalgebraicstack,andingeneralfcanisnotanisomorphism.)Lemma4.7.04XLThereexistsauniquetopologyonthesetsofpointsofalgebraicstackswiththefollowingproperties:(1)foreverymorphismofalgebraicstacksX!YthemapjXj!jYjiscontinuous,and(2)foreverymorphismU!Xwhichisatandlocallyofnitepresenta-tionwithUanalgebraicspacethemapoftopologicalspacesjUj!jXjiscontinuousandopen.Proof.Chooseamorphismp:U!Xwhichissurjective,at,andlocallyofnitepresentationwithUanalgebraicspace.Suchexistbythedenitionofanalgebraicstack,asasmoothmorphismisatandlocallyofnitepresentation(seeMorphismsofSpaces,Lemmas37.5and37.7).WedeneatopologyonjXjbytherule:WjXjisopenifandonlyifjpj�1(W)isopeninjUj.Toshowthatthisisindependentofthechoiceofp,letp0:U0!Xbeanothermorphismwhich PROPERTIESOFALGEBRAICSTACKS11issurjective,at,locallyofnitepresentationfromanalgebraicspacetoX.SetU00=UXU0sothatwehavea2-commutativediagramU00//  U0 U// XAsU!XandU0!Xaresurjective,at,locallyofnitepresentationweseethatU00!U0andU00!Uaresurjective,atandlocallyofnitepresentation,seeLemma3.2.HencethemapsjU00j!jU0jandjU00j!jUjarecontinuous,openandsurjective,seeMorphismsofSpaces,Denition5.2andLemma30.6.Thisclearlyimpliesthatourdenitionisindependentofthechoiceofp:U!X.Letf:X!Ybeamorphismofalgebraicstacks.ByAlgebraicStacks,Lemma15.1wecannda2-commutativediagramUx a// Vy Xf// Ywithsurjectivesmoothverticalarrows.Considertheassociatedcommutativedia-gramjUjjxj jaj// jVjjyj jXjjfj// jYjofsets.IfWjYjisopen,thenbythedenitionabovethismeansexactlythatjyj�1(W)isopeninjVj.Sincejajiscontinuousweconcludethatjaj�1jyj�1(W)=jxj�1jfj�1(W)isopeninjWjwhichmeansbydenitionthatjfj�1(W)isopeninjXj.Thusjfjiscontinuous.Finally,wehavetoshowthatifUisanalgebraicspace,andU!Xisatandlocallyofnitepresentation,thenjUj!jXjisopen.LetV!Xbesurjective,at,andlocallyofnitepresentationwithVanalgebraicspace.ConsiderthecommutativediagramjUXVje// f&& jUjjXjjVjc d// jVjb jUja// jXjNowthemorphismUXV!Uissurjective,i.e,f:jUXVj!jUjissurjective.Thelefttophorizontalarrowissurjective,seeLemma4.3.ThemorphismUXV!Visatandlocallyofnitepresentation,hencede:jUXVj!jVjisopen,seeMorphismsofSpaces,Lemma30.6.PickWjUjopen.Thepropertiesaboveimplythatb�1(a(W))=(de)(f�1(W))isopen,whichbyconstructionmeansthata(W)isopenasdesired.Denition4.8.04Y8LetXbeanalgebraicstack.TheunderlyingtopologicalspaceofXisthesetofpointsjXjendowedwiththetopologyconstructedinLemma4.7. PROPERTIESOFALGEBRAICSTACKS12ThisdenitiondoesnotconictwiththealreadyexistingtopologyonjXjifXisanalgebraicspace.Lemma4.9.04Y9LetXbeanalgebraicstack.EverypointofjXjhasafundamentalsystemofquasi-compactopenneighbourhoods.InparticularjXjislocallyquasi-compactinthesenseofTopology,Denition13.1.Proof.ThisfollowsformallyfromthefactthatthereexistsaschemeUandasurjective,open,continuousmapU!jXjoftopologicalspaces.Namely,ifU!Xissurjectiveandsmooth,thenLemma4.7guaranteesthatjUj!jXjiscontinuous,surjective,andopen.5.Surjectivemorphisms04ZRLetf:X!Ybeamorphismofalgebraicstackswhichisrepresentablebyalgebraicspaces.InSection3wehavealreadydenedwhatitmeansforftobesurjective.InLemma4.4wehaveseenthatthisisequivalenttorequiringjfj:jXj!jYjtobesurjective.Thisclearsthewayforthefollowingdenition.Denition5.1.04ZSLetf:X!Ybeamorphismofalgebraicstacks.Wesayfissurjectiveifthemapjfj:jXj!jYjofassociatedtopologicalspacesissurjective.Herearesomelemmas.Lemma5.2.04ZTThecompositionofsurjectivemorphismsissurjective.Proof.Omitted.Lemma5.3.04ZUThebasechangeofasurjectivemorphismissurjective.Proof.Omitted.Hint:UseLemma4.3.Lemma5.4.06PMLetf:X!Ybeamorphismofalgebraicstacks.LetY0!Ybeasurjectivemorphismofalgebraicstacks.Ifthebasechangef0:Y0YX!Y0offissurjective,thenfissurjective.Proof.ImmediatefromLemma4.3.Lemma5.5.06PNLetX!Y!Zbemorphismsofalgebraicstacks.IfX!ZissurjectivesoisY!Z.Proof.Immediate.6.Quasi-compactalgebraicstacks04YAThefollowingdenitionisequivalentwiththedenitionforalgebraicspacesbyPropertiesofSpaces,Lemma5.2.Denition6.1.04YBLetXbeanalgebraicstack.WesayXisquasi-compactifandonlyifjXjisquasi-compact.Lemma6.2.04YCLetXbeanalgebraicstack.Thefollowingareequivalent:(1)Xisquasi-compact,(2)thereexistsasurjectivesmoothmorphismU!XwithUaquasi-compactscheme,(3)thereexistsasurjectivesmoothmorphismU!XwithUaquasi-compactalgebraicspace,and PROPERTIESOFALGEBRAICSTACKS13(4)thereexistsasurjectivemorphismU!XofalgebraicstackssuchthatUisquasi-compact.Proof.WewilluseLemma4.4.SupposeUandU!Xareasin(4).ThensincejUj!jXjissurjectiveandcontinuousweconcludethatjXjisquasi-compact.Thus(4)implies(1).Theimplications(2))(3))(4)areimmediate.Assume(1),i.e.,Xisquasi-compact,i.e.,thatjXjisquasi-compact.ChooseaschemeUandasurjectivesmoothmorphismU!X.ThensincejUj!jXjisopenweseethatthereexistsaquasi-compactopenU0UsuchthatjU0j!jXjissurjective(andstillsmooth).Hence(2)holds.Lemma6.3.04YDAnitedisjointunionofquasi-compactalgebraicstacksisaquasi-compactalgebraicstack.Proof.Thisisclearfromthecorrespondingtopologicalfact.7.Propertiesofalgebraicstacksdenedbypropertiesofschemes04YEAnysmoothlocalpropertyofschemesgivesrisetoacorrespondingpropertyofalgebraicstacksviathefollowinglemma.Notethatapropertyofschemeswhichissmoothlocalisalsoétalelocalasanyétalecoveringisalsoasmoothcovering.HenceforasmoothlocalpropertyPofschemesweknowwhatitmeanstosaythatanalgebraicspacehasP,seePropertiesofSpaces,Section7.Lemma7.1.04YFLetPbeapropertyofschemeswhichislocalinthesmoothtopol-ogy,seeDescent,Denition12.1.LetXbeanalgebraicstack.Thefollowingareequivalent(1)forsomeschemeUandsomesurjectivesmoothmorphismU!XtheschemeUhaspropertyP,(2)foreveryschemeUandeverysmoothmorphismU!XtheschemeUhaspropertyP,(3)forsomealgebraicspaceUandsomesurjectivesmoothmorphismU!XthealgebraicspaceUhaspropertyP,and(4)foreveryalgebraicspaceUandeverysmoothmorphismU!Xthealge-braicspaceUhaspropertyP.IfXisaschemethisisequivalenttoP(U).IfXisanalgebraicspacethisisequivalenttoXhavingpropertyP.Proof.LetU!XsurjectiveandsmoothwithUanalgebraicspace.LetV!XbeasmoothmorphismwithVanalgebraicspace.ChooseschemesU0andV0andsurjectiveétalemorphismsU0!UandV0!V.Finally,chooseaschemeWandasurjectiveétalemorphismW!V0XU0.ThenW!V0andW!U0aresmoothmorphismsofschemesascompositionsofétaleandsmoothmorphismsofalgebraicspaces,seeMorphismsofSpaces,Lemmas39.6and37.2.Moreover,W!V0issurjectiveasU0!Xissurjective.Hence,wehaveP(U),P(U0))P(W))P(V0),P(V)wheretheequivalencesarebydenitionofpropertyPforalgebraicspaces,andthetwoimplicationscomefromDescent,Denition12.1.Thisproves(3))(4).Theimplications(2))(1),(1))(3),and(4))(2)areimmediate. PROPERTIESOFALGEBRAICSTACKS14Denition7.2.04YGLetXbeanalgebraicstack.LetPbeapropertyofschemeswhichislocalinthesmoothtopology.WesayXhaspropertyPifanyoftheequivalentconditionsofLemma7.1hold.Remark7.3.04YHHereisalistofpropertieswhicharelocalforthesmoothtopology(keepinmindthatthefpqc,fppf,andsyntomictopologiesarestrongerthanthesmoothtopology):(1)locallyNoetherian,seeDescent,Lemma13.1,(2)Jacobson,seeDescent,Lemma13.2,(3)locallyNoetherianand(Sk),seeDescent,Lemma14.1,(4)Cohen-Macaulay,seeDescent,Lemma14.2,(5)reduced,seeDescent,Lemma15.1,(6)normal,seeDescent,Lemma15.2,(7)locallyNoetherianand(Rk),seeDescent,Lemma15.3,(8)regular,seeDescent,Lemma15.4,(9)Nagata,seeDescent,Lemma15.5.Anysmoothlocalpropertyofgermsofschemesgivesrisetoacorrespondingprop-ertyofalgebraicstacks.Notethatapropertyofgermswhichissmoothlocalisalsoétalelocal.HenceforasmoothlocalpropertyofgermsofschemesPweknowwhatitmeanstosaythatanalgebraicspaceXhaspropertyPatx2jXj,seePropertiesofSpaces,Section7.Lemma7.4.04YILetXbeanalgebraicstack.Letx2jXjbeapointofX.LetPbeapropertyofgermsofschemeswhichissmoothlocal,seeDescent,Denition18.1.Thefollowingareequivalent(1)foranysmoothmorphismU!XwithUaschemeandu2Uwitha(u)=xwehaveP(U;u),(2)forsomesmoothmorphismU!XwithUaschemeandsomeu2Uwitha(u)=xwehaveP(U;u),(3)foranysmoothmorphismU!XwithUanalgebraicspaceandu2jUjwitha(u)=xthealgebraicspaceUhaspropertyPatu,and(4)forsomesmoothmorphismU!XwithUaanalgebraicspaceandsomeu2jUjwitha(u)=xthealgebraicspaceUhaspropertyPatu.IfXisrepresentable,thenthisisequivalenttoP(X;x).IfXisanalgebraicspacethenthisisequivalenttoXhavingpropertyPatx.Proof.Leta:U!Xandu2jUjasin(3).Letb:V!XbeanothersmoothmorphismwithVanalgebraicspaceandv2jVjwithb(v)=xalso.ChooseaschemeU0,anétalemorphismU0!Uandu02U0mappingtou.ChooseaschemeV0,anétalemorphismV0!Vandv02V0mappingtov.ByLemma4.3thereexistsapoint w2jV0XU0jmappingtou0andv0.ChooseaschemeWandasurjectiveétalemorphismW!V0XU0.Wemaychooseaw2jWjmappingto w(seePropertiesofSpaces,Lemma4.4).ThenW!V0andW!U0aresmoothmorphismsofschemesascompositionsofétaleandsmoothmorphismsofalgebraicspaces,seeMorphismsofSpaces,Lemmas39.6and37.2.HenceP(U;u),P(U0;u0),P(W;w),P(V0;v0),P(V;v)TheoutertwoequivalencesbyPropertiesofSpaces,Denition7.5andtheothertwobywhatitmeanstobeasmoothlocalpropertyofgermsofschemes.Thisproves(4))(3). PROPERTIESOFALGEBRAICSTACKS15Theimplications(1))(2),(2))(4),and(3))(1)areimmediate.Denition7.5.04YJLetPbeapropertyofgermsofschemeswhichissmoothlocal.LetXbeanalgebraicstack.Letx2jXj.WesayXhaspropertyPatxifanyoftheequivalentconditionsofLemma7.4holds.8.Monomorphismsofalgebraicstacks04ZVWedeneamonomorphismofalgebraicstacksinthefollowingway.WewillseeinLemma8.4thatthisiscompatiblewiththecorresponding2-categorytheoreticnotion.Denition8.1.04ZWLetf:X!Ybeamorphismofalgebraicstacks.WesayfisamonomorphismifitisrepresentablebyalgebraicspacesandamonomorphisminthesenseofSection3.Firstsomebasiclemmas.Lemma8.2.04ZXLetX!Ybeamorphismofalgebraicstacks.LetZ!Ybeamonomorphism.ThenZYX!Xisamonomorphism.Proof.ThisfollowsfromthegeneraldiscussioninSection3.Lemma8.3.04ZYCompositionsofmonomorphismsofalgebraicstacksaremonomor-phisms.Proof.ThisfollowsfromthegeneraldiscussioninSection3andMorphismsofSpaces,Lemma10.4.Lemma8.4.04ZZLetf:X!Ybeamorphismofalgebraicstacks.Thefollowingareequivalent:(1)fisamonomorphism,(2)fisfullyfaithful,(3)thediagonal
f:X!XYXisanequivalence,and(4)thereexistsanalgebraicspaceWandasurjective,atmorphismW!YwhichislocallyofnitepresentationsuchthatV=XYWisanalgebraicspace,andthemorphismV!Wisamonomorphismofalgebraicspaces.Proof.Theequivalenceof(1)and(4)followsfromthegeneraldiscussioninSection3andinparticularLemmas3.1and3.3.Theequivalenceof(2)and(3)isCategories,Lemma35.9.Assumetheequivalentconditions(2)and(3).ThenfisrepresentablebyalgebraicspacesaccordingtoAlgebraicStacks,Lemma15.2.Moreover,the2-YonedalemmacombinedwiththefullyfaithfulnessimpliesthatforeveryschemeTthefunctorMor(T;X)�!Mor(T;Y)isfullyfaithful.Hencegivenamorphismy:T!Ythereexistsuptounique2-isomorphismatmostonemorphismx:T!Xsuchthaty=fx.Inparticular,givenamorphismofschemesh:T0!Tthereexistsatmostonelift~h:T0!TYXofh.ThusTYX!Tisamonomorphismofalgebraicspaces,whichprovesthat(1)holds.Finally,assumethat(1)holds.ThenforanyschemeTandmorphismy:T!YthebreproductTYXisanalgebraicspace,andTYX!Tisamonomorphism. PROPERTIESOFALGEBRAICSTACKS16Hencethereexistsuptouniqueisomorphismexactlyonepair(x;)wherex:T!Xisamorphismand:fx!yisa2-morphism.Applyingthe2-Yonedalemmathissaysexactlythatfisfullyfaithful,i.e.,that(2)holds.Lemma8.5.0500Amonomorphismofalgebraicstacksinducesaninjectivemapofsetsofpoints.Proof.Letf:X!Ybeamonomorphismofalgebraicstacks.Supposethatxi:Spec(Ki)!Xbemorphismssuchthatfx1andfx2denethesameelementofjYj.Applyingthedenitionwendacommonextension withcorrespondingmorphismsci:Spec( )!Spec(Ki)anda2-isomorphism:fx1c1!fx1c2.Asfisfullyfaithful,seeLemma8.4,wecanlifttoanisomorphism:fx1c1!fx1c2.Hencex1andx2denethesamepointofjXjasdesired.Lemma8.6.0CBBLetX!X0!Ybemorphismsofalgebraicstacks.IfX!X0isamonomorphismthenthecanonicaldiagramX//  XYX X0// X0YX0isabreproductsquare.Proof.WehaveX=XX0XbyLemma8.4.ThustheresultbyapplyingCategories,Lemma31.13.9.Immersionsofalgebraicstacks04YKImmersionsofalgebraicstacksaredenedasfollows.Denition9.1.04YLImmersions.(1)Amorphismofalgebraicstacksiscalledanopenimmersionifitisrepre-sentable,andanopenimmersioninthesenseofSection3.(2)Amorphismofalgebraicstacksiscalledaclosedimmersionifitisrepre-sentable,andaclosedimmersioninthesenseofSection3.(3)Amorphismofalgebraicstacksiscalledanimmersionifitisrepresentable,andanimmersioninthesenseofSection3.Thisisnotthemostconvenientwaytothinkaboutimmersionsforus.ForusitisalittlebitmoreconvenienttothinkofanimmersionasamorphismofalgebraicstackswhichisrepresentablebyalgebraicspacesandisanimmersioninthesenseofSection3.Similarlyforclosedandopenimmersions.Sincethisisclearlyequivalenttothenotionjustdenedweshallusethischaracterizationwithoutfurthermention.Weproveafewsimplelemmasaboutthisnotion.Lemma9.2.0501LetX!Ybeamorphismofalgebraicstacks.LetZ!Ybea(closed,resp.open)immersion.ThenZYX!Xisa(closed,resp.open)immersion.Proof.ThisfollowsfromthegeneraldiscussioninSection3.Lemma9.3.0502Compositionsofimmersionsofalgebraicstacksareimmersions.Similarlyforclosedimmersionsandopenimmersions. PROPERTIESOFALGEBRAICSTACKS17Proof.ThisfollowsfromthegeneraldiscussioninSection3andSpaces,Lemma12.2.Lemma9.4.0503Letf:X!Ybeamorphismofalgebraicstacks.letWbeanalgebraicspaceandletW!Ybeasurjective,atmorphismwhichislocallyofnitepresentation.Thefollowingareequivalent:(1)fisan(open,resp.closed)immersion,and(2)V=WYXisanalgebraicspace,andV!Wisan(open,resp.closed)immersion.Proof.ThisfollowsfromthegeneraldiscussioninSection3andinparticularLemmas3.1and3.3.Lemma9.5.0504Animmersionisamonomorphism.Proof.SeeMorphismsofSpaces,Lemma10.7.Thefollowingtwolemmasexplainhowtothinkaboutimmersionsintermsofpresentations.Lemma9.6.0505Let(U;R;s;t;c)beasmoothgroupoidinalgebraicspaces.Leti:Z![U=R]beanimmersion.ThenthereexistsanR-invariantlocallyclosedsubspaceZUandapresentation[Z=RZ]!ZwhereRZistherestrictionofRtoZsuchthat[Z=RZ] $$ // Z i}} [U=R]is2-commutative.Ifiisaclosed(resp.open)immersionthenZisaclosed(resp.open)subspaceofU.Proof.ByLemma3.6wegetacommutativediagram[U0=R0] $$ // Z}} [U=R]whereU0=Z[U=R]UandR0=Z[U=R]R.SinceZ![U=R]isanimmersionweseethatU0!Uisanimmersionofalgebraicspaces.LetZUbethelocallyclosedsubspacesuchthatU0!UfactorsthroughZandinducesanisomorphismU0!Z.ItisclearfromtheconstructionofR0thatR0=U0U;tR=Rs;UU0.ThisimpliesthatZ=U0isR-invariantandthattheimageofR0!RidentiesR0withtherestrictionRZ=s�1(Z)=t�1(Z)ofRtoZ.Hencethelemmaholds.Lemma9.7.04YNLet(U;R;s;t;c)beasmoothgroupoidinalgebraicspaces.LetX=[U=R]betheassociatedalgebraicstack,seeAlgebraicStacks,Theorem17.3.LetZUbeanR-invariantlocallyclosedsubspace.Then[Z=RZ]�![U=R]isanimmersionofalgebraicstacks,whereRZistherestrictionofRtoZ.IfZUisopen(resp.closed)thenthemorphismisanopen(resp.closed)immersionofalgebraicstacks. PROPERTIESOFALGEBRAICSTACKS18Proof.RecallthatbyGroupoidsinSpaces,Denition17.1(seealsodiscussionfollowingthedenition)wehaveRZ=s�1(Z)=t�1(Z)aslocallyclosedsubspacesofR.HencethetwomorphismsRZ!Zaresmoothasbasechangesofsandt.Hence(Z;RZ;sjRZ;tjRZ;cjRZs;Z;tRZ)isasmoothgroupoidinalgebraicspaces,andweseethat[Z=RZ]isanalgebraicstack,seeAlgebraicStacks,Theorem17.3.TheassumptionsofGroupoidsinSpaces,Lemma24.3areallsatisedanditfollowsthatwehavea2-bresquareZ // [Z=RZ] U// [U=R]ItfollowsfromthisandLemma3.1that[Z=RZ]![U=R]isrepresentablebyalgebraicspaces,whereuponitfollowsfromLemma3.3thattherightverticalarrowisanimmersion(resp.closedimmersion,resp.openimmersion)ifandonlyiftheleftverticalarrowis.Wecandeneopen,closed,andlocallyclosedsubstacksasfollows.Denition9.8.04YMLetXbeanalgebraicstack.(1)AnopensubstackofXisastrictlyfullsubcategoryX0XsuchthatX0isanalgebraicstackandX0!Xisanopenimmersion.(2)AclosedsubstackofXisastrictlyfullsubcategoryX0XsuchthatX0isanalgebraicstackandX0!Xisaclosedimmersion.(3)AlocallyclosedsubstackofXisastrictlyfullsubcategoryX0XsuchthatX0isanalgebraicstackandX0!Xisanimmersion.Thisdenitionshouldbeusedwithcaution.Namely,iff:X!YisanequivalenceofalgebraicstacksandX0Xisanopensubstack,thenitisnotnecessarilythecasethatthesubcategoryf(X0)isanopensubstackofY.Theproblemisthatitmaynotbeastrictlyfullsubcategory;butthisisalsotheonlyproblem.Hereisaformalstatement.Lemma9.9.0506Foranyimmersioni:Z!XthereexistsauniquelocallyclosedsubstackX0Xsuchthatifactorsasthecompositionofanequivalencei0:Z!X0followedbytheinclusionmorphismX0!X.Ifiisaclosed(resp.open)immersion,thenX0isaclosed(resp.open)substackofX.Proof.Omitted.Lemma9.10.0507Let[U=R]!Xbeapresentationofanalgebraicstack.ThereisacanonicalbijectionlocallyclosedsubstacksZofX�!R-invariantlocallyclosedsubspacesZofUwhichsendsZtoUXZ.Moreover,amorphismofalgebraicstacksf:Y!XfactorsthroughZifandonlyifYXU!UfactorsthroughZ.Similarlyforclosedsubstacksandopensubstacks.Proof.ByLemmas9.6and9.7wendthatthemapisabijection.IfY!XfactorsthroughZthenofcoursethebasechangeYXU!UfactorsthroughZ.Converse,supposethatY!XisamorphismsuchthatYXU!UfactorsthroughZ.WewillshowthatforeveryschemeTandmorphismT!Y,givenby PROPERTIESOFALGEBRAICSTACKS19anobjectyofthebrecategoryofYoverT,theobjectyisinfactinthebrecategoryofZoverT.Namely,thebreproductTXUisanalgebraicspaceandTXU!Tisasurjectivesmoothmorphism.HencethereisanfppfcoveringfTi!TgsuchthatTi!TfactorsthroughTXU!Tforalli.ThenTi!XfactorsthroughYXUandhencethroughZU.ThusyjTiisanobjectofZ(asZisthebreproductofUwithZoverX).SinceZisastrictlyfullsubstack,weconcludethatyisanobjectofZasdesired.Lemma9.11.06FJLetXbeanalgebraicstack.TheruleU7!jUjdenesaninclusionpreservingbijectionbetweenopensubstacksofXandopensubsetsofjXj.Proof.Chooseapresentation[U=R]!X,seeAlgebraicStacks,Lemma16.2.ByLemma9.10weseethatopensubstackscorrespondtoR-invariantopensubschemesofU.OntheotherhandLemmas4.5and4.7guaranteethesecorrespondbijectivelytoopensubsetsofjXj.Lemma9.12.05UPLetXbeanalgebraicstack.LetUbeanalgebraicspaceandU!Xasurjectivesmoothmorphism.ForanopenimmersionV,!U,thereexistsanalgebraicstackY,anopenimmersionY!X,andasurjectivesmoothmorphismV!Y.Proof.WedeneacategorybredingroupoidsYbylettingthebercategoryYToveranobjectTof(Sch=S)fppfbethefullsubcategoryofXTconsistingofally2Ob(XT)suchthattheprojectionmorphismVX;yT!Tsurjective.Nowforanymorphismx:T!X,the2-bredproductTx;XYhasbercategoryoverT0consistingoftriples(f:T0!T;y2XT0;fx'y)suchthatVX;yT0!T0issurjective.NotethatTx;XYisberedinsetoidssinceY!Xisfaithful(seeStacks,Lemma6.7).Nowtheisomorphismfx'ygivesthediagramVX;yT0 // VX;xT//  V T0f// Tx// Xwherebothsquaresarecartesian.ThemorphismVX;xT!Tissmoothbybasechange,andhenceopen.LetT0Tbeitsimage.FromthecartesiansquareswededucethatVX;yT0!T0issurjectiveifandonlyifflandsinT0.ThereforeTx;XYisrepresentablebyT0,sotheinclusionY!Xisanopenimmersion.ByAlgebraicStacks,Lemma15.5weconcludethatYisanalgebraicstack.LastlyifwedenotethemorphismV!Xbyg,wehaveVXV!Vissurjective(thediagonalgivesasection).HencegisintheimageofYV!XV,i.e.,weobtainamorphismg0:V!YttingintothecommutativediagramV// g0 U Y// XSinceVg;XY!Visamonomorphism,itisinfactanisomorphismsince(1;g0)denesasection.Thereforeg0:V!Yisasmoothmorphism,asitisthebasechangeofthesmoothmorphismg:V!X.ItissurjectivebyourconstructionofYwhichnishestheproofofthelemma. PROPERTIESOFALGEBRAICSTACKS20Lemma9.13.05UQLetXbeanalgebraicstackandXiXacollectionofopensubstacksindexedbyi2I.Thenthereexistsanopensubstack,whichwedenoteSi2IXiX,suchthattheXiareopensubstackscoveringit.Proof.WedeneabredsubcategoryX0=Si2IXibylettingthebercategoryoveranobjectTof(Sch=S)fppfbethefullsubcategoryofXTconsistingofallx2Ob(XT)suchthatthemorphism`i2I(XiXT)!Tissurjective.Letxi2Ob((Xi)T).Then(xi;1)givesasectionofXiXT!T,sowehaveanisomorphism.ThusXiX0isafullsubcategory.Nowletx2Ob(XT).ThenXiXTisrepresentablebyanopensubschemeTiT.The2-bredproductX0XThasberoverT0consistingof(y2XT0;f:T0!T;fx'y)suchthat`(XiX;yT0)!T0issurjective.Theisomorphismfx'yinducesanisomorphismXiX;yT0'TiTT0.ThentheTiTT0coverT0ifandonlyifflandsinSTi.ThereforewehaveadiagramTi//  STi//  T Xi// X0// Xwithbothsquarescartesian.ByAlgebraicStacks,Lemma15.5weconcludethatX0Xisalgebraicandanopensubstack.Itisalsoclearfromthecartesiansquaresabovethatthemorphism`i2IXi!X0whichnishestheproofofthelemma.Lemma9.14.05URLetXbeanalgebraicstackandX0Xaquasi-compactopensubstack.SupposethatwehaveacollectionofopensubstacksXiXindexedbyi2IsuchthatX0Si2IXi,wherewedenetheunionasinLemma9.13.ThenthereexistsanitesubsetI0IsuchthatX0Si2I0Xi.Proof.SinceXisalgebraic,thereexistsaschemeUwithasurjectivesmoothmorphismU!X.LetUiUbetheopensubschemerepresentingXiXUandU0UtheopensubschemerepresentingX0XU.Byhypothesis,U0Si2IUi.FromtheproofofLemma6.2,thereisaquasi-compactopenVU0suchthatV!X0isasurjectivesmoothmorphism.ThereforethereexistsanitesubsetI0IsuchthatVSi2I0Ui.WeclaimthatX0Si2I0Xi.Takex2Ob(X0T)forT2Ob((Sch=S)fppf).SinceX0!Xisamonomorphism,wehavecartesiansquaresVXT//  Tx Tx V// X0// XBybasechange,VXT!Tissurjective.ThereforeSi2I0UiXT!Tisalsosurjective.LetTiTbetheopensubschemerepresentingXiXT.Byaformalargument,wehaveaCartesiansquareUiXiTi//  UXT Ti// T PROPERTIESOFALGEBRAICSTACKS21wheretheverticalarrowsaresurjectivebybasechange.SinceUiXiTi'UiXT,wendthatSi2I0Ti=T.Hencexisanobjectof(Si2I0Xi)Tbydenitionoftheunion.ObservethattheinclusionX0Si2I0Xiisautomaticallyanopensubstack.Lemma9.15.05WELetXbeanalgebraicstack.LetXi,i2IbeasetofopensubstacksofX.Assume(1)X=Si2IXi,and(2)eachXiisanalgebraicspace.ThenXisanalgebraicspace.Proof.ApplyStacks,Lemma6.10tothemorphism`i2IXi!Xandthemor-phismid:X!XtoseethatXisastackinsetoids.HenceXisanalgebraicspace,seeAlgebraicStacks,Proposition13.3.Lemma9.16.05WFLetXbeanalgebraicstack.LetXi,i2IbeasetofopensubstacksofX.Assume(1)X=Si2IXi,and(2)eachXiisaschemeThenXisascheme.Proof.ByLemma9.15weseethatXisanalgebraicspace.Sinceanyalgebraicspacehasalargestopensubspacewhichisascheme,seePropertiesofSpaces,Lemma13.1weseethatXisascheme.ThefollowinglemmaistheanalogueofMoreonGroupoids,Lemma6.1.Lemma9.17.06M3LetP;Q;Rbepropertiesofmorphismsofalgebraicspaces.Assume(1)P;Q;Rarefppflocalonthetargetandstableunderarbitrarybasechange,(2)smooth)R,(3)foranymorphismf:X!YwhichhasQthereexistsalargestopensubspaceW(P;f)XsuchthatfjW(P;f)hasP,and(4)foranymorphismf:X!YwhichhasQ,andanymorphismY0!YwhichhasRwehaveY0YW(P;f)=W(P;f0),wheref0:XY0!Y0isthebasechangeoff.Letf:X!Ybeamorphismofalgebraicstacksrepresentablebyalgebraicspaces.AssumefhasQ.Then(A)thereexistsalargestopensubstackX0XsuchthatfjX0hasP,and(B)ifZ!YisamorphismofalgebraicstacksrepresentablebyalgebraicspaceswhichhasRthenZYX0isthelargestopensubstackofZYXoverwhichthebasechangeidZfhaspropertyP.Proof.ChooseaschemeVandasurjectivesmoothmorphismV!Y.SetU=VYXandletf0:U!Vbethebasechangeoff.Themorphismofalgebraicspacesf0:U!VhaspropertyQ.ThusweobtaintheopenW(P;f0)Ubyassumption(3).NotethatUXU=(VYV)YXhencethemorphismf00:UXU!VYVisthebasechangeoffviaeitherprojectionVYV!V.ByourchoiceofVtheseprojectionsaresmooth,hencehavepropertyRby(2).Thusby(4)weseethattheinverseimagesofW(P;f0)underthetwoprojectionspri:UXU!Uagree.Inotherwords,W(P;f0)isanR-invariantsubspaceofU(whereR=UXU).LetX0betheopensubstackofXcorrespondingtoW(P;f) PROPERTIESOFALGEBRAICSTACKS22viaLemma9.6.ByconstructionW(P;f0)=X0YVhencefjX0haspropertyPbyLemma3.3.Also,X0isthelargestopensubstacksuchthatfjX0hasPasthesamemaximalityholdsforW(P;f).Thisproves(A).Finally,ifZ!YisamorphismofalgebraicstacksrepresentablebyalgebraicspaceswhichhasRthenwesetT=VYZandweseethatT!VisamorphismofalgebraicspaceshavingpropertyR.Setf0T:TVU!Tthebasechangeoff0.By(4)againweseethatW(P;f0T)istheinverseimageofW(P;f)inTVU.Thisimplies(B);somedetailsomitted.Remark9.18.06M4Warning:Lemma9.17shouldbeusedwithcare.Forexample,itappliestoP=at,Q=empty,andR=atandlocallyofnitepresentation.Butgivenamorphismofalgebraicspacesf:X!YthelargestopensubspaceWXsuchthatfjWisatisnotthesetofpointswherefisat!Remark9.19.06M5NotwithstandingthewarninginRemark9.18therearesomecaseswhereLemma9.17canbeusedwithoutcausingambiguity.Wegivealist.Ineachcaseweomitthevericationofassumptions(1)and(2)andwegivereferenceswhichimply(3)and(4).Hereisthelist:(1)06M6Q=locallyofnitetype,R=;,andP=relativedimensiond.SeeMorphismsofSpaces,Denition33.2andMorphismsofSpaces,Lemmas34.4and34.3.(2)06M7Q=locallyofnitetype,R=;,andP=locallyquasi-nite.Thisisthecased=0ofthepreviousitem,seeMorphismsofSpaces,Lemma34.6.Ontheotherhand,properties(3)and(4)arespelledoutinMorphismsofSpaces,Lemma34.7.(3)06M8Q=locallyofnitetype,R=;,andP=unramied.ThisisMor-phismsofSpaces,Lemma38.10.(4)06M9Q=locallyofnitepresentation,R=atandlocallyofnitepresenta-tion,andP=at.SeeMoreonMorphismsofSpaces,Theorem22.1andLemma22.2.NotethathereW(P;f)isalwaysexactlythesetofpointswherethemorphismfisatbecauseweonlyconsiderthisopenwhenfhasQ(seeloc.cit.).(5)06MAQ=locallyofnitepresentation,R=atandlocallyofnitepresenta-tion,andP=étale.Thisfollowsoncombining(3)and(4)becauseanunramiedmorphismwhichisatandlocallyofnitepresentationisétale,seeMorphismsofSpaces,Lemma39.12.(6)Addmorehereasneeded(comparewiththelongerlistatMoreonGroupoids,Remark6.3).10.Reducedalgebraicstacks0508WehavealreadydenedreducedalgebraicstacksinSection7.Lemma10.1.0509LetXbeanalgebraicstack.LetTjXjbeaclosedsubset.ThereexistsauniqueclosedsubstackZXwiththefollowingproperties:(a)wehavejZj=T,and(b)Zisreduced.Proof.LetU!Xbeasurjectivesmoothmorphism,whereUisanalgebraicspace.SetR=UXU,sothatthereisapresentation[U=R]!X,seeAlgebraicStacks,Lemma16.2.Asusualwedenotes;t:R!Uthetwosmoothprojectionmorphisms.ByLemma4.5weseethatTcorrespondstoaclosedsubsetT0jUj PROPERTIESOFALGEBRAICSTACKS23suchthatjsj�1(T0)=jtj�1(T0).LetZUbethereducedinducedalgebraicspacestructureonT0,seePropertiesofSpaces,Denition12.5.ThebreproductsZU;tRandRs;UZareclosedsubspacesofR(Spaces,Lemma12.3).TheprojectionsZU;tR!ZandRs;UZ!ZaresmoothbyMorphismsofSpaces,Lemma37.3.ThusasZisreduced,itfollowsthatZU;tRandRs;UZarereduced,seeRemark7.3.SincejZU;tRj=jtj�1(T0)=jsj�1(T0)=Rs;UZweconcludefromtheuniquenessinPropertiesofSpaces,Lemma12.3thatZU;tR=Rs;UZ.HenceZisanR-invariantclosedsubspaceofU.Bythecorrespon-denceofLemma9.10weobtainaclosedsubstackZXwithZ=ZXU.Then[Z=RZ]!Zisapresentation(Lemma9.6).ThenjZj=jZj=jRZj=jT0j=isthegivenclosedsubsetT.Weomittheproofofunicity.Lemma10.2.050ALetXbeanalgebraicstack.IfX0Xisaclosedsubstack,XisreducedandjX0j=jXj,thenX0=X.Proof.Chooseapresentation[U=R]!XwithUascheme.AsXisreduced,weseethatUisreduced(bydenitionofreducedalgebraicstacks).ByLemma9.10X0correspondstoanR-invariantclosedsubschemeZU.ButnowjZjjUjistheinverseimageofjX0j,andhencejZj=jUj.HenceZisaclosedsubschemeofUwhoseunderlyingsetsofpointsagree.BySchemes,Lemma12.7themapidU:U!UfactorsthroughZ!U,andhenceZ=U,i.e.,X0=X.Lemma10.3.050BLetX,Ybealgebraicstacks.LetZXbeaclosedsubstackAssumeYisreduced.Amorphismf:Y!XfactorsthroughZifandonlyiff(jYj)jZj.Proof.Assumef(jYj)jZj.ConsiderYXZ!Y.ThereisanequivalenceYXZ!Y0whereY0isaclosedsubstackofY,seeLemmas9.2and9.9.UsingLemmas4.3,8.5,and9.5weseethatjY0j=jYj.HencewehavereducedthelemmatoLemma10.2.Denition10.4.050CLetXbeanalgebraicstack.LetZjXjbeaclosedsubset.AnalgebraicstackstructureonZisgivenbyaclosedsubstackZofXwithjZjequaltoZ.ThereducedinducedalgebraicstackstructureonZistheoneconstructedinLemma10.1.ThereductionXredofXisthereducedinducedalgebraicstackstructureonjXj.Infactwecanusethistodenethereducedinducedalgebraicstackstructureonalocallyclosedsubset.Remark10.5.06FKLetXbeanalgebraicstack.LetTjXjbealocallyclosedsubset.Let@TbetheboundaryofTinthetopologicalspacejXj.Inaformula@T= TnT:LetUXbetheopensubstackofXwithjUj=jXjn@T,seeLemma9.11.LetZbethereducedclosedsubstackofUwithjZj=Tobtainedbytakingthereducedinducedclosedsubspacestructure,seeDenition10.4.ByconstructionZ!UisaclosedimmersionofalgebraicstacksandU!Xisanopenimmersion,henceZ!XisanimmersionofalgebraicstacksbyLemma9.3.NotethatZisareducedalgebraicstackandthatjZj=TassubsetsofjXj.WesometimessayZisthereducedinducedsubstackstructureonT. PROPERTIESOFALGEBRAICSTACKS2411.Residualgerbes06MLIntheStacksprojectwewouldliketodenetheresidualgerbeofanalgebraicstackXatapointx2jXjtobeamonomorphismofalgebraicstacksmx:Zx!XwhereZxisareducedalgebraicstackhavingauniquepointwhichismappedbymxtox.Itturnsoutthattherearemanyissueswiththisnotion;existenceisnotclearingeneralandneitherisuniqueness.WeresolvetheuniquenessissuebyimposingaslightlystrongerconditiononthealgebraicstacksZx.Wediscussthisinmoredetailbyworkingthroughafewsimplelemmasregardingreducedalgebraicstackshavingauniquepoint.Lemma11.1.06MMLetZbeanalgebraicstack.LetkbeaeldandletSpec(k)!Zbesurjectiveandat.ThenanymorphismSpec(k0)!Zwherek0isaeldissurjectiveandat.Proof.ConsiderthebresquareT // Spec(k) Spec(k0)// ZNotethatT!Spec(k0)isatandsurjectivehenceTisnotempty.OntheotherhandT!Spec(k)isataskisaeld.HenceT!Zisatandsurjective.ItfollowsfromMorphismsofSpaces,Lemma31.5(viathediscussioninSection3)thatSpec(k0)!Zisat.ItisclearthatitissurjectiveasbyassumptionjZjisasingleton.Lemma11.2.06MNLetZbeanalgebraicstack.Thefollowingareequivalent(1)ZisreducedandjZjisasingleton,(2)thereexistsasurjectiveatmorphismSpec(k)!Zwherekisaeld,and(3)thereexistsalocallyofnitetype,surjective,atmorphismSpec(k)!Zwherekisaeld.Proof.Assume(1).LetWbeaschemeandletW!Zbeasurjectivesmoothmorphism.ThenWisareducedscheme.Let2Wbeagenericpointofanirre-duciblecomponentofW.SinceWisreducedwehaveOW;=().Itfollowsthatthecanonicalmorphism=Spec(())!Wisat.Weseethatthecomposition!Zisat(seeMorphismsofSpaces,Lemma30.3).ItisalsosurjectiveasjZjisasingleton.Inotherwords(2)holds.Assume(2).LetWbeaschemeandletW!Zbeasurjectivesmoothmorphism.ChooseaeldkandasurjectiveatmorphismSpec(k)!Z.ThenWZSpec(k)isanalgebraicspacesmoothoverk,henceregular(seeSpacesoverFields,Lemma16.1)andinparticularreduced.SinceWZSpec(k)!WissurjectiveandatweconcludethatWisreduced(DescentonSpaces,Lemma8.2).Inotherwords(1)holds.Itisclearthat(3)implies(2).Finally,assume(2).PickanonemptyaneschemeWandasmoothmorphismW!Z.Pickaclosedpointw2Wandsetk=(w).ThecompositionSpec(k)w�!W�!Z PROPERTIESOFALGEBRAICSTACKS25islocallyofnitetypebyMorphismsofSpaces,Lemmas23.2and37.6.ItisalsoatandsurjectivebyLemma11.1.Hence(3)holds.Thefollowinglemmasinglesoutaslightlybetterclassofsingletonalgebraicstacksthantheprecedinglemma.Lemma11.3.06MPLetZbeanalgebraicstack.Thefollowingareequivalent(1)Zisreduced,locallyNoetherian,andjZjisasingleton,and(2)thereexistsalocallynitelypresented,surjective,atmorphismSpec(k)!Zwherekisaeld.Proof.Assume(2)holds.ByLemma11.2weseethatZisreducedandjZjisasingleton.LetWbeaschemeandletW!Zbeasurjectivesmoothmor-phism.Chooseaeldkandalocallynitelypresented,surjective,atmorphismSpec(k)!Z.ThenWZSpec(k)isanalgebraicspacesmoothoverk,hencelocallyNoetherian(seeMorphismsofSpaces,Lemma23.5).SinceWZSpec(k)!Wisat,surjective,andlocallyofnitepresentation,weseethatfWZSpec(k)!WgisanfppfcoveringandweconcludethatWislocallyNoetherian(DescentonSpaces,Lemma8.3).Inotherwords(1)holds.Assume(1).PickanonemptyaneschemeWandasmoothmorphismW!Z.Pickaclosedpointw2Wandsetk=(w).BecauseWislocallyNoetherianthemorphismw:Spec(k)!Wisofnitepresentation,seeMorphisms,Lemma21.7.HencethecompositionSpec(k)w�!W�!ZislocallyofnitepresentationbyMorphismsofSpaces,Lemmas28.2and37.5.ItisalsoatandsurjectivebyLemma11.1.Hence(2)holds.Lemma11.4.06MQLetZ0!Zbeamonomorphismofalgebraicstacks.Assumethereexistsaeldkandalocallynitelypresented,surjective,atmorphismSpec(k)!Z.TheneitherZ0isemptyorZ0!Zisanequivalence.Proof.WemayassumethatZ0isnonempty.InthiscasethebreproductT=Z0ZSpec(k)isnonempty,seeLemma4.3.NowTisanalgebraicspaceandtheprojectionT!Spec(k)isamonomorphism.HenceT=Spec(k),seeMorphismsofSpaces,Lemma10.8.WeconcludethatSpec(k)!ZfactorsthroughZ0.Supposethemorphismz:Spec(k)!ZisgivenbytheobjectoverSpec(k).Wehavejustseenthatisisomorphictoanobject0ofZ0overSpec(k).Sincezissurjective,at,andlocallyofnitepresentationweseethateveryobjectofZoveranyschemeisfppflocallyisomorphictoapullbackof,hencealsotoapullbackof0.BydescentofobjectsforstacksingroupoidsthisimpliesthatZ0!Zisessentiallysurjective(aswellasfullyfaithful,seeLemma8.4).Hencewewin.Lemma11.5.06MRLetZbeanalgebraicstack.AssumeZsatisestheequivalentconditionsofLemma11.2.ThenthereexistsauniquestrictlyfullsubcategoryZ0ZsuchthatZ0isanalgebraicstackwhichsatisestheequivalentconditionsofLemma11.3.TheinclusionmorphismZ0!Zisamonomorphismofalgebraicstacks.Proof.ThelastpartisimmediatefromtherstpartandLemma8.4.PickaeldkandamorphismSpec(k)!Zwhichissurjective,at,andlocallyofnitetype.SetU=Spec(k)andR=UZU.Theprojectionss;t:R!Uarelocallyofnite PROPERTIESOFALGEBRAICSTACKS26type.SinceUisthespectrumofaeld,itfollowsthats;tareatandlocallyofnitepresentation(byMorphismsofSpaces,Lemma28.7).WeseethatZ0=[U=R]isanalgebraicstackbyCriteriaforRepresentability,Theorem17.2.ByAlgebraicStacks,Lemma16.1weobtainacanonicalmorphismf:Z0�!Zwhichisfullyfaithful.Hencethismorphismisrepresentablebyalgebraicspaces,seeAlgebraicStacks,Lemma15.2andamonomorphism,seeLemma8.4.ByCriteriaforRepresentability,Lemma17.1themorphismU!Z0issurjective,at,andlocallyofnitepresentation.HenceZ0isanalgebraicstackwhichsatisestheequivalentconditionsofLemma11.3.ByAlgebraicStacks,Lemma12.4wemayreplaceZ0byitsessentialimageinZ.HencewehaveprovedalltheassertionsofthelemmaexceptfortheuniquenessofZ0Z.SupposethatZ00Zisasecondsuchalgebraicstack.ThentheprojectionsZ0 �Z0ZZ00�!Z00aremonomorphisms.ThealgebraicstackinthemiddleisnonemptybyLemma4.3.HencethetwoprojectionsareisomorphismsbyLemma11.4andwewin.Example11.6.06MSHereisanexamplewherethemorphismconstructedinLemma11.5isn'tanisomorphism.ThisexampleshowsthatimposingthatresidualgerbesarelocallyNoetherianisnecessaryinDenition11.8.Infact,theexampleisevenanalgebraicspace!LetGal( Q=Q)betheabsoluteGaloisgroupofQwiththepro-nitetopology.LetU=Spec( Q)Spec(Q)Spec( Q)=Gal( Q=Q)Spec( Q)(weomitapreciseexplanationofthemeaningofthelastequalsign).LetGdenotetheabsoluteGaloisgroupGal( Q=Q)withthediscretetopologyviewedasaconstantgroupschemeoverSpec( Q),seeGroupoids,Example5.6.ThenGactsfreelyandtransitivelyonU.LetX=U=G,seeSpaces,Denition14.4.ThenXisanon-noetherianreducedalgebraicspacewithexactlyonepoint.Furthermore,Xhasa(locally)nitetypepoint:x:Spec( Q)�!U�!XIndeed,everypointofUisactuallyclosed!AsXisanalgebraicspaceover Qitfollowsthatxisamonomorphism.SoxisthemorphismconstructedinLemma11.5butxisnotanisomorphism.InfactSpec( Q)!XistheresidualgerbeofXatx.ItwillturnoutlaterthatundermildassumptionsonthealgebraicstackXtheequivalentconditionsofthefollowinglemmaaresatisedforeverypointx2jXj(seeMorphismsofStacks,Section31).Lemma11.7.06MTLetXbeanalgebraicstack.Letx2jXjbeapoint.Thefollowingareequivalent(1)thereexistsanalgebraicstackZandamonomorphismZ!XsuchthatjZjisasingletonandsuchthattheimageofjZjinjXjisx,(2)thereexistsareducedalgebraicstackZandamonomorphismZ!XsuchthatjZjisasingletonandsuchthattheimageofjZjinjXjisx, PROPERTIESOFALGEBRAICSTACKS27(3)thereexistsanalgebraicstackZ,amonomorphismf:Z!X,andasurjectiveatmorphismz:Spec(k)!Zwherekisaeldsuchthatx=f(z).Moreover,iftheseconditionshold,thenthereexistsauniquestrictlyfullsubcategoryZxXsuchthatZxisareduced,locallyNoetherianalgebraicstackandjZxjisasingletonwhichmapstoxviathemapjZxj!jXj.Proof.IfZ!Xisasin(1),thenZred!Xisasin(2).(SeeSection10forthenotionofthereductionofanalgebraicstack.)Hence(1)implies(2).Itisimmediatethat(2)implies(1).Theequivalenceof(2)and(3)isimmediatefromLemma11.2.Atthispointwe'veseentheequivalenceof(1)(3).Pickamonomorphismf:Z!Xasin(2).Notethatthisimpliesthatfisfullyfaithful,seeLemma8.4.DenoteZ0Xtheessentialimageofthefunctorf.Thenf:Z!Z0isanequivalenceandhenceZ0isanalgebraicstack,seeAlgebraicStacks,Lemma12.4.ApplyLemma11.5togetastrictlyfullsubcategoryZxZ0asinthestatementofthelemma.Thisprovesallthestatementsofthelemmaexceptforuniqueness.InordertoprovetheuniquenesssupposethatZxXandZ0xXaretwostrictlyfullsubcategoriesasinthestatementofthelemma.ThentheprojectionsZ0x �Z0xXZx�!Zxaremonomorphisms.ThealgebraicstackinthemiddleisnonemptybyLemma4.3.HencethetwoprojectionsareisomorphismsbyLemma11.4andwewin.Havingexplainedtheabovewecannowmakethefollowingdenition.Denition11.8.06MULetXbeanalgebraicstack.Letx2jXj.(1)WesaytheresidualgerbeofXatxexistsiftheequivalentconditions(1),(2),and(3)ofLemma11.7hold.(2)IftheresidualgerbeofXatxexists,thentheresidualgerbeofXatx1isthestrictlyfullsubcategoryZxXconstructedinLemma11.7.InparticularweknowthatZx(ifitexists)isalocallyNoetherian,reducedalgebraicstackandthatthereexistsaeldandasurjective,at,locallynitelypresentedmorphismSpec(k)�!Zx:WewillseeinMorphismsofStacks,Lemma28.12thatZxisagerbe.ExistenceofresidualgerbesisdiscussedinMorphismsofStacks,Section31.ItturnsoutthatZxisaregularalgebraicstackasfollowsfromthefollowinglemma.Lemma11.9.06MVAreduced,locallyNoetherianalgebraicstackZsuchthatjZjisasingletonisregular. 1Thisclasheswith[LMB00]inspirit,butnotinfact.Namely,inChapter11theyassociatetoanypointonanyquasi-separatedalgebraicstackagerbe(notnecessarilyalgebraic)whichtheycalltheresidualgerbe.WewillseeinMorphismsofStacks,Lemma31.1thatonaquasi-separatedalgebraicstackeverypointhasaresidualgerbeinoursensewhichisthenequivalenttotheirs.Formoreinformationonthistopicsee[Ryd10,AppendixB]. PROPERTIESOFALGEBRAICSTACKS28Proof.LetW!ZbeasurjectivesmoothmorphismwhereWisascheme.LetkbeaeldandletSpec(k)!Zbesurjective,at,andlocallyofnitepresentation(seeLemma11.3).ThealgebraicspaceT=WZSpec(k)issmoothoverkinparticularregular,seeSpacesoverFields,Lemma16.1.SinceT!Wislocallyofnitepresentation,at,andsurjectiveitfollowsthatWisregular,seeDescentonSpaces,Lemma8.4.BydenitionthismeansthatZisregular.Lemma11.10.06MWLetXbeanalgebraicstack.Letx2jXj.AssumethattheresidualgerbeZxofXexists.Letf:Spec(K)!XbeamorphismwhereKisaeldintheequivalenceclassofx.ThenffactorsthroughtheinclusionmorphismZx!X.Proof.ChooseaeldkandasurjectiveatlocallynitepresentationmorphismSpec(k)!Zx.SetT=Spec(K)XZx.ByLemma4.3weseethatTisnonempty.AsZx!XisamonomorphismweseethatT!Spec(K)isamonomorphism.HencebyMorphismsofSpaces,Lemma10.8weseethatT=Spec(K)whichprovesthelemma.Lemma11.11.06MXLetXbeanalgebraicstack.Letx2jXj.LetZbeanalgebraicstacksatisfyingtheequivalentconditionsofLemma11.3andletZ!XbeamonomorphismsuchthattheimageofjZj!jXjisx.ThentheresidualgerbeZxofXatxexistsandZ!XfactorsasZ!Zx!Xwheretherstarrowisanequivalence.Proof.LetZxXbethefullsubcategorycorrespondingtotheessentialimageofthefunctorZ!X.ThenZ!Zxisanequivalence,henceZxisanalgebraicstack,seeAlgebraicStacks,Lemma12.4.SinceZxinheritsallthepropertiesofZfromthisequivalenceitisclearfromtheuniquenessinLemma11.7thatZxistheresidualgerbeofXatx.Lemma11.12.0DTHLetf:X!Ybeamorphismofalgebraicstacks.Letx2jXjwithimagey2jYj.IftheresidualgerbesZxXandZyYofxandyexist,thenfinducesacommutativediagramXf Zxoo  YZyoo Proof.Chooseaeldkandasurjective,at,locallynitelypresentedmorphismSpec(k)!Zx.ThemorphismSpec(k)!YfactorsthroughZybyLemma11.10.ThusZxYZyisanonemptysubstackofZxhenceequaltoZxbyLemma11.4.Lemma11.13.0DTILetf:X!Ybeamorphismofalgebraicstacks.Letx2jXjwithimagey2jYj.AssumetheresidualgerbesZxXandZyYofxandyexistandthatthereexistsamorphismSpec(k)!XintheequivalenceclassofxsuchthatSpec(k)XSpec(k)�!Spec(k)YSpec(k)isanisomorphism.ThenZx!Zyisanisomorphism.Proof.Letk0=kbeanextensionofelds.ThenSpec(k0)XSpec(k0)�!Spec(k0)YSpec(k0) PROPERTIESOFALGEBRAICSTACKS29isthebasechangeofthemorphisminthelemmabythefaithfullyatmorphismSpec(k0 k0)!Spec(k k).Thusthepropertydescribedinthelemmaisinde-pendentofthechoiceofthemorphismSpec(k)!Xintheequivalenceclassofx.ThuswemayassumethatSpec(k)!Zxissurjective,at,andlocallyofnitepresentation.InthissituationwehaveZx=[Spec(k)=R]withR=Spec(k)XSpec(k).SeeproofofLemma11.5.SincealsoR=Spec(k)YSpec(k)weconcludethatthemorphismZx!ZyofLemma11.12isfullyfaithfulbyAlgebraicStacks,Lemma16.1.WeconcludeforexamplebyLemma11.11.12.Dimensionofastack0AFLWecandenethedimensionofanalgebraicstackXatapointx,usingthenotionofdimensionofanalgebraicspaceatapoint(PropertiesofSpaces,Denition9.1).Inthefollowinglemmatheoutputmaybe1eitherbecauseXisnotquasi-compactorbecausewerunintothephenomenondescribedinExamples,Section14.Lemma12.1.0AFMLetXbealocallyNoetherianalgebraicstackoveraschemeS.Letx2jXjbeapointofX.Let[U=R]!Xbeapresentation(AlgebraicStacks,Denition16.5)whereUisascheme.Letu2Ubeapointthatmapstox.Lete:U!Rbetheidentitymapandlets:R!Ubethesourcemap,whichisasmoothmorphismofalgebraicspaces.LetRubetheberofs:R!Uoveru.Theelementdimx(X)=dimu(U)�dime(u)(Ru)2Z[1isindependentofthechoiceofpresentationandthepointuoverx.Proof.SinceR!Uissmooth,theschemeRuissmoothover(u)andhencehasnitedimension.Ontheotherhand,theschemeUislocallyNoetherian,butthisdoesnotguaranteethatdimu(U)isnite.ThusthedierenceisanelementofZ[f1g.Let[U0=R0]!Xandu02U0beasecondpresentationwhereU0isaschemeandu0mapstox.ConsiderthealgebraicspaceP=UXU0.ByLemma4.3thereexistsap2jPjmappingtouandu0.SinceP!UandP!U0aresmoothweseethatdimp(P)=dimu(U)+dimp(Pu)anddimp(P)=dimu0(U0)+dimp(Pu0),seeMorphismsofSpaces,Lemma37.10.NotethatR0u0=Spec((u0))XU0andPu=Spec((u))XU0Letusrepresentp2jPjbyamorphismSpec( )!P.Sincepmapstobothuandu0itinducesa2-morphismbetweenthecompositionsSpec( )!Spec((u0))!XandSpec( )!Spec((u))!XwhichinturndenesanisomorphismSpec( )Spec((u0))R0u0=Spec( )Spec((u))PuasalgebraicspacesoverSpec( )mappingthe -rationalpoint(1;e0(u0))to(1;p)(somedetailsomitted).Weconcludethatdime0(u0)(R0u0)=dimp(Pu)byMorphismsofSpaces,Lemma34.3.Bysymmetrywehavedime(u)(Ru)=dimp(Pu0).Puttingeverythingtogetherweobtaintheindependenceofchoices.Wecanusethelemmaabovetomakethefollowingdenition. PROPERTIESOFALGEBRAICSTACKS30Denition12.2.0AFNLetXbealocallyNoetherianalgebraicstackoveraschemeS.Letx2jXjbeapointofX.Let[U=R]!Xbeapresentation(AlgebraicStacks,Denition16.5)whereUisaschemeandletu2Ubeapointthatmapstox.WedenethedimensionofXatxtobetheelementdimx(X)2Z[1suchthatdimx(X)=dimu(U)�dime(u)(Ru):withnotationasinLemma12.1.ThedimensionofastackatapointagreeswiththeusualnotionwhenXisascheme(Topology,Denition10.1),ormoregenerallywhenXisalocallyNoether-ianalgebraicspace(PropertiesofSpaces,Denition9.1).Denition12.3.0AFPLetSbeascheme.LetXbealocallyNoetherianalgebraicstackoverS.Thedimensiondim(X)ofXisdenedtobedim(X)=supx2jXjdimx(X)ThisdenitionofdimensionagreeswiththeusualnotionifXisascheme(Prop-erties,Lemma10.2)oranalgebraicspace(PropertiesofSpaces,Denition9.2).Remark12.4.0AFQIfXisanonemptystackofnitetypeoveraeld,thendim(X)isaninteger.ForanarbitrarylocallyNoetherianalgebraicstackX,dim(X)isinZ[f1g,anddim(X)=�1ifandonlyifXisempty.Example12.5.0AFRLetXbeaschemeofnitetypeoveraeldk,andletGbeagroupschemeofnitetypeoverkwhichactsonX.Thenthedimensionofthequotientstack[X=G]isequaltodim(X)�dim(G).Inparticular,thedimensionoftheclassifyingstackBG=[Spec(k)=G]is�dim(G).Thusthedimensionofanalgebraicstackcanbeanegativeinteger,incontrasttowhathappensforschemesoralgebraicspaces.13.Localirreducibility0DQGWehavedenedthegeometricnumberofbranchesofaschemeatapointinProp-erties,Section15andforanalgebraicspaceatapointinPropertiesofSpaces,Section23.Letn2N.ForalocalringAsetPn(A)=thenumberofgeometricbranchesofAisnForasmoothringmapA!BandaprimeidealqofBlyingoverpofAwehavePn(Ap),Pn(Bq)byMoreonAlgebra,Lemma98.8.AsinPropertiesofSpaces,Remark7.6wemayusePntodeneanétalelocalpropertyPnofgerms(U;u)ofschemesbysettingPn(U;u)=Pn(OU;u).ThecorrespondingpropertyPnofanalgebraicspacesXatapointx(seePropertiesofSpaces,Denition7.5)isjustthepropertythenumberofgeometricbranchesofXatxisn,seePropertiesofSpaces,Denition23.4.Moreover,thepropertyPnissmoothlocal,seeDescent,Denition18.1.ThisfollowseitherfromtheequivalencedisplayedaboveorMoreonMorphisms,Lemma33.4.ThusDenition7.5appliesandweobtainanotionforalgebraicstacksatapoint.Denition13.1.0DQHLetXbeanalgebraicstack.Letx2jXj.(1)ThenumberofgeometricbranchesofXatxiseithern2NiftheequivalentconditionsofLemma7.4holdforPndenedabove,orelse1. PROPERTIESOFALGEBRAICSTACKS31(2)WesayXisgeometricallyunibranchatxifthenumberofgeometricbranchesofXatxis1.14.Finitenessconditionsandpoints0DTJThissectionistheanalogueofDecentSpaces,Section4forpointsofalgebraicstacks.Lemma14.1.0DTKLetXbeanalgebraicstack.Letx2jXjbeapoint.Thefollowingareequivalent(1)somemorphismSpec(k)!Xintheequivalenceclassofxisquasi-compact,and(2)anymorphismSpec(k)!Xintheequivalenceclassofxisquasi-compact.Proof.LetSpec(k)!Xbeintheequivalenceclassofx.Letk0=kbeaeldextension.ThenwehavetoshowthatSpec(k)!Xisquasi-compactifandonlyifSpec(k0)!Xisquasi-compact.ThisfollowsfromMorphismsofSpaces,Lemma8.6andtheprincipleofAlgebraicStacks,Lemma10.9.15.OtherchaptersPreliminaries(1)Introduction(2)Conventions(3)SetTheory(4)Categories(5)Topology(6)SheavesonSpaces(7)SitesandSheaves(8)Stacks(9)Fields(10)CommutativeAlgebra(11)BrauerGroups(12)HomologicalAlgebra(13)DerivedCategories(14)SimplicialMethods(15)MoreonAlgebra(16)SmoothingRingMaps(17)SheavesofModules(18)ModulesonSites(19)Injectives(20)CohomologyofSheaves(21)CohomologyonSites(22)DierentialGradedAlgebra(23)DividedPowerAlgebra(24)DierentialGradedSheaves(25)HypercoveringsSchemes(26)Schemes(27)ConstructionsofSchemes(28)PropertiesofSchemes(29)MorphismsofSchemes(30)CohomologyofSchemes(31)Divisors(32)LimitsofSchemes(33)Varieties(34)TopologiesonSchemes(35)Descent(36)DerivedCategoriesofSchemes(37)MoreonMorphisms(38)MoreonFlatness(39)GroupoidSchemes(40)MoreonGroupoidSchemes(41)ÉtaleMorphismsofSchemesTopicsinSchemeTheory(42)ChowHomology(43)IntersectionTheory(44)PicardSchemesofCurves(45)WeilCohomologyTheories(46)AdequateModules(47)DualizingComplexes(48)DualityforSchemes(49)DiscriminantsandDierents(50)deRhamCohomology(51)LocalCohomology(52)AlgebraicandFormalGeometry(53)AlgebraicCurves(54)ResolutionofSurfaces(55)SemistableReduction PROPERTIESOFALGEBRAICSTACKS32(56)DerivedCategoriesofVarieties(57)FundamentalGroupsofSchemes(58)ÉtaleCohomology(59)CrystallineCohomology(60)Pro-étaleCohomology(61)MoreÉtaleCohomology(62)TheTraceFormulaAlgebraicSpaces(63)AlgebraicSpaces(64)PropertiesofAlgebraicSpaces(65)MorphismsofAlgebraicSpaces(66)DecentAlgebraicSpaces(67)CohomologyofAlgebraicSpaces(68)LimitsofAlgebraicSpaces(69)DivisorsonAlgebraicSpaces(70)AlgebraicSpacesoverFields(71)TopologiesonAlgebraicSpaces(72)DescentandAlgebraicSpaces(73)DerivedCategoriesofSpaces(74)MoreonMorphismsofSpaces(75)FlatnessonAlgebraicSpaces(76)GroupoidsinAlgebraicSpaces(77)MoreonGroupoidsinSpaces(78)Bootstrap(79)PushoutsofAlgebraicSpacesTopicsinGeometry(80)ChowGroupsofSpaces(81)QuotientsofGroupoids(82)MoreonCohomologyofSpaces(83)SimplicialSpaces(84)DualityforSpaces(85)FormalAlgebraicSpaces(86)RestrictedPowerSeries(87)ResolutionofSurfacesRevisitedDeformationTheory(88)FormalDeformationTheory(89)DeformationTheory(90)TheCotangentComplex(91)DeformationProblemsAlgebraicStacks(92)AlgebraicStacks(93)ExamplesofStacks(94)SheavesonAlgebraicStacks(95)CriteriaforRepresentability(96)Artin'sAxioms(97)QuotandHilbertSpaces(98)PropertiesofAlgebraicStacks(99)MorphismsofAlgebraicStacks(100)LimitsofAlgebraicStacks(101)CohomologyofAlgebraicStacks(102)DerivedCategoriesofStacks(103)IntroducingAlgebraicStacks(104)MoreonMorphismsofStacks(105)TheGeometryofStacksTopicsinModuliTheory(106)ModuliStacks(107)ModuliofCurvesMiscellany(108)Examples(109)Exercises(110)GuidetoLiterature(111)Desirables(112)CodingStyle(113)Obsolete(114)GNUFreeDocumentationLi-cense(115)AutoGeneratedIndexReferences[LMB00]GérardLaumonandLaurentMoret-Bailly,Champsalgébriques,ErgebnissederMathe-matikundihrerGrenzgebiete.3.Folge.,vol.39,Springer-Verlag,2000.[Ryd10]DavidRydh,étaledévissage,descentandpushoutsofstacks,math.AG/1005.2171v1(2010).