; TeX output 2000.08.30:1123 Kv{p0J cmsl10TheoryUUandApplicationsofCategories,K`y cmr10V*ol.q7,No.11,pp.227{235.uKl&t4K`y ff cmr10ONTHEOBJECT-WISETENSORPRODUCTOFFUNCTORSTO &MODULES \+XQ cmr12MAREKGOLASIwwxNSKI jT*ransmittedUUbyMichaelBarr"$A |{Y cmr8BSTRAÎCT. W*einvestigatepreservingofpro 8jectivityandinjectivitybytheob 8ject-wise $tensorlproGductofb> cmmi10RP msbm10PC-modules,rwherePCisasmallcategory*.{Inparticular,let!", cmsy10OG(G;X )$bGethecategoryofcanonicalorbitsofadiscretegroupG,overaG-setX .TW*eshowthat$pro 8jectivity|ofROG(G;X )-modules|ispreservedbythistensorproGduct.L*Moreover,ifGis$aDcnitegroup,GX EaniteG-setandRX*isanintegraldomainthensuchatensorproGduct$ofUUtwoinjectiveROG(G;X )-modulesisagaininjective.,1.34IntrofgductionS\It*iswrell-known*thatthetensorproSductoftrwo*pro jective,g cmmi12RJ-moSdulesispro jective.One can1easilycrheck1thatthetensorproSductoftrwo1injectiveRJ-moSdulesisinjectiveforRbSeinganinrtegraldomainormoregenerallyV,- theproductofanitenrumberofinrtegraldomains.gBut<(evrenforacommutativeringRJ)thisisnotthecase,ingeneral.gBy[3],foraUcommrutativenoSetherianringRJ,pothetensorproductofanrytwoinjectiveRJ-moSdulesisinjectivreifandonlyiftheloScalringRUX& ff eufm7Up@isquasi-FVrobeniusforanryprimeidealT%n eufm10Tp ֹinRJ.`LetGbSeadiscretegroupand-!", cmsy10OUV(G)theassociatedcategoryofcanonicalorbits.5Theinjectivitryoftheob ject-wisetensorproSductofinjectivefunctorsfromsomecategoriesassoSciatedPwithOUV(G)tovrectorspaceshasbeenrstextensivrelyusedbutnotprovedinc[4,11#]tostudytheequivXarianrtrationalhomotopytheoryandthenappliedin[5,10#]for'ifurthergeneralizedinrvestigations.$The'ibalanceofthepapSerisdevrotedtopreservingof΅thepro jectivitryandinjectivitybysuchatensorproSductoffunctorsfromanysmallcategoryZ msbm10ZCtoRJ-moSdules,calledRZC-moSdules.`Section,2dealswithconrtravXariant,functorsfromasmallcategoryZCtoRJ-moSdules.WVestartKfromtheilluminatingcounrterexampleshowingthattheob ject-wisetensorproSductofmtrwopro jectiveRJZC-moSdulesisnotpro jectiveevenforthecategoryZCassoSciatedwithanitepartiallyorderedset.WVeobservrethatthetensorproSductoftwosuchpro jectivefunctorsEispro jectivreifandonlythefunctorRJ(ZC(C5; )zZC(DS; ))Eispro jectivreforanyob jectsC5;D|inthecategoryZC.)By[2],itissucienrtthatthecompSonentsofthecommacategorygY#*ZC(C5; )ZC(DS; )gharverightzeros,HwhereY:*ZC!ZSet2[qy ff msbm7[C-:; cmmi6op istheYVonedaimrbSeddingOandZSetthecategoryofsets. gThenwededucethattheob ject-wisetensor ff HE ReceivedUUbytheeditors2000March14. PublishedUUon2000SeptembGer1. 2000UUMathematicsSub 8jectClassication:qPrimary18G05;secondary16W50,55P91. KeyUUwordsandphrases:qcategoryofcanonicalorbits,injective(pro 8jective)RPC-moGdule,linearly tcompactUUkP-moGdule,tensorproduct. c UYMarekUUGolasiGnski,2000.qPermissiontocopyforprivqateusegranted. ֔227 *Kv{TheoryUUandApplicationsofCategories,V*ol.q7,No.11 [4228uK{proSduct"oftrwo"pro jectiveRJOUV(G;X )-moSdulesispro jective, whereOUV(G;X )denotesthe categoryofcanonicalorbitsofadiscretegroupG,orveraG-setX .In!section3wremovetocovXariantfunctorsfromZCtoRJ-moSdulesandexaminethedual problem,7preservingoftheinjectivitrybytheob ject-wisetensorproSductofRJZC-moSdules. WVeeshorwthatsuchatensorproSductoftwoinjectiveRJOUV(G;X )-moSdulesisinjectivre;providedthatthegroupGisnite,PSX-aniteG-setandthetensorproSductofanrytwoinjectiveRJ-moSdulesisinjective.1KThenwededucethattheexterior,symmetricandtensorpSorwerconstructionspreservreinjectivityofRJOUV(G;X )-moSdules,DforR beingaeldofzerocrharacteristic. 6Attheend,6LforRbSeingaeld,wreconcludethatthepro jectivitry2(resp.injectivity)offunctorsfromOUV(G;X )tolinearly-compactvectorRJ-spacesD_arepreservred.FThatisthecrucialfactin[5]toextendtheequivXarianthomotopytheoryonthedisconnectedcase.TheauthorisdeeplygratefultoProfessorA.K.BouseldforhiscommrunicationandExample2.1.&<+2.34Pro jectivefunctorsLetb2 cmmi8R VEMoSd%7bSebthecategoryofleftRJ-modulesorverbacommrutativebringR{withidenrtitybandZC*asmallcategorywiththesetofob jectsjZCj.A*corvXariant*functorZCC!̹R MoSd&_^is*calledasf.@ cmti12leftRJZC-moffduleandthefunctorcategory̹R[CxMoSd-jKofallleftsucrhmoSdulesiscalledthecffategory35ofleftZC-moffdules.The)ob jectofthissectionisthecategoryMoSd}̹R[C*ֻofconrtravXariant)functorsZCUR!̹R HMoSd!,alias`right9RJZC-moffdules,|#calledinthesequelsimplyRZC-moSdules. Notionslikrecoproducts,proSducts,pinjectivre,pro jectiveUetc.aredenedasusual.zInparticular,panRJZC-moSdulePispro jectivreifthefollowingproblem8g kP =I5D xyatip10IJ5D xybtip10J ӟ= 6 fd ת=I~J~ ת