; TeX output 2001.11.06:1405ǭ7o][o]N cmbx12MathematischesInstitut 8WS2001/02derUniversitXatM`unchenSet4Prof.Dr.B.Pareigis;KrNff cmbx12ProblemffsetforAdvancedffAlgebra! XQ cmr12(13)%Let%g cmmi12XبbSeasetandV¹:=UR( msbm10KXbSethefreeK-moduleorver%X._ShowthatXF!", cmsy10``!V%' !8Tƹ(Vp)ade nesa+@ cmti12frffeealgebraaorverX,Si.e.foreveryK-algebraAandevery%mapff8:9X)G!GAthereisauniquehomomorphismofK-algebrasgr:Tƹ(Vp)%' !7R_A꨹sucrhthatthediagramH)HXHTƹ(Vp)&<{fd ဍO line10-H`Xfׁ @ɏ @ӏ @ݏ @l>@l>RHWAۚǠ*Ffe̟Ǡ? Lge%〹commrutes.1WVewriteKhXi:=Tƹ(KX)andcallitthepffolynomialKringoverKinthe%non-cffommuting35variablesX. (14)%Let%XبbSeasetandV¹:=URKXbSethefreeK-moduleorver%X._ShowthatXF``!V%' !7R_S׹(Vp)de nesafrffeeSMcommutativealgebraorverX,'i.e.foreverycommutative%K-algebra5Aandevrerymapf%:&XƩ4!\Athereisauniquehomomorphismof%K-algebrasgË:URS׹(Vp)n!1AsucrhthatthediagramIՍH+XHS׹(Vp)ST{fd!;ά-H`ƅfׁ @ɽ @ӽ @ݽ @>@>RHoAǠ*Ffe;Ǡ? dg%〹commrutes. (15))((a)=ѬLetS׹(Vp)and_&:V !!tIS(Vp)bSeasymmetricalgebra. fShorwthatthere=ѬisxauniquehomomorphismofalgebrasF:S׹(Vp)`Q!S(Vp) = S(Vp)xwith=Ѭ(vn9)UR=v 1+1 vXforallvË2URVp.)' (b)=ѬShorwthat( 1)UR=(1 )UR:S׹(Vp)n!1S(Vp) S(Vp) S(Vp). (16)%LetVbSeaK-moduleandAbeaK-algebra.)((a)=ѬLet^7fb:Vt!AbSeahomomorphismofK-modulessatisfyingfG(vn9)2|{Ycmr82=0=ѬforallvË2URVp.8ThenfG(vn9)f(v2 K cmsy80