; TeX output 2001.11.13:09337 V6N cmbx12MathematischesInstitutmWS2001/026derUniversitXatM`unchenSet56Prof.Dr.B.Pareigis;KNff cmbx12ProblemffsetforAdvancedffAlgebra=}BXQ cmr12(17)\$Shorwthatg cmmi12V!", cmsy10 OVp2 K cmsy8 9Yisacoalgebraforevery nitedimensional\$vrector&spaceVCovera eld( msbm10Kifthecomultiplicationisde ned\$bry(vP vn92.=)UR:="u cmex10P*2cmmi8n U_i|{Ycmr8=1 ASv v2n9RAiT vi vn92 wherefvidgandfv2n9RAi.=gare\$dualbasesofVresp.Vp2\t.ʍB(18)\$Shorw thata nitedimensionalvectorspaceVzisacomoSdule\$orverVthecoalgebraV~ 8Vp2 ʹasde nedinproblem17withthe\$coactions2(vn9)UR:=Pvm v2RAi- vi,2UR(V2 Vp2\t) V3mwherePAv2RAi- vi\$isthedualbasisofVinVp2  Vp.B(19)\$ShorwthatthefreeK-moSdulesKXkwiththebasisXandthe\$comrultiplication7(x)&E=x xisacoalgebra. /Whatisthe\$counit?8Isthecounitunique?B(20)\$ShorwthatthetensorproSductoftwocommutativeK-algebras\$isacoproSduct.=}6Duedate:8TVuesdary,20.11.2001,16:15inLectureHall138*;7 ( msbm10"u cmex10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12