Exercise 22 1 Show that IF D CM E RCM El is a homomorphism endomorphism 2 Show that it satisfies 4 is also needed for 1 Icw 2 C l P w n k r 3 Show that the correspondence Kate defines a bijection SPLM End LED End cm DCM El where End Prem DCU E denotes the space of all endomorphisms SCM E arches that rise with degree p 4 prove that the wedge operation 1.29 becomes the wedge operators Fru Koti for all k k e REM EndCEI proof Note 1 Take 2 Er M E and hied M E Futa module morphism we need Then re ICM E to show flats feast fly If 4 22 K 2,1 22 f q k 7 We write Kd 4 X Xp g Sgnco k Xous dogs NXocptn X.ocp.gg Now we investigate 4 We write Wearegoing surpress this temporarily Xocpti a y X.ocp.gg I I t 42 Xocpti a y Xocp.gg 1 Ii Xo Cpt i Xocptg t 22 Xocpi.is XGcp Cdefl So Kd 4 X Xp q Sync o N 2 Xocp Xocp g t 22 Xocpi.is Xocpx.ci Sgh o k 2 Xocp Xocp KC I C End E and is thus linear sgnCoJk zCXocpi.n Xocpt.ci he Y C X Xp g t K 12 X Xp tg So we have shown tech Uz th Y th na f e E i thmh i where g 9 t 92 So k war X Xp Wcw 4 X Xp g l syn o k Koch Xocptp WA h Koopa i Roopa Let us investigate war We write w N X ocp i i I Xoc ptg W d U C Xocpt I Xoc ptg It Ptl l l XOLPH 1 42 Sgh p w Xp crept Xpocptg Y Xp ocp g ti l I XP OCP 1 41 42 with this obtained we write k wage X Xp i inininitii h Z X Xptp I Sync o k Xocy I Xocpj N Tep ti l I kept 921 1 So w r te U X XP ta th sync p w Xp Xp can I 9 CX 2 Now that we have shown it is a group homorphism we want to show Te wry C 1 w k n for all we M We know be corn h w 7 aw ne e Stevie gate do C is w n k n The only thing left to prove is h Cw 9 Ww 2 Let W E SEM Ke DP CM End E and h E REM E 1 Then hd WAN Xi Xp tr sign co k Koe Koop w 4 Xo ptg i i 1 Xo Cpt r l Now for Ww 2 let us first look at Cknw We know that h w X Xp g Sgh o k Koe Xocpj w Xocp Roc ptg I