# A-modules over A-algebras and Hochschild cohomology for by Ladoshkin M.V.

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Additional info for A-modules over A-algebras and Hochschild cohomology for modules over algebras

Example text

Sind (Basiserg~nzungs­ satz ). Setzt man speziell der Reihe nach u = o,al, ••• ,an_l' so ist notwendig (an = a) + cpc(a n ) = an' cpc(an+ak) = CPc(an)+CPc(ak) = an + cak' k E In_l' also notwendig cpc(al) = cal"'" CPc(an-l) = can_l' cpc(a n ) = an' und damit liegt eine lineare Abbildung CPc eindeutig fest. u. sind. ~ xio i wird durch ~c abgebildet in 1. 1 n-l ~c(on+u) = ~c(on + ~ n-l ~ i .. 1 i .. d. n Ein beliebiger Vektor , E Vn mit, = ~ xio i wird abgebildet i=1 in n n-l ~c(,) ... ~c( ~ xio i ) ..

H. ~~_~_:_~. u. sein). Dies steht in Ubereinstimmung zu der Formel ~ + Rang ~. Di~ U1. 5), aus der weiter Dim(Kern Kern ~ = \0\ folgt. ~) 0, also b) Da wir uns im R3 befinden, erkennt man fur die Dimension der Verbindung U1 + U2 zunachst Dim(U 1 + U3 ) ~ 3. u. sind. B. u. und es ist Dim(U 1 + U2 ) = 3. Die Dimension des Durchschnittes U1 n U2 erha1t man aus der Dimensionsformel Dim(U1 + U2 ) + Dim(U 1 n U2 ) 2 3 + ? + zu Dim(U 1 n U2 ) = 1. h. 03 spannt den gesamten Durchschnitt U1 n U2 auf. 3.

Also ist Dim ~(y3) = 2 und Rang ~ = 2. ~_~ = ziehung Dim y3 Dim (Kern ~) + Rang ~ 3 ? + 2 Dim (Kern ~) = 1. h. die einparametrige LOsungs- - 46 schar xl = -2t, x 2 = 0, x3 = t, t E R. ~~ von t;1,t;2: Nach Aufgabe sind die Bilder der Vektoren t;1 = 301 + 02' t;2 = 601 + 202 + 03 zu bestimmen. , und es ist Dim [CP(t;1),CP(t;2)] = 2. Bemerkung: Mit der Matrizenrechnung (siehe § 5) rechnet man eleganter cp(t;) = At; mit A= (: 1 2 wobei in den Spalten von A die Koordinaten der Bilder der Basisvektoren 01'02'03 stehen.