Vertices of Integral Representations 163

commutative , then there exists an RG-homomorphism

z M-^N^ with Ni—^N2 commutative

( vi ) (Dual to (v) ) If A/i —^ N2 IS a diagram as in (v) and

M if for any UeU there exists an RU-homomorphism

Tu . N2-^M with N^-^^N2

commutative , then there exists an RG-homomorphism X N2-^M with N^-^^N2

r Ф Ут

M

commutative

( vil ) If Фи M ^^ {M \u)^^^ IS the canonical RG-homomorphism

Фи { т ) = Z g®g-'m, фи=®Фи M->e(M|^f-^,

gUeGU Veil UeU

then there exists cp ф [M\u)^~*^-^ M with (рф11 = г Idj^

UeU

The proof again is an easy generalisation of the proof of Theorem 1 We state some corollaries

Corollary ! If UçïB, then P(M, U) ç F(M, 2B), especially P(M,U) = P(M, U)

Proof Use (11)

Corollary T P(M,Ui) P(M,ll2)^P(M, О^пИг)

Proof Use (11)

Corollary 3 For any ideal 51Ç Я (including 9X = P) there exists - up to equivalence - a unique minimal и = ЩМ,Щ such that 5t"çP(M,ll) for some power 51" of 51 XI (M, 51) could be called the set of 5l-vertices of M

Proof This is an immediate consequence of Corollary 2

Corollary 4' If l/6U,then(G C/)gP(M, IT)

Proof Define J^i; = Idjvf and X^ =0 for U'eU,U+U' and use (iv)