By Zhao W., Liu P.
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2. 3. 4. 5. 6. h(k) = 1. If G ∈ (E) then each group that is locally isomorphic to G is isomorphic to G. Each G ∈ (E) has the power cancellation property. Some G ∈ (E) has the power cancellation property. Each G ∈ (E) has a -unique decomposition. Some G ∈ (E) has a -unique decomposition. Proof: 1 ⇔ 2. Let G ∈ (E). 16), h(G) = h(E(G)) = h(E) = 1 iff part 2 is true. 1 ⇒ 3. Let G ∈ (E). 16 and part 1, h(G) = h(E(G)) = h(E) = h(k) = 1. 30, G has the power cancellation property. This is part 3. 38. 4 ⇒ 1.
Since (AG (G)) = (E(G)) is the identity in Pic(E(G)), λn ((G n−1 ⊕ H )) = (E(G)) · · · (E(G)) (AG (H )) = (U ). n−1 Thus λn is a surjection. Let (H ), (K) ∈ (G n ) be such that λn ((H )) = λn ((K)). 1 we can write AG (H ) ∼ = U1 ⊕ · · · ⊕ Ur and AG (K) ∼ = W1 ⊕ · · · ⊕ Ws for some invertible fractional ideals Ui and Wj . 2, AG (H ), AG (K), and AG (G n ) = E(G)n are locally isomorphic. Uniqueness of rank over the integral domain E(G) implies that n = r = s. 7 implies that AG (H ) ∼ = U1 ⊕ · · · ⊕ Un ∼ = W1 ⊕ · · · ⊕ Wn ∼ = AG (K).
13. 15. Let G be a cocommutative strongly indecomposable rtffr group, and let n > 0 be an integer. There is a bijection of finite sets, λn : (G n ) −→ Pic(E(G)). 9 that there are at most finitely many isomorphism classes of fractional ideals of E(G). Thus Pic(E(G)) is a finite multiplicative abelian group. Let n > 0 be an integer. To define λn , let H be locally isomorphic to G n . 4, AG (H ) is a finitely generated projective right E(G)-module of rank n over the integral domain E(G). 1, there are invertible fractional right ideals U1 , .
A Biplurisubharmonic Characterization of AUMD Spaces by Zhao W., Liu P.