Solutions for the Suggested Problems 1. Suppose that R and S are
... Solution. Let s = ϕ(e), which is an element in S. Since e is an idempotent of R, we have ee = e. Thus, we have ss = ϕ(e)ϕ(e) = ϕ(ee) = ϕ(e) = s . This proves that ss = s and hence that s is an idempotent in the ring S. Now suppose that R = Z and that ϕ : Z → S is a ring homomorphism. Note that 1 is ...
... Solution. Let s = ϕ(e), which is an element in S. Since e is an idempotent of R, we have ee = e. Thus, we have ss = ϕ(e)ϕ(e) = ϕ(ee) = ϕ(e) = s . This proves that ss = s and hence that s is an idempotent in the ring S. Now suppose that R = Z and that ϕ : Z → S is a ring homomorphism. Note that 1 is ...
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... The question of whether or not X and X φ are birationally isomorphic over k is delicate in general. Birational isomorphism over K = k(X)G is more accessible because it has a natural interpretation in terms of Galois cohomology. In this section we will to show that in many cases X and X φ are, indeed ...
... The question of whether or not X and X φ are birationally isomorphic over k is delicate in general. Birational isomorphism over K = k(X)G is more accessible because it has a natural interpretation in terms of Galois cohomology. In this section we will to show that in many cases X and X φ are, indeed ...
Smoothness of Schubert varieties via patterns in root subsystems
... Theorem 2.2. Let G be any semisimple simply-connected Lie group, B be any Borel subgroup, with corresponding root system Φ and Weyl group W = WΦ . For w ∈ W , the Schubert variety Xw ⊂ G/B is smooth (rationally smooth) if and only if, for every stellar root subsystem ∆ in Φ, the pair (∆+ , f∆ (w)) i ...
... Theorem 2.2. Let G be any semisimple simply-connected Lie group, B be any Borel subgroup, with corresponding root system Φ and Weyl group W = WΦ . For w ∈ W , the Schubert variety Xw ⊂ G/B is smooth (rationally smooth) if and only if, for every stellar root subsystem ∆ in Φ, the pair (∆+ , f∆ (w)) i ...
