There is no sharp transitivity on q6 when q is a type of Morley rank 2
There is no sharp transitivity on q6 when q is a type of Morley rank 2
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当 q 是 Morley 2 阶类型时，q6 上不存在尖锐传递性
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发表时间：
1992
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通讯作者：
Ursula Gropp
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作者：
Ursula Gropp
在本文中，我们研究传递群行为：G × X → X，可在 ω 稳定理论中定义，其中 G 是一个连通群，X 是莫利秩 2 的集合，相对于 qα 上的尖锐传递性。这里 q 是 X 的泛型类型（根据命题 (1)，X 的度为 1），对于序数 α > 0，qα 是 q 的 α 次幂，即 (aβ)β < α, ⊨ qα iff (aβ)β < α 是 q 的实现的独立序列（在分叉的意义上），并且 G 被定义为在 qα 上锐传递，当且仅当对于所有 (aβ)β < α 时， (bβ)β < α ⊨ qα 对于所有 β < α，存在且仅有一个 g ∈ G，且 g.aβ = bβ。这里研究的问题是：对于 q 的哪些幂 α，存在满足上述条件且 G 在 qα 上急剧传递的群作用？ In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q.此外，如果对于某些 n ≥ 2，G 在 qn 上急剧传递，则稳定器 Ga 对 X 的某个子集 Y 的作用满足上述条件，其中 Ga 在 qm−1 上急剧传递，其中 q′ 是 Y 的泛型（命题（8））。因此，如果我们能找到某个 n < ω 使得不存在上述 G 在 qn 上急剧传递的群作用，但对于 n – 1 则存在，那么这个问题就会有一个完整的答案。在本文中，我们证明这样的 n 存在，并且它要么是 5，要么是 6。更准确地说，在第 2 节中，我们证明不存在满足上述条件且 G 在 q6 上急剧传递的群作用。这是本文的主要结果。作者不知道同样的情况是否也适用于 q5 而不是 q6。然而，它不适用于 q4，如第 3 节所示。我们给出了射影几何提供的一个例子，对于满足上述条件的群作用，G 在 q4 上急剧传递； for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.
In this paper we study transitive group actions.:G × X → X, definable in an ω-stable theory, where G is a connected group and X a set of Morley rank 2, with respect to sharp transitivity on qα. Here q is the generic type of X (X is of degree 1 by Proposition (1)), for ordinals α > 0, qα is the αth power of q, i.e. (aβ)β < α, ⊨ qα iff (aβ)β < α is an independent sequence (in the sense of forking) of realizations of q, and G is defined to be sharply transitive on qα iff for all (aβ)β < α, (bβ)β < α ⊨ qα there is one and only one g ∈ G with g.aβ = bβ for all β < α. The question studied here is: For which powers α of q are there group actions subject to the above conditions with G sharply transitive on qα? In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q. Moreover, if G is sharply transitive on qn for some n ≥ 2, then the action of the stabilizer Ga on a certain subset Y of X satisfies the conditions above with Ga being sharply transitive on qm−1, where q′ is the generic type of Y (Proposition (8)). Thus, there would be a complete answer to the question if one could find some n < ω such that there is no group action as above with G sharply transitive on qn, but for n – 1 there is. In this paper we prove that such n exists and that it is either 5 or 6. More precisely, in §2 we prove that there is no group action satisfying the above conditions with G sharply transitive on q6. This is the main result of this paper. It is not known to the author whether the same also holds for q5 instead of q6. However, it does not hold for q4, as is seen in §3. There we give an example provided from projective geometry, for a group action satisfying the above conditions with G sharply transitive on q4; for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.