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Osee Posted 19 years ago
Vocabulary
part of my talk --1
Hi, the following is what I wrote for a talk, can you make comments regarding any aspect of the language used here? Thanks a lot!
In modular invariant theory there is a celebrated theorem, due to Nakajima, which says that a group over a prime field is a Nakajima group if and only if its ring of invariants is a polynomial ring. Although, due to an example of Stong, we know that a group over a field of positive characteristic which admits the polynomial ring of invariants may not be a Nakajima group, yet it is not hard to observe that a Nakajima group over a field of positive characteristic always admits the polynomial ring of invariants which is generated by some fancy polynomials called norms. It is this observation that reminds us to consider whether its reverse holds, namely, whether a group over a field of positive characteristic admitting the polynomial ring of invariants that is generated by norms has to be a Nakajima group. It turns out that we fully solve this problem and give a confirmative answer.
In modular invariant theory there is a celebrated theorem, due to Nakajima, which says that a group over a prime field is a Nakajima group if and only if its ring of invariants is a polynomial ring. Although, due to an example of Stong, we know that a group over a field of positive characteristic which admits the polynomial ring of invariants may not be a Nakajima group, yet it is not hard to observe that a Nakajima group over a field of positive characteristic always admits the polynomial ring of invariants which is generated by some fancy polynomials called norms. It is this observation that reminds us to consider whether its reverse holds, namely, whether a group over a field of positive characteristic admitting the polynomial ring of invariants that is generated by norms has to be a Nakajima group. It turns out that we fully solve this problem and give a confirmative answer.
Top answer
Hi, It'sobviously very technical material that I don't understand, but I've tried to suggest a few changes. Please read carefully. In modular invariant theory there is a celebrated theorem by Nakajima, which says that a group over a prime field is a Nakajima group if and only if its ring of invariants is a polynomial ring.
- Hi, It'sobviously very technical material that I don't understand, but I've tried to suggest a few changes.
- Please read carefully.
- In modular invariant theory there is a celebrated theorem by Nakajima, which says that a group over a prime field is a Nakajima group if and only if its ring of invariants is a polynomial ring.
- Although, from a example by Stong, we know that a group over a field of positive characteristic which admits the polynomial ring of invariants may not be a Nakajima group, it is not hard to observe that a Nakajima group over a field of positive characteristic always admits the polynomial ring of invariants which is generated by some complex polynomials called 'norms'.
- It is this observation that reminds us to consider whether its reverse holds, namely, whether a group over a field of positive characteristic admitting the polynomial ring of invariants that is generated by norms has to be a Nakajima group.
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2 Answers
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Hi,
It'sobviously very technical material that I don't understand, but I've tried to suggest a few changes. Please read carefully.
It'sobviously very technical material that I don't understand, but I've tried to suggest a few changes. Please read carefully.
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In modular invariant theory, there is a celebrated theorem, byNakajima. He said, “a group over a prime field is a Nakajima group, if and only if its ring of invariants is a polynomial ring”. Because of an example of Stong, we know that a group over a field of positive characteristic that admits the polynomial ring of invariants may not be the Nakajima group. Yet,
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