Separating invariants. (English) Zbl 1172.13001
The author defines a general notion of a separating subset. Explicitly, let \(X\) and \(K\) be sets (\(K\) will be a field or an integral domain in most situations), and let \(K^X\) be the set of all functions from \(X\) to \(K\). Let \(F\) be any subset of \(K^X\). A subset \(S\) of \(F\) is called an \(F\)-separating set if, for any \(x,y \in X\), if \(g(x)=g(y)\) for all \(g \in S\), then \(f(x)=f(y)\) for all \(f \in F\). This notion is useful in the study of modular invariant theory.
Among other results in this paper, the following theorem is proved.
Theorem. Let \(X\) be a set, \(K\) be a commutative noetherian ring, \(A\) be a finitely generated \(K\)-algebra contained in \(K^X\). For any subset \(F\) of \(A\), there exists a finite \(F\)-separating subset \(S\) for \(F\). Many examples are exhibited also.
Among other results in this paper, the following theorem is proved.
Theorem. Let \(X\) be a set, \(K\) be a commutative noetherian ring, \(A\) be a finitely generated \(K\)-algebra contained in \(K^X\). For any subset \(F\) of \(A\), there exists a finite \(F\)-separating subset \(S\) for \(F\). Many examples are exhibited also.
Reviewer: Ming-chang Kang (Taipei)
MSC:
| 13A50 | Actions of groups on commutative rings; invariant theory |
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