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Skolem-Noether theorems and coalgebra actions. (English) Zbl 0793.16030
Let \(k\) be a semilocal commutative ring, let \(A\) be an algebra over \(k\), let \(B\) be a subalgebra of \(A\), and let \(C\) be a coalgebra over \(k\). A measuring \(C \otimes B \to A\) is equivalent to an algebra homomorphism of \(B\) into \(\text{Hom}(C,A)\). If \(B\) is a central, separable algebra over \(k\) and \(\text{Hom}(C,A)\) is a finitely generated module over \(k\), the author shows that every measuring \(C \otimes B \to A\) is inner by proving a Skolem-Noether type theorem for algebra homomorphisms from a central, separable algebra into an algebra which is finitely generated as a module. He also indicates how many of the known results about when measurings are inner may be derived by his methods. Of particular interest in this paper is the demonstration that when \(k\) is a field and \(B\) is finite dimensional, but not central, over \(k\), there exists a coalgebra \(C\) and a measuring \(C \otimes B \to A\) which is not inner.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W20 Automorphisms and endomorphisms
16S40 Smash products of general Hopf actions
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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