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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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