zbMATH — the first resource for mathematics

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.

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.)
Full Text: DOI
[1] DOI: 10.1080/00927879008824104 · Zbl 0760.16014 · doi:10.1080/00927879008824104
[2] DOI: 10.1090/S0002-9947-1986-0860387-X · doi:10.1090/S0002-9947-1986-0860387-X
[3] Bourbaki N., Algèbre VIII (1958)
[4] Childs L.N., Pacific J.Math 23 pp 25– (1967)
[5] DOI: 10.1007/BF01168681 · Zbl 0604.16005 · doi:10.1007/BF01168681
[6] Knus Et M.A., Théory de la Descente et Algèbres d’Azumaya (1974) · Zbl 0284.13002
[7] DOI: 10.1007/BF01200036 · Zbl 0731.16026 · doi:10.1007/BF01200036
[8] DOI: 10.1016/0021-8693(76)90146-0 · Zbl 0343.16025 · doi:10.1016/0021-8693(76)90146-0
[9] Masuoka A., Tsukuba J.Math 14 pp 107– (1990)
[10] DOI: 10.1007/BF02764617 · Zbl 0765.16012 · doi:10.1007/BF02764617
[11] Roy A., J.Math.Kyoto Univ 7 pp 161– (1967)
[12] Schneider H.J., On inner actions of Hopf algebras and stabilzers of representations (1991)
[13] Sweedler M.E., Hopf algebras (1969)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.