Galois and Hopf-Galois Theory for Associative S-Algebras

Abstract

We define and investigate Galois and Hopf-Galois extensions of associative S-algebras, generalizing both the algebraic notions and the notions introduced by John Rognes for commutative S-algebras. We provide many examples such as matrix extensions, Thom spectra and extensions of Morava-K-Theory spectra induced from Lubin-Tate extensions. We show three applications. First, we show the existence of associative S-algebras which have as homotopy groups a finite possibly associative Galois extension of the homotopy groups of a commutative S-algebra. Second, we show that B defines an element in the Picard group Pic(A[G]) whenever A->B is a Galois extension of associative S-algebras with finite abelian Galois group G. A third application concerns the calculation of the topological Hochschild homology of a Hopf-Galois extension of commutative S-algebras which we relate to the topological Hochschild homology of the Hopf-algebra involved. The appendix contains a Galois correspondence for extensions of associative rings, generalizing at least two main theorems from literature.

Wir definieren und untersuchen Galois und Hopf-Galois Erweiterungen für assoziative S-Algebren. Wir zeigen drei Anwendungen. Erstens die Existenz gewisser assoziative S-Algebren. Zweitens, dass Galoiserweiterungen invertierbare Bimoduln definieren. Drittens untersuchen wir die Hochschild Homologie von Hopf-Galois Erweiterungen kommutative S-Algebren, die wir mit der Hochschild Homologie der Hopf-Algebra in Verbindung setzen. Der Anhang enthält eine Galois Korrespondenz für Erweiterungen von assoziativen Ringen.