On constructions similar to the Burnside ring for commutative rings and profinite groups
Let R be a commutative ring with separable closure \(\bar R\) and let \(\Gamma\) be the Galois group of \(\bar R\) over R [cf. \textit{G. Janusz}, Trans. Am. Math. Soc. 122, 461-479 (1966; Zbl 0141.034)]. Let K be a Galois extension of R with group \(\Delta\). Let \(Gal(K/R,G)\) be the group [under the Harrison product \(*_ G\), cf. \textit{D. K. Harrison}, Mem. Am. Math. Soc. 52, 1-14 (1965; Zbl 0143.060)] of Galois extensions of R with group G which become trivial as Galois extensions when base changed to K. The set of \((R,G)\)-algebras \((A,\vartheta)\), that is, separable, commutative R-algebras A which as R-modules are finitely generated and projective and have a G-action \(\vartheta: G\to Aut_ R(A)\), is closed under direct products and under the Harrison product. The set of \((R,G)\)- algebras, modulo a Grothendieck-group-like equivalence, forms a commutative ring \(W(R,G;K)\), where \((A,\vartheta)\) becomes trivial in \(W(R,G;K)\) if it is fixed under the action (via \(*_ G)\) of \(Gal(K/R,G)\). There is a well-known category equivalence between separable R-algebras and finite \(\Gamma\)-sets. Under this equivalence, \((R,G)\)-algebras correspond to finite, continuous \(\Gamma\times G\)-sets, and translating the construction of \(W(R,G;K)\) into the latter category yields an isomorphic ring \(W(\Gamma,G;J)\) for suitable \(J\subseteq Hom_{cont}(\Gamma,G)\). For G trivial this ring is related to the Burnside ring of \(\Gamma\) [cf. \textit{C. W. Curtis} and \textit{I. Reiner}, ``Methods of representation theory'', Vol. II (1987; Zbl 0616.20001); chapter 11], although the authors do not make this connection explicit. - The authors carry out the theory in the setting of \(\Gamma\times G\)-sets. The paper concludes by computing \(W(\Gamma,G;J)\) when \(\Gamma ={\hat {\mathbb{Z}}}\), G is cyclic of order p, prime, and \(J=Hom_{cont}(\Gamma,G).\) The introduction hints at connections of this theory to algebraic number theory, but any such connections are left for the reader to ponder.
scientific article; zbMATH DE number 4208224