A GALOIS EXTENSION THAT IS NOT FAITHFUL

John Rognes

Ben Wieland wrote to me on August 14th 2008, pointing out that not all G-Galois extensions A → B of commutative S-algebras are faithful, thus answering Question 4.3.6 of [1] in the negative. In fact, a counterexample already appears in Proposition 5.6.3(b) of that memoir, as the case G = ℤ/p where p is prime, A = F (BG_{+}, Hℤ/p) and B = F (EG_{+}, Hℤ/p) ≃ Hℤ/p. Then A → B is a G-Galois extension by the cited proposition. However, this extension is not faithful by the criterion given in Proposition 6.3.3 of the memoir, since the Tate construction B^{tG} has homotopy π_{∗}(B^{tG}) = Ĥ^{−∗}(G; ℤ/p), which is not zero. Hence A → B cannot be faithful, by the latter proposition.

Reference

[1] John Rognes, Galois extensions of structured ring spectra, Mem. Amer. Math. Soc. 192 (2008), no. 898, 1–97.

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