Some cyclotomic additive Galois cohomology rings
نویسنده
چکیده
Let p ≥ 3 be a prime. We consider the cyclotomic extension Z(p)[ζp2] of Z(p), with Galois group G := (Z/p2)∗. Since this extension is wildly ramified, Z(p)[ζp2] is not projective as a module over the group ring Z(p)G (Speiser). Extending this module structure, we can regard Z(p)[ζp2 ] as a module over the twisted group ring Z(p)[ζp2] ≀G; as such, it remains faithful and non projective. We calculate its cohomology ring Ext∗Z(p)[ζp2 ]≀G (Z(p)[ζp2],Z(p)[ζp2 ]), which is, as a graded Z(p)module, isomorphic to the additive Galois cohomology H(G,Z(p)[ζp2 ];Z(p)). To obtain a projective resolution that allows to conveniently calculate the multiplication thereupon, we embed the twisted group ring Z(p)[ζp2] ≀ G into EndZ(p)Z(p)[ζp2] by sending an element to its module operation, and describe the embedded isomorphic copy via ties, that is, via congruences of matrix entries.
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