[Show abstract][Hide abstract] ABSTRACT: The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [Rocky Mountain J. Math. 39 (2009), no. 3, 765–787] and [J. Number Theory 129 (2009), no. 6, 1350–1365]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field ℚ, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [Invent. Math. 153 (2003), no. 2, 303–359] and Flach completed their results at 2 in [Recent work and new directions, Contemp. Math. 358, American Mathematical Society, Providence (2004), 79–125] and [J. reine angew. Math. 661 (2011), 1–36].
Journal für die reine und angewandte Mathematik (Crelles Journal) 01/2015; DOI:10.1515/crelle-2015-0014 · 1.30 Impact Factor
"If C is any algebra (assumed graded here), and if M; N; and R are (graded) left C – modules, Yoneda defines in  the " composition pairing " . This is a degree-preserving map of bigraded vector spaces: "
"and 1st cohomology groups H° and H l (the dimension indices in triple cohomology being one less than usual) were discussed by J. M. Beck in his dissertation . The purpose of the present paper is to interpret the second cohomology, H 2 (A, M), of an algebra A with coefficients in an ^4-module M as the set of equivalence classes after Yoneda  of two term extensions of A by M (see §3, Lichtenbaum-Schlessinger  or Gerstenhaber  for two term extensions). Our interpretation appears to be more direct than those through classical obstruction theory for algebra extensions (MacLane , , Hochschild , Shukla , Barr ) and suggests a close relationship between H n and n term extensions for n>2 (see §4). "
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