Article
ON THE HOMOLOGY THEORY OF MODULES
Journal of the Faculty of Science. Section I 01/1954;
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 "One defines Ext 0 R (M, N ) := Ker(∂ * 1 ) and for n ≥ 1 Ext n R (M, N ) := Ker(∂ * n+1 )/Im(∂ * n ). In [29] "
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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/crelle20150014 · 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 [27] the " composition pairing " . This is a degreepreserving map of bigraded vector spaces: "
Algebraic & Geometric Topology 05/2008; 8(1):541562. DOI:10.2140/agt.2008.8.541 · 0.68 Impact Factor 
 "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 [5]. The purpose of the present paper is to interpret the second cohomology, H 2 (A, M), of an algebra A with coefficients in an ^4module M as the set of equivalence classes after Yoneda [29] of two term extensions of A by M (see §3, LichtenbaumSchlessinger [23] or Gerstenhaber [13] for two term extensions). Our interpretation appears to be more direct than those through classical obstruction theory for algebra extensions (MacLane [24], [10], Hochschild [18], Shukla [27], Barr [4]) and suggests a close relationship between H n and n term extensions for n>2 (see §4). "
Publications of the Research Institute for Mathematical Sciences 01/1969; DOI:10.2977/prims/1195194633 · 0.61 Impact Factor
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