On the Homology Theory of Modules

Journal of the Faculty of Science. Section I 01/1954; 7(2).
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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].
    Preview · Article · Jan 2015 · Journal für die reine und angewandte Mathematik (Crelles Journal)
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    • "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 degree-preserving map of bigraded vector spaces: "
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    ABSTRACT: We write P⊗s for the polynomial ring on s letters over the field Z/2, equipped with the standard action of σs, the symmetric group on s letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring (P⊗s)σs as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces Z/2⊗A (P⊗s)σs. Our main result is stated in terms of a "bigraded Steenrod algebra" H. The generators of this algebra H, like the generators of the classical Steenrod algebra A, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that Sq0 is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces Z/2⊗A (P⊗s)σs, for all s ≥ 0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over H.
    Preview · Article · May 2008 · Algebraic & Geometric Topology
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    • "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 ^4-module M as the set of equivalence classes after Yoneda [29] of two term extensions of A by M (see §3, Lichtenbaum-Schlessinger [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). "

    Preview · Article · Jan 1969 · Publications of the Research Institute for Mathematical Sciences
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