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ON THE HOMOLOGY THEORY OF MODULES

Journal of the Faculty of Science. Section I 01/1954;
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    ABSTRACT: In this thesis, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the so called Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasitriangular Hopf algebras, we further determine a significant part of the Lie bracket's kernel. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties.
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    Journal of the Mathematical Society of Japan 01/1956; 8(1956). · 0.51 Impact Factor
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    ABSTRACT: jects, the unity was more abstractly philosophical and logical than mathematical. It was based more on the philosophy behind Russell and Whitehead’s Principia Mathematica than on experience in geometry, algebra, or analysis. Yet he gained experience. Then came his collaboration with Samuel Eilenberg on one specic problem. They worked to explain how some pure algebra by Mac Lane had arrived at the cohomology of thep-adic solenoid, an innitely tangled compact metric space. To do this they created category theory. They organized the functorial basis for homology and cohomology in topology. And they created group cohomology in its full functorial form. Over the next 60 years these tools gave precise mathematical unity to Mac Lane’s work. His 1930s work in arithmetic algebraic geometry took on new importance in light of his homological algebra of the 1950s. These ideas joined with his ear- liest interest in logic in his last mathematical book (Mac Lane & Moerdijk 1992). Mac Lane expressed all of this experience, plus what he learned from Emmy Noether and Hermann Weyl when he was a student in the last great days of David Hilbert’s G ottingen, in his reections on \mathematical truth and beauty" (Mac Lane 1986, p. 409). As an undergraduate at Yale University from 1926 to 1930 Mac Lane was at- tracted to the new symbolic logic, especially Principia Mathematica, and he studied Hausdor’s,set theory and topology. He also learned that Emmy Noether was pro- ducing great new mathematical results by thinking about the foundations of algebra although he did not yet learn much of her work. As a graduate student at the Uni- versity of Chicago he was deeply impressed by the founder of the mathematics department, Eliakim Hastings Moore. Moore followed the Chicago tradition by
    Scientiae Mathematicae Japonicae. 01/2006; 63(1).