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nLab > Latest Changes: Emmy Noether

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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 4th 2019

    Added some more content, most particularly the abstract of her talk from 1925 introducing homology groups, as a form of categorification:

    Ableitung der Elementarteilertheorie aus den Gruppentheorie. Die Elementarteilertheorie gibt bekanntlich für Moduln aus ganzzahligen Linearformen eine Normalbasis von der Form (e 1y 1,e 2y 2,...,e ryr)(e_1y_1, e_2y_2, ..., e_ry_r), wo jedes ee durch das folgende teilbar ist; die ee sind dadurch bis aufs Vorzeichen eindeutig festgelegt. Da jede Abelsche Gruppe mit endlich vielen Erzeugenden dem Restklassensystem nach einem solchen Modul isomorph ist, ist dadurch der Zerlegungssatz dieser Gruppen als direkte Summe größter zyklischer mitbewiesen. Es wird nun umgekehrt der Zerlegungssatz rein gruppentheoretisch direkt gewonnen, in Verallgemeinerung des für endliche Gruppen üblichen Beweises, und daraus durch Übergang vom Restklassensystem zum Modul selbst die Elementarteilertheorie abgeleitet. Der Gruppensatz erweist sich so als der einfachere Satz; in den Anwendungen des Gruppensatzes — z.B. Bettische und Torsionszahlen in der Topologie — is somit ein Zurückgehen auf die Elementarteilertheorie nich erforderlich.

    diff, v2, current

  2. fix a few typos

    Anonymous

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 4th 2019

    Thanks!

    • CommentRowNumber4.
    • CommentAuthorPaoloPerrone
    • CommentTimeNov 5th 2019

    Can a German-speaking algebrist add a translation? I could attempt myself, but for example I have no clue what a Restklassensystem is (residue class system maybe?).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2019

    The answer to both questions is yes.

    • CommentRowNumber6.
    • CommentAuthorPaoloPerrone
    • CommentTimeNov 6th 2019
    • (edited Nov 6th 2019)

    Here is my attempt. I’m not a native German speaker nor an algebrist, please correct any mistakes. Thanks!

    Derivation of the theory of elementary divisors from group theory. As it is known, the theory of elementary divisors gives for [modules from integral linear forms?] a normal basis in the form (e 1y 1,e 2y 2,…,e ry r)(e_1 y_1, e_2 y_2, \dots, e_r y_r), where each ee is divisible by the next one; this way, all the ee are uniquely determined up to sign. Using the fact that [according to the residue class system?] every finitely generated Abelian group is isomorphic to such a module, one can prove the decomposition theorem, i.e. that every such group can be expressed as the direct sum of cyclic groups. Here, conversely, we obtain the decomposition theorem purely from group theory, generalizing the usual proof for finite groups, and by passing from the residue class system to the module we derive the theory of elementary divisors itself. This way, the group-theoretical statement turns out to be the simpler one; and so, in the applications of this group-theoretical statement - such as Betti numbers and torsion coefficients in topology - it is not necessary to revert to elementary divisors.

    • CommentRowNumber7.
    • CommentAuthorNikolajK
    • CommentTimeNov 6th 2019
    Sounds about right. Where you say "one can prove the decomposition theorem", it actually says that this already proves it, but that's maybe an overstatement.
    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 7th 2023

    Additional details.

    diff, v4, current

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