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nLab > Latest Changes: Brauer group

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2012
    • (edited Dec 12th 2012)

    started Brauer group, collecting some references on the statement that/when Br(X)≃H2et(X,𝔾m)tor and moved notes from a talk by David Gepner on ∞-Brauer groups to there.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 13th 2012
    • (edited Jun 15th 2012)

    I added the insightful references of Street at Azumaya algebra and at Brauer group:

    • Ross Street, Descent, Oberwolfach preprint (sec. 6, Brauer groups) pdf; Some combinatorial aspects of descent theory, Applied categorical structures 12 (2004) 537-576, math.CT/0303175 (sec. 12, Brauer groups)
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2012

    Thanks. I have changed “Brower group” to “Brauer group”, okay?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2012

    So I have added a remark in a new section Relation to categories of modules.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2012

    Thanks for Brauer, Urs.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2012

    added to Brauer group references on Brauer groups for superalgebras.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeDec 12th 2012

    added Duskin’s historical paper

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2014
    • (edited May 22nd 2014)

    have added to Brauer group a paragraph (here) on its “bigger” version (thanks toDavid C. for the pointer to Heinloth’s article)

    It is therefore natural to regard all of H2et(R,𝔾m) as the “actual” Brauer group. This has been called the “bigger Brauer group” (Taylor 82, Caenepeel-Grandjean 98, Heinloth-Schöer 08). the bigger Brauer group has actually traditionally been implicit already in the term “formal Brauer group”, which is really the formal geometry-version of the bigger Brauer group.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2014

    Have added at the references at Brauer group:

    The observation that passing to derived algebraic geometry makes also the non-torsion elements in H2et(−,𝔾m) be represented by (derived) Azumaya algebras is due to

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 17th 2020

    Added the reference:

    diff, v32, current

    • CommentRowNumber11.
    • CommentAuthorStableHolonomy
    • CommentTimeOct 19th 2024
    The quaternions are an algebra, but the octonions only an A-infinity algebra.The quaternions form a central simple extension of the real numbers since H tensor Hop is Mat2(R), the 2 by 2 matrices over the reals. I have read that the property of a finite dimensional k-algebra that tensoring with the separable closure of k produces a finite dimensional matrix ring Matn(ksep) is equivalent to being a central simple k-algebra.

    But the octonions form an A-infinity algebra and so it would be nice to know about a lax analogue. Is there a "lax Brauer group" which features the octonions as an A-infinity algebra? It would maybe feature Mat4(R)
    • CommentRowNumber12.
    • CommentAuthorStableHolonomy
    • CommentTimeOct 19th 2024
    • (edited Oct 19th 2024)

    (tests)

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