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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)≃H et 2(X,𝔾 m) torBr(X) \simeq H^2_{et}(X, \mathbb{G}_m)_{tor} and moved notes from a talk by David Gepner on ∞\infty-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 H et 2(R,𝔾 m)H^2_{et}(R, \mathbb{G}_m) 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 H et 2(−,𝔾 m)H^2_{et}(-,\mathbb{G}_m) 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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