symmetric monoidal (∞,1)-category of spectra
This may be understood as the 2-groupoid of (generalized) line 2-bundles over (for instance holomorphic line 2-bundles in the case of higher complex analytic geometry), inside that of all 2-vector bundles.
The homotopy groups of are the following:
See for instance (Street).
This was first stated in (Grothendieck 68) (see also Grothendieck 64, prop. 1.4 and see at algebraic line n-bundle – Properties). Review discussion is in (Milne, chapter IV). A detailed discussion in the context of nonabelian cohomology is in (Giraud).
A theorem stating conditions under which the Brauer group is precisely the torsion subgroup of is due to (Gabber), see also the review in (de Jong). For more details and more literature on this see (Bertuccioni).
This fits into the following pattern
(Picard group: iso classes of invertible -modules)
It is therefore natural to regard all of 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.
where the top morphism is the inclusion of locally free -modules.
so is not equal to , but it is off only by .
There is a notion of -enriched (infinity,1)-category, of “-linear -categories”.
-modiles in presentable (infinity,1)-categories.
Forming module -categories is then an (infinity,1)-functor
One shows (Antieau-Gepner) that this is exactly the Azumaya -algebras modulo Morita equivalence.
Theorem (B. Antieau, D. Gepner)
For a connective ring, any Azumaya -algebra is étale locally trivial: there is an etale cover such that .
(Think of this as saying that an Azumaya -algebra is étale-locally a Matric algebra, hence Morita-trivial: a “bundle of compact operators” presenting a (torsion) -2-bundle).
is a sheaf for the etale cohomology.
is connected. Hence .
Postnikov tower for :
hence for eétale
This is a quasi-coherent sheaf on of the form (quasicoherent sheaf associated with a module), for an -module. By vanishing theorem of higher cohomology for quasicoherent sheaves
(the second argument on the left denotes the Postnikov stage). From this one gets the following.
is quasicoherent for .
there is an exact sequence
(notice the inclusion )
this is split exact and so computes for connective .
Now some more on the case that is not connective.
Give , have a fiber sequence
Theorem (descent theorems) (Tyler Lawson, David Gepner) Given -Galois extension (homotopy fixed points)
it follows that there is a homotopy fixed points spectral sequence
Conjecture The spectral sequence gives an Azumaya -algebra which is a nontrivial element in but becomes trivial in .
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|
Brauer groups are named after Richard Brauer.
Original discussion includes
Alexandre Grothendieck, Le groupe de Brauer, Dix exposés sur la cohomologie des schémas_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.
An introduction is in
John Duskin, The Azumaya complex of a commutative ring, in: Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107–117, Lecture Notes in Math. 1348, Springer 1988.
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)
Ofer Gabber, Some theorems on Azumaya algebras, Ph. D. Thesis, Harvard University, 1978, Groupe de Brauer, Lecture Notes in Mathematics, vol. 844, Springer-Verlag, Berlin, 1981, pp. 129–209.
Brauer groups of superalgebras are discussed in
C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187-199.
The “bigger Brauer group” is discussed in
J. Taylor, A bigger Brauer group Pacic J. Math. 103 (1982), 163-203
S. Caenepeel, F. Grandjean, A note on Taylor’s Brauer group. Pacific J. Math. 186 (1998), 13-27
Related MO discussion includes