Algebras and modules
Model category presentations
Geometry on formal duals of algebras
For a ring, the Brauer group is the group of Morita equivalence classes of Azumaya algebras over .
Relation to categories of modules
The homotopy groups of are the following:
is the Brauer group of ;
is the Picard group of ;
is the group of units of .
See for instance (Street).
Relation to étale cohomology
The Brauer group of a ring is a torsion subgroup of the second etale cohomology group of with values in the multiplicative group
This was first stated in (Grothendieck 68), a discussion is in chapter IV of (Milne). 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
Relation to derived étale cohomology
More generally, this works for a (connective) E-infinity ring (the following is due to Benjamin Antieau and David Gepner).
Let be its infinity-group of units. If is connective, then the first Postikov stage of the Picard infinity-groupoid
where the top morphism is the inclusion of locally free -modules.
so is not equal to , but it is off only by .
Let be the (infinity,1)-category of -modules.
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
Write for the image of . Then define the Brauer infinity-group to be
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
For every (infinity,1)-sheaf of infinity-groups, there is a spectral sequence
(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.
Suppose there exists which is a faithful Galois extension for a finite group.
(real into complex K-theory spectrum) (this is )
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 .
Brauer groups are named after Richard Brauer.
An introduction is in
Pete Clark, On the Brauer group (2003) (pdf)
Alexandre Grothendieck, Le groupe de Brauer, Dix exposés sur la cohomologie des schémas_, Masson and North-Holland, Paris and Amsterdam, (1968), pp. 46–66.
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)
The relation to cohomology/etale cohomology is discussed in
- James Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, New Jersey (1980)
- Jean Giraud, Cohomologie non abelienne, Die Grundlehren der mathematischen Wissenschaften, vol. 179, Springer- Verlag, Berlin, 1971.
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.
Aise Johan de Jong, A result of Gabber (pdf)
- Inta Bertuccioni, Brauer groups and cohomology, Archiv der Mathematik, vol. 84 Number 5 (2005)
Brauer groups of superalgebras are discussed in
Refinement to stable homotopy theory and Brauer ∞-groups is discussed in
Related MO discussion includes