symmetric monoidal (∞,1)-category of spectra
An Azumaya algebra over a commutative unital ring is an algebra over that has an inverse up to Morita equivalence. That is, is an Azumaya algebra if there is an -algebra such that is Morita equivalent to , which is the unit for the tensor product of -algebras. Thus, Morita equivalence classes of Azumaya algebras over form a group, which is called the Brauer group of .
In what follows, is a commutative unital ring and algebras over are assumed associative and unital but not necessarily commmutative. An Azumaya algebra over is an algebra over obeying any of the following equivalent conditions:
There exists an -algebra such that the tensor product of -algebras is Morita equivalent to .
The -algebra is Morita equivalent to , where is the opposite algebra of .
The center of is , and is a separable algebra.
As a left -module, finitely generated, faithful and projective, and the canonical morphism is an isomorphism.
When is a field, an Azumaya algebra is the same as a central simple algebra over .
For any commutative ring there is a monoidal bicategory with
Given any monoidal bicategory we can take its core: that is, the sub-monoidal bicategory where we only keep objects invertible up to equivalence, morphisms invertible up to 2-isomorphism, and invertible 2-morphisms. This core is a 3-group, sometimes called the Picard 3-group, and it has Azumaya algebras over as its objects.
More generally, Grothendieck defines an Azumaya algebra over a scheme as a sheaf of -algebras such that for each point , the corresponding stalk is an Azumaya -algebra.
The Brauer group classifies Azumaya algebras over up to a suitably defined equivalence relation: if for some locally free sheaves of -modules and of finite rank. The group operation of is induced by the tensor product. The Brauer group can be reexpressed in terms of second nonabelian cohomology; indeed a sheaf of Azumaya algebras over determines an -gerbe (or -gerbe in the manifold context).
Karoubi K-theory involves an element in a Brauer group and in the original Karoubi–Donovan paper is related to a twisting with a “local system” which involves Azumaya algebras.
For a ring and the etale cohomology, the multiplicative group of the affine line; then
(Picard group: iso classes of invertible -modules)
(Brauer group Morita classes of Azumaya -algebras)
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
is
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.
Corollary
is connected. Hence .
, hence
Postnikov tower for :
hence for étale
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.
Examples
(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 .
Borceux and Vitale have noted that the monoidal bicategory of -algebras, bimodules and bimodule morphisms can be generalized in the context of enriched category theory, leading to a concept of “Azumaya category”. An Azumaya algebra over the commutative ring is then a one-object Azumaya category enriched over .
More precisely, they consider an arbitrary Benabou cosmos , meaning a complete and cocomplete symmetric monoidal closed category. This gives a monoidal bicategory with
The core of this monoidal bicategory is a 3-group, and they call the objects of the core Azumaya categories.
G. Cortiñas, Charles Weibel, Homology of Azumaya algebras, Proc. AMS 121, 1, pp. 1994 (jstor)
John Duskin, The Azumaya complex of a commutative ring, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) doi:10.1007/BFb0081352
Alexander Grothendieck, Le groupe de Brauer I, II, III, in Dix exposes sur la cohomologie des schemas (I: Algèbres d’Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)
Max Karoubi, Peter Donovan, Graded Brauer groups and -theory with local coefficients (pdf)
M-A. Knus, M. Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lec. Notes in Math. 389, Springer 1974, doi:10.1007/BFb0057799, MR0417149
J. Milne, Étale cohomology, Princeton Univ. Press
Ross Street, Descent, Oberwolfach preprint (sec. 6, Brower groups) pdf; Some combinatorial aspects of descent theory, Applied categorical structures 12 (2004) 537-576, math.CT/0303175 (sec. 12, Brower groups)
Enrico Vitale, A Picard-Brauer exact sequence of categorical groups, pdf
Ana-L. Agore, Stefan Caenepeel?, Gigel Militaru, Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings, Appl. Categor. Struct. 22, 29–42 (2014) doi
The observation that passing to derived algebraic geometry makes also the non-torsion elements in the “bigger Brauer group” be represented by (derived) Azumaya algebras is due to
Bertrand Toën, Derived Azumaya algebras and generators for twisted derived categories (arXiv:1002.2599)
Michael Artin, On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11, 532–563 doi
The comparison of the Artin’s theorem on characterization of Azumaya algebras and Tomiyama-Takesaki’s theorem on -homogeneous C*-algebras is in chapter 9 of
See also
Wikipedia, Azumaya algebra
Urs Schreiber, Picard and Brauer 2-groups, String Theory Coffee Table, 2006.
John Baez, The Brauer 3-group, -Category Café, 2020.
Last revised on July 10, 2024 at 15:15:43. See the history of this page for a list of all contributions to it.