1,210 research outputs found
Azumaya Objects in Triangulated Bicategories
We introduce the notion of Azumaya object in general homotopy-theoretic
settings. We give a self-contained account of Azumaya objects and Brauer groups
in bicategorical contexts, generalizing the Brauer group of a commutative ring.
We go on to describe triangulated bicategories and prove a characterization
theorem for Azumaya objects therein. This theory applies to give a homotopical
Brauer group for derived categories of rings and ring spectra. We show that the
homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the
homotopical Brauer group of its underlying commutative ring. We also discuss
tilting theory as an application of invertibility in triangulated bicategories.Comment: 23 pages; final version; to appear in Journal of Homotopy and Related
Structure
Noncommutative motives of Azumaya algebras
Let k be a base commutative ring, R a commutative ring of coefficients, X a
quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of
rank r, and Hmo(R) the category of noncommutative motives with R-coefficients.
Assume that 1/r belongs to R. Under this assumption, we prove that the
noncommutative motives with R-coefficients of X and A are isomorphic. As an
application, we show that all the R-linear additive invariants of X and A are
exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic
homology (and all its variants), topological Hochschild homology, etc. Making
use of these isomorphisms, we then computer the R-linear additive invariants of
differential operators in positive characteristic, of cubic fourfolds
containing a plane, of Severi-Brauer varieties, of Clifford algebras, of
quadrics, and of finite dimensional k-algebras of finite global dimension.
Along the way we establish two results of independent interest. The first one
asserts that every element of the Grothendieck group of X which has rank r
becomes invertible in the R-linearized Grothendieck group, and the second one
that every additive invariant of finite dimensional algebras of finite global
dimension is unaffected under nilpotent extensions.Comment: 22 pages; revised versio
Topological Hochschild cohomology and generalized Morita equivalence
We explore two constructions in homotopy category with algebraic precursors
in the theory of noncommutative rings and homological algebra, namely the
Hochschild cohomology of ring spectra and Morita theory. The present paper
provides an extension of the algebraic theory to include the case when is
not necessarily a progenerator. Our approach is complementary to recent work of
Dwyer and Greenlees and of Schwede and Shipley. A central notion of
noncommutative ring theory related to Morita equivalence is that of central
separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild
cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of
A. We introduce a notion of topological Azumaya algebra and show that in the
case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a
commutative ring this notion specializes to classical Azumaya algebras. A
canonical example of a topological Azumaya R-algebra is the endomorphism
R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod
2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over
the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the
determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2.
As far as we know this is the first calculation of THH(A,A) for a
noncommutative S-algebra A.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-29.abs.htm
Structure theorems of H4-Azumaya algebras
AbstractLet k be a field and H4 be Sweedler's 4-dimensional algebra over k. It is well known that H4 has a family of triangular structures Rt indexed by the ground field k and each triangular structure Rt makes the H4-module category MH4 a braided monoidal category, denoted MRtH4. In this paper, we study the Azumaya algebras in the categories MRtH4. We obtain the structure theorems for Azumaya algebras in each braided monoidal category MRtH4, t∈k. Utilizing the structure theorems we obtain a scalar invariant for each Azumaya algebra in the aforementioned categories
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