Back to Search
Start Over
Noncommutative motives of separable algebras
- Source :
- Vrije Universiteit Brussel, arXiv
-
Abstract
- In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (=CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Hochschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras.<br />Comment: 30 pages. Revised version. A strong relation between the birationality of Severi-Brauer varieties and NC motives was added
- Subjects :
- Hecke algebra
Pure mathematics
math.AT
General Mathematics
Cyclic homology
math.RT
01 natural sciences
noncommutative motives
separable algebra
Brauer group
twisted flag variety
convolution
cyclic sieving phenomenon
dg Azumaya algebra
math.AG
Noncommutative motives
Separable algebra
Twisted flag variety
Convolution
Mathematics - Algebraic Geometry
math.KT
Mathematics::K-Theory and Homology
FOS: Mathematics
Algebraic Topology (math.AT)
Mathematics - Algebraic Topology
0101 mathematics
Representation Theory (math.RT)
Algebraic Geometry (math.AG)
math.RA
Mathematics
Subcategory
Hochschild homology
010102 general mathematics
K-Theory and Homology (math.KT)
Mathematics - Rings and Algebras
16. Peace & justice
Noncommutative geometry
010101 applied mathematics
Field extension
Rings and Algebras (math.RA)
Mathematics - K-Theory and Homology
16H05, 16K50, 14M15, 18D20, 20C08
Mathematics - Representation Theory
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Vrije Universiteit Brussel, arXiv
- Accession number :
- edsair.doi.dedup.....c03bb118fc437daf1e1a168d780e4351