Topological K-theory of complex noncommutative spaces
By A. Blanc. Compositio Mathematica, 2012
96 citations recorded by the source corpus.
The purpose of this work is to give a definition of a topological K-theory for dg-categories over $\mathbb{C}$ and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum $\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with $\mathbb{C}$ is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0279227ed755a3f68b44ecf2dbd40a8c525dced3
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
- P. Stellari
- Emanuele Macrì
- D. Huybrechts
- Arend Bayer
- Chunyi Li
- Alexander Kuznetsov
- E. Markman
- Benjamin Schmidt
In Mathematical Physics
- P. Stellari
- Emanuele Macrì
- D. Huybrechts
- Arend Bayer
- Alexander Kuznetsov
- E. Markman
- Chunyi Li
- Benjamin Schmidt
Citations in the corpus
- The K3 category of a cubic fourfold (2015, methodology): “cubics is abstractly isomorphic to Λr. Remark 3.3. Via the fully faithful embedding AX ˜• /D/ bpXq the category A X inherits a dgenhancement which in turn can be used to define KtoppAXq directly using [7]. However, this a priori still depends on the embedding A˜ • /D/ bpXq and so does not really lead”
- Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, background): “apter 10] for a summary of results on K3 surfaces. Topological K-theory. Let D be a non-commutative smooth projective variety. A general construction of the topological K-theory associated to D is in [Bla16]. In our setting, this can be introduced in a way more closely related to the usual K-theory ”