A splitting theorem and the Künneth formula in algebraic K-theory

1969 ◽  
pp. 72-77 ◽  
Author(s):  
W. C. Hsiang
Keyword(s):  
Splitting Theorem ◽  
Kunneth Formula ◽  
K Theory

scholarly journals Quantitative K-theory and the Künneth formula for operator algebras

2019 ◽  
Vol 277 (7) ◽  
pp. 2003-2091 ◽  
Author(s):  
Hervé Oyono-Oyono ◽  
Guoliang Yu
Keyword(s):  
Operator Algebras ◽  
Kunneth Formula ◽  
K Theory

scholarly journals Topological K-theory of complex noncommutative spaces

2015 ◽  
Vol 152 (3) ◽  
pp. 489-555 ◽  
Author(s):  
Anthony Blanc
Keyword(s):  
Chern Character ◽  
Cyclic Homology ◽  
Finite Dimensional ◽  
Perfect Complexes ◽  
Topological Realization ◽  
Kunneth Formula ◽  
K Theory ◽  
Simplicial Presheaves ◽  
Definition Of ◽  
Dg Algebra

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.

scholarly journals Real C*-Algebras, United K-Theory, and the Künneth Formula

K-Theory ◽  
2002 ◽  
Vol 26 (4) ◽  
pp. 345-402 ◽  
Author(s):  
Jeffrey L. Boersema
Keyword(s):  
C Algebras ◽  
Kunneth Formula ◽  
K Theory

On the Kunneth formula spectral sequence in equivariant K- theory

1972 ◽  
Vol 72 (2) ◽  
pp. 167-177 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Vector Bundle ◽  
Spectral Sequence ◽  
Lie Group ◽  
Complex Vector Bundle ◽  
Complex Vector ◽  
Kunneth Formula ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

Let G be a compact, connected Lie group such that π2(G) is torsion free. Throughout this paper a vector bundle (representation) will mean a complex vector bundle (representation) and KG will denote the equivariant K-theory functor associabed with the group, G.

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory

On a question of swan in algebraic k-theory

1973 ◽  
Vol 6 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Pramod K. Sharma ◽  
Jan R. Strooker
Keyword(s):  
K Theory

scholarly journals Wilson loop algebras and quantum K-theory for Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hans Jockers ◽  
Peter Mayr ◽  
Urmi Ninad ◽  
Alexander Tabler
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Wilson Line ◽  
Quantum Cohomology ◽  
Three Dimensional ◽  
Loop Algebra ◽  
Loop Algebras ◽  
Verlinde Algebra ◽  
K Theory ◽  
Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

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