topological cyclic homology
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2020 ◽  
pp. 619-656
Author(s):  
Lars Hesselholt ◽  
Thomas Nikolaus

2019 ◽  
Vol 2019 (755) ◽  
pp. 247-277 ◽  
Author(s):  
Wolfgang Lück ◽  
Holger Reich ◽  
John Rognes ◽  
Marco Varisco

AbstractWe use assembly maps to study \mathbf{TC}(\mathbb{A}[G];p), the topological cyclic homology at a prime p of the group algebra of a discrete group G with coefficients in a connective ring spectrum \mathbb{A}. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective.


2019 ◽  
Vol 15 (2) ◽  
pp. 805-940 ◽  
Author(s):  
Lars Hesselholt ◽  
Peter Scholze

2019 ◽  
Vol 222 (1) ◽  
pp. 215-218
Author(s):  
Thomas Nikolaus ◽  
Peter Scholze

2018 ◽  
Vol 221 (2) ◽  
pp. 203-409 ◽  
Author(s):  
Thomas Nikolaus ◽  
Peter Scholze

2014 ◽  
Vol 15 (3) ◽  
pp. 539-567 ◽  
Author(s):  
Matthew Morrow

In this paper, we prove that cyclic homology, topological cyclic homology, and algebraic $K$-theory satisfy a pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of $K$-theory with compact support.


Author(s):  
Vigleik Angeltveit ◽  
Teena Gerhardt ◽  
Michael A. Hill ◽  
Ayelet Lindenstrauss

AbstractWe consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.


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