Questions tagged [kt.k-theory-and-homology]
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
914
questions
4
votes
1
answer
122
views
The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$
$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
- 43
1
vote
0
answers
96
views
Lengths and additive invariants which preserve positivity
The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
- 58.7k
12
votes
1
answer
299
views
Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?
There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
- 54.6k
5
votes
0
answers
83
views
Realize a $K_0$-group homomorphism by a unital $\ast$-homomorphism
This question is inspired by Exercise $7.7$ in *An Introduction to $K$-theory for $C^*$-algebras (available here). Given a unital AF-algebra $A$ and another unital $C^*$-algebra $B$ that has ...
- 219
5
votes
0
answers
140
views
Equivalent descriptions of equivariant K-theory
I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
- 51
9
votes
0
answers
880
views
Some questions about Clausen's third IHES lecture on Efimov K-theory
I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
- 2,015
14
votes
1
answer
932
views
What is decategorification?
A decategorification is, roughly, some procedure $\Phi$ which inputs some sort of $n$-categorical data and outputs some sort of $(n-1)$-categorical data. Whatever this means, categorification is ...
- 58.7k
2
votes
2
answers
228
views
The complex $K$-theory of the Thom spectrum $MU$
The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
- 21
2
votes
1
answer
108
views
$K_0$ group of an infinite factor
The following question was already posted in this link but I could not understand hints given in this post.
Let $\mathcal{M}$ be an infinite factor and my question is how to prove that $K_0(\mathcal{M}...
- 219
6
votes
0
answers
226
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 4,447
4
votes
0
answers
144
views
K-theory of toric varieties
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. Is there a good presentation for the K-theory ring $K_0(X)$ in terms of the corresponding fan, analogous to the presentation of the Chow ...
- 2,775
6
votes
0
answers
119
views
Maps in the Künneth theorem for K-theory of C*-algebras
The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
- 2,896
3
votes
0
answers
78
views
Explicit computation of the transfer in the representation ring for unitary groups
For a compact Lie group $G$ we let $R(G)$ be the ring of finite dimensional complex $G$-representations studied by Segal in http://www.numdam.org/item/PMIHES_1968__34__113_0.pdf.
This comes with extra ...
- 73
0
votes
1
answer
137
views
Equivariant sheaves on $\mathbb P^1$
Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
- 2,471
3
votes
0
answers
78
views
Cohomological counterpart of the K-theory of the Roe C*-algebra in non-periodic systems
In crystalline insulating quantum systems the dynamics of electrons is governed by a Schrödinger operator which is periodic with respect to a Bravais lattice $\Gamma\cong \mathbb{Z}^d$ and whose ...
- 31
2
votes
0
answers
91
views
etale cohomology and algebric K theory for algebraic stack
Let $X$ be a smooth variety over a perfect field $k$. Fix a prime $p$ which is invertible in $k$.
Thomason proved that there is Atiyah-Hirzebruch type spectral sequence that computes $K(1)$-local $K$ ...
- 357
4
votes
1
answer
234
views
Can one bypass the geometric realization in the definition of algebraic $K$-theory?
I believe there is no good notion of homotopy groups for an arbitrary simplicial set $S$. However, when $S$ is fibrant - meaning that $S\to *$ is a fibration - there is a definition. The singular ...
- 1,419
3
votes
0
answers
109
views
Cyclic K-theory as cyclic nerve in a letter of Goodwillie
Kaledin mentioned in https://arxiv.org/abs/2004.04279 Remark 11.5 that, in a letter to Waldhausen by Goodwillie in 1988, Goodwillie showed that the cyclic K-theory can be computed by the geometric ...
- 1,603
3
votes
0
answers
146
views
Which "tensor" endofunctors on triangulated categories are essentially exact?
Assume that $T$ is a symmetric monoidal triangulated category, and $X$ is an object in it. Then the functors $X\otimes -$ and $-\otimes X: T\to T$ are not necessarily exact since they send ...
- 16.3k
3
votes
0
answers
86
views
Is there a reasonable K-grroup of Behrend’s absolutely convergent complexes?
Let $\mathfrak X$ be an algebraic stack over $\mathbb F_q$ and let $D_{\mathrm{abs}}(\mathfrak X)$ be the derived category of complexes of $\overline{\mathbb Q}_\ell$-sheaves which are absolutely ...
- 137
2
votes
0
answers
105
views
About Atiyah-Segal Localization Theorem
In $K$-Theory, actually also in equivariant cohomology theory, there exists a useful theorem as known Borel-Hsiang-Atiyah-Segal Localization theorem. For $K$-Theory
Theorem: Let $G$ be a compact Lie ...
- 997
4
votes
1
answer
161
views
Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?
Let $A$ be an $\mathbb{E}_\infty$-ring spectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras.
We assume there is a homotopy fibre sequence
$$
R_1\to R_2 \to R_3
$$
in the stable ...
- 489
3
votes
0
answers
85
views
What does homotopy invariance mean for twisted K-theory?
In ordinary K-theory, homotopy invariance means that if $f,g \colon X \to Y$ are homotopic maps then their induced maps on K-theory are equal: $f^* = g^* \colon K(Y) \to K(X)$.
My question is how to ...
- 293
3
votes
1
answer
204
views
"High-dimensional" classes in topological $K$-theory
I am looking for a sequence of topological spaces $X_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X_n)$ be the complex reduced $K$-theory group of $X_n$ (with respect to some ...
- 1,851
7
votes
1
answer
223
views
$K_1$ of Categories for intuition
Maybe there is no good answer to this, but I'm trying to get a feel for what the $K$-theory of a (permutative or symmetric monoidal $\infty$-)category computes.
In algebraic $K$-theory, we have ...
- 4,136
3
votes
0
answers
123
views
Which spectra have a homotopy-universal connective quotient?
Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...
- 52.2k
10
votes
1
answer
289
views
Which spectra have a universal connective quotient?
Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...
- 52.2k
5
votes
0
answers
140
views
Questions about the $K$-theory of the algebraic standard Podleś sphere
Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
- 2,135
2
votes
0
answers
122
views
Algebra of finite width matrices
$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
- 4,254
0
votes
0
answers
72
views
Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$
The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...)
By ...
- 219
9
votes
0
answers
253
views
Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
- 191
5
votes
1
answer
373
views
The graded multiplication on topological $K$-theory
In every reference I have looked at (the books by Atiyah, Karoubi, Lawson--Michelsohn, Hatcher's unpublished book) the exterior multiplication on (reduced, negative) $K$-theory is given by the ...
- 17.5k
0
votes
0
answers
68
views
Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group
Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
- 219
1
vote
0
answers
71
views
Hochschild homology computation of certain type
I know that in general Hochschild homology is not very computable. However, this certain type of Hoshschild homology shows up and I feel like there could be existing result.
Let $k$ be a field and $A$ ...
- 369
2
votes
1
answer
102
views
K-ring of splittable bundles
Is there something non-trivial to be said about the subring of $K(X)$ spanned by one-dimensional bundles?
If I am not mistaken, it is still a functor into commutative rings from the category of ...
- 1,182
2
votes
0
answers
184
views
The trigonometric $C^*$-algebra
The trigonometric $C^*$-algebra is the universal $C^*$-algebra generated by $\mathcal{G}=\{x,y,z\}$ subject to relations \begin{align}x^2=x=x^*, &\quad y^2=y=y^*\\ [x, z]=y, &\quad [y,z]=...
- 270
2
votes
0
answers
68
views
The $K_0$ mapping of an automorphism induced by a derivation
Let $\mathfrak{A}$ be a unital $C^*$-Algebra and let $\delta: \mathfrak{A} \rightarrow \mathfrak{A}$ be a linear map that is not constantly zero and satisfies, for every $A, B\in\mathfrak{A}$, $\delta(...
- 219
0
votes
0
answers
51
views
A smooth algebra over a $K_i$-regular ring is $K_i$-regular?
Let $A$ be a smooth algebra over a commutative ring $R$. If we assume that $R$ is $K_i$-regular for any $i\geq 0$, then $A$ is also $K_i$-regular?
- 91
2
votes
0
answers
88
views
The group of quasi unitary elements of a (simple) Banach algebra
For a Banach algebra $A$ with invertible group $G(A)$ we define the following group:
$$QG(A)=\{u\in G(A)\mid \;\text{the mapping}\; a\mapsto u^{-1} a u \;\text{is an isometry}\}$$
What is an ...
- 270
4
votes
0
answers
142
views
base change property of Topological Hochschild homology
What is the "base change property" of topological Hochschild homology?
In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...
- 91
8
votes
0
answers
128
views
Behavior of $K_0$ towers
In the spirit of the automorphism tower problem, I've been thinking about "$K_0$ towers." Since $K_0$ may be imbued with a ring structure by the tensor product, it makes sense to ask what ...
- 410
1
vote
0
answers
85
views
bott element in periodic cyclic homology
I am reading a paper by Thomason "Algebraic K-theory and étale cohomology", which deals with algebraic K theory localized by inverting the bott element $\beta \in K_2(\overline{k})^{\wedge}...
- 91
1
vote
0
answers
116
views
K-theory of l-adic sheaves of a curve
I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...
- 11
6
votes
0
answers
92
views
$C(X)$-Fredholm operators and Atiyah-Jänich theorem
Let $X$ be a compact Hausdorff topological space and consider the Hilbert space $\ell^2(\mathbb N)$. As shown here, any $T\in C(X,\ell^2(\mathbb N))$ induces a $C(X)$-Fredholm operator
$$
\begin{array}...
- 61
4
votes
0
answers
400
views
In which "sense" unramified Milnor-Witt K-groups are unramified
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
- 5,668
0
votes
0
answers
170
views
Stable homotopy group of K(1)-local spectra
Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$,...
- 127
2
votes
0
answers
101
views
Did anybody study split homotopy cartesian squares in triangulated categories?
Let us call a commutative square
$$ \require{AMScd}
\begin{CD}
A @>{g'}>> B \\
@V{f'}VV @VV{f}V \\
C @>>{g}> D
\end{CD}
$$
in a triangulated category split homotopy ...
- 16.3k
4
votes
1
answer
302
views
Equivariant K-theory for products of groups?
Let $X$ be a $(G \times H)$-space. What is known about the connection between the groups $K_G(X)$, $K_H(X)$ and $K_{G \times H}(X)$? The $G$ and $H$ action on $X$ come from the canonical inclusions $G ...
- 293
1
vote
0
answers
136
views
$K_1(k[x]/(x^2))$ for a field $k$
$\DeclareMathOperator\GL{GL}$The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $\GL(R)$, where $\GL(R)$ is the union of the sequence $R^{ \times} = \GL_1(R)...
- 167
2
votes
1
answer
286
views
Grothendieck group of triangulated categories
Let $A$ be a full triangulated subcategory of $B$, $u:A\rightarrow B$ the corresponding embedding. Let $f:B\rightarrow A$ be a triangulated functor
satisfying:
$f\circ u = id$
Let $b \in B $, if $f(b)...
- 169