Questions tagged [kt.k-theory-and-homology]

Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras

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19 votes
2 answers
649 views

Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$? Here, by "trivial examples" ...
7 votes
1 answer
87 views

Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
6 votes
0 answers
139 views

$K_0$ of arithmetic surfaces

In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
1 vote
0 answers
88 views

Computing the induced homomorphisms of derived functors using acyclic resolutions

Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
0 votes
0 answers
73 views

Projectivity of equivariant K-theory of toric variety

I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups. In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
5 votes
1 answer
341 views

Model structures on the category of unbounded chain complexes

In his book "Model Categories" Mark Hovey constructs both projective and injective model structures on unbounded chain complexes of $R$-modules. For what kinds of abelian categories does ...
17 votes
1 answer
1k views

Application of higher categories in algebra

Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
2 votes
0 answers
101 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
9 votes
1 answer
679 views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
2 votes
1 answer
87 views

Stabilizing conjugacy classes of integer matrices

$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$ For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$ be the ...
6 votes
1 answer
332 views

Integral group rings on which stably free modules are free

Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...
4 votes
0 answers
93 views

category of vector bundles with connections and its K-theory

For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
13 votes
1 answer
411 views

Does Grayson/Quillen's "pre group completion" have a universal property?

Algebraic $K$-theory of a symmetric monoidal category $C$ is defined in two steps: first take geometric realization, then group complete: $K(C) = \Omega B |C|$. In HAK II, Grayson (following Quillen)...
3 votes
1 answer
273 views

Is there a notion of injective, projective, flat, dimension for a differential graded algebra?

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
3 votes
0 answers
76 views

When does homology preserve inverse limits of Eilenberg-MacLane spaces?

Let $... \to G_3 \to G_2 \to G_1$ be an inverse system of abelian groups and $G$ the limit of the system. By a theorem of Goerss the integral homology of the Eilenberg-MacLane space $K(G,n)$ for $n &...
4 votes
1 answer
323 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 ...
4 votes
1 answer
157 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 ...
1 vote
0 answers
104 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 \...
13 votes
1 answer
336 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 ...
5 votes
0 answers
98 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 ...
9 votes
0 answers
1k 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 ...
5 votes
0 answers
147 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 $G/H\cong S^{2k-1}$ ...
14 votes
1 answer
1k 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 ...
2 votes
2 answers
241 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 ...
2 votes
1 answer
119 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}...
6 votes
0 answers
235 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 votes
0 answers
168 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 ...
6 votes
0 answers
127 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 ...
3 votes
0 answers
81 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 ...
0 votes
1 answer
161 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 ...
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 ...
2 votes
0 answers
94 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 $K$ ...
4 votes
1 answer
240 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 ...
6 votes
1 answer
518 views

Cohomology classes coming from algebraic K-theory

Suppose $X$ is a smooth variety over $\mathbb{C}$. I believe there is a Chern class map $\operatorname{ch}:K_*(X) \otimes \mathbb{Q} \to H_{sing}^*(X(\mathbb{C}),\mathbb{Q})$ for algebraic $K$-theory. ...
3 votes
0 answers
115 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 ...
3 votes
0 answers
160 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 ...
2 votes
3 answers
879 views

Easy way to define positive higher K groups?

I find in my books it is given by Bott periodicity, but this is not direct and Bott periodicity is not easy. Is there an easy and direct way to define $K^n(X)$, like $K^{-n}(X)$? I just start to learn ...
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 ...
13 votes
2 answers
643 views

The category theory of Span-enriched categories / 2-Segal spaces

The category $\mathsf{Span}$ of spans of sets is symmetric monoidal closed under $\times$ (the cartesian product from $\mathsf{Set}$, which is not the categorical product in $\mathsf{Span}$), ...
19 votes
2 answers
3k views

Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((...
2 votes
0 answers
113 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 ...
4 votes
1 answer
163 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 ...
3 votes
0 answers
87 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 ...
3 votes
1 answer
207 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 ...
56 votes
7 answers
5k views

What are surprising examples of Model Categories?

Background Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three ...
7 votes
1 answer
237 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 ...
10 votes
1 answer
300 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 ...
3 votes
0 answers
127 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 ...
5 votes
0 answers
145 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 votes
0 answers
127 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 ...

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