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

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

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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 ...
onefishtwofish's user avatar
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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 ...
fool rabbit's user avatar
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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 ...
Ingrid's user avatar
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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 ...
Song Ye's user avatar
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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 ...
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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 &...
willie's user avatar
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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 ...
atinag's user avatar
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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 \...
Tim Campion's user avatar
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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 ...
Neil Strickland's user avatar
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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 ...
Sanae Kochiya's user avatar
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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 $G/H\cong S^{2k-1}$ ...
Yun Liu's user avatar
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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 ...
davik's user avatar
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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 ...
Tim Campion's user avatar
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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 ...
Plius's user avatar
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$K_0$ 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}...
Sanae Kochiya's user avatar
6 votes
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234 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}),\...
Jared Weinstein's user avatar
4 votes
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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 ...
Antoine Labelle's user avatar
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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 ...
AlexE's user avatar
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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 ...
MLV's user avatar
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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 ...
Yellow Pig's user avatar
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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 ...
Rosencrantz's user avatar
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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 $K$ ...
OOOOOO's user avatar
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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 ...
Stabilo's user avatar
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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 ...
Z. M's user avatar
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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 ...
Mikhail Bondarko's user avatar
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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 ...
rrrrrttttttt's user avatar
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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 ...
Mehmet Onat's user avatar
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4 votes
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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 ...
user145752's user avatar
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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 ...
Motmot's user avatar
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"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 ...
geometricK's user avatar
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7 votes
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$K_1$ 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 ...
curious math guy's user avatar
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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 ...
Theo Johnson-Freyd's user avatar
10 votes
1 answer
299 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 ...
Theo Johnson-Freyd's user avatar
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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}^\...
Branimir Ćaćić's user avatar
2 votes
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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 ...
Denis T's user avatar
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9 votes
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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 ...
Alexandar Ruño's user avatar
5 votes
1 answer
396 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 ...
Oscar Randal-Williams's user avatar
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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 ...
Sanae Kochiya's user avatar
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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 $A$ ...
Li Guanyu's user avatar
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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 ...
Grisha Taroyan's user avatar
2 votes
0 answers
191 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]=...
Ali Taghavi's user avatar
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(...
Sanae Kochiya's user avatar
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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 ...
Ali Taghavi's user avatar
4 votes
0 answers
157 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 ...
K.M.'s user avatar
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8 votes
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130 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 ...
tox123's user avatar
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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}...
K.M.'s user avatar
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1 vote
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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)...
Жека's user avatar
6 votes
0 answers
102 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}...
Mezzovilla's user avatar
4 votes
0 answers
406 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. ...
user267839's user avatar
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0 votes
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174 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 $M$,...
Fredy's user avatar
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