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

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

679
questions

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### elementary matrices over a regular ring

Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...

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123 views

### Equivariant Venice Lemma

In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...

**3**

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**0**answers

166 views

### K-theoretic derivation of Bézout theorem

In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...

**1**

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76 views

### $C^*$-algebras appearance in study of Lie groupoids and Differentiable stacks

I am reading Differentiable Stacks, Gerbes, and Twisted K-Theory by Ping Xu.
To talk about (twisted) K-theory of Differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...

**4**

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131 views

### K-theory for a (geometric) stack

There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...

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82 views

### definition of $G$-equivariant abstract elliptic operator [closed]

The following definition is from Baum, Connes and Higson' notes on classifying space for proper actions.
Let $X$ be a $G$-compact, proper $G$-space. A $G$-equivariant abstract elliptic operator on ...

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214 views

### Applications of Jordan-Holder theorem in an abelian category

The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.
This theorem holds ...

**11**

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**1**answer

618 views

### “a sign that one should be computing K-theory”

Allen Knutson said here in comments below the question that
I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it.
I know one ...

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votes

**3**answers

435 views

### The algebro-geometrical version of K-theory

It has been proved that the higher K groups of a (possible noncommutative, but here only comm. for convenience) ring $R$ are correctly defined by Q-construction or + construction.
Recently I'm ...

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**1**answer

197 views

### Status of the extended form of the Lichtenbaum conjecture

The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$.
...

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**2**answers

319 views

### The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $KU$ be the complex $K$-theory spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum.
For $n\in \mathbb{Z}$, it is known what the homology groups $KU_{n}(H\mathbb{Z})$ are?

**11**

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349 views

### The homotopy theory presented by a Waldhausen category

Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once:
...

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**1**answer

253 views

### $K_3(\mathbb{Z})$ and $\pi ^S_3$

This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the ...

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199 views

### Funtoriality of twisted K-theory

I posted this question on math.stackexchange, but received no answer there.
In order to avoid the XY problem I will first state what I want, then what I think is the solution and how that failed ...

**10**

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**1**answer

408 views

### Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?

Does the following diagram commute?
$$
\require{AMScd}
\begin{CD}
BU @>{\psi^k}>> BU \\
@VVV @VVV \\
BO @>{\psi^k}>> BO
\end{CD}
$$
Evidence for: $rc = 2$, it works for $BU(1) \...

**18**

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**0**answers

370 views

### A curious switch in infinite dimensions

Let $V$ be a finite dimensional real vector space. Let $GL(V)$ be the set of invertible linear transformations, and $\Phi(V)$ be the set of all linear transformations. We can also characterize $\Phi(V)...

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81 views

### Failure of devissage vs link topology in algebraic K-theory

This is somehow related to (or maybe a simplified version of) an earlier question (see here) regarding Gersten complexes for singular varieties. The Gersten complexes arise from the coniveau spectral ...

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174 views

### Extending $K$-theory classes

Let $X$ be a $G$-space, for a compact Lie group $G$. If $U\subset X$ is a $G$-invariant open subspace, is it true that the restriction map on equivariant $K$-theory $$K_G(X)\rightarrow K_G(U)$$ is ...

**3**

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**1**answer

212 views

### Description of higher chow groups

In the literature there are several descriptions of motivic cohomology groups, some of them rather explicit, but I don't always understand why they are equivalent. The simplest example I have in mind ...

**6**

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**1**answer

182 views

### Equivalence between categories of coherent sheaf of codimension p

Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \...

**7**

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**1**answer

222 views

### Reference request: mod 2 cohomology of periodic KO theory

The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...

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votes

**1**answer

96 views

### Kuenneth short exact sequence for K-homology

Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?
Using general spectra stuff, one gets a ...

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193 views

### Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?

By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\...

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251 views

### Reference request: complex K-theory as a commutative ring spectrum

Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum?
For real $K$-theory
I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...

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**1**answer

120 views

### Coarse index of Dirac operator on $\mathbb{R}$

Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...

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162 views

### Geometric motivation behind the Fredholm module definition

If $A$ is an involutive algebra over the complex numbers $\mathbb{C}$, then a Fredholm module over $A$ consists of an involutive representation of $A$ on a Hilbert space $H$, together with a self-...

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**1**answer

189 views

### A question about the group $[HZ,KU]$

I don't know if the following question is obvious, but can't figure it out.
I want to ask if it is known what $[HZ,KU]$ is? Here $KU$ is the complex $K$-theory.

**4**

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**1**answer

123 views

### Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules
For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...

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459 views

### Algebraic K-theory of schemes and cohomology

Are there examples of:
two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives;
two smooth projective schemes ...

**10**

votes

**1**answer

282 views

### Where does the $\hat A$ class get its name?

In K-theory we have the Todd class and the $\hat A$ class.
The Todd class is named after the Cambridge geometer John Arthur Todd.
Where does the name $\hat A$ come from? Does the A stand for Atiyah?...

**3**

votes

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42 views

### Minimum rank of inverse complex vector bundles

When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...

**5**

votes

**2**answers

280 views

### Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...

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211 views

### The term “absolute geometry”

My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...

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59 views

### Closable operators on Hilbert modules

For $T:{\frak{Dom}}(T) \to H$ a densely defined operator, with $H$ a (separable) Hilbert space, we know that $T$ is closable if its adjoint $T^*$ has dense domain in $H$.
Does this extend to the (...

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**1**answer

164 views

### Producing $K$-homology cycles from $KK$-cycles

For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :)
I wonder if there us a natural way to "forget" the ...

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546 views

### Maps from mod-$p$ Eilenberg-MacLane spectrum to connective $K$-theory spectrum

Let $ku$ be the connective cover of the complex $K$-theory spectrum $KU$. Consider the mod-$p$ Eilenberg-MacLane spectrum $H\mathbb{Z}/p$.
I want to see that $[H\mathbb{Z}/p,ku]=0$.
Since $H\mathbb{...

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**0**answers

117 views

### When does the canonical $t$-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...

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157 views

### Fundamental group of the Grothendieck ring scheme

Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties with $k$ a field. Let S$_k$ be its affine scheme. Is this a connected scheme for any field ? (I understand that this could be a very naive ...

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99 views

### Tensor product of compact operators on Banach modules

Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...

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**1**answer

180 views

### On definitions and explicit examples of pure-injective modules

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the ...

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**1**answer

224 views

### Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra

In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^*_r(\Gamma)$-module $L^...

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**1**answer

301 views

### Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras

In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.
Roughly, the group $K_0(A)$ is given by the ...

**7**

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**1**answer

413 views

### The connective $k$-theory cohomology of Eilenberg-MacLane spectra

Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
Is it known what $ku^{*}(H\...

**2**

votes

**1**answer

130 views

### A question on the ring structure of topological K-theory and Chern character

Let $X$ and $Y$ be compact spaces (or closed manifolds if you want). I have two questions relating the ring structure of topological $K$-theory. To motivate my questions let me give some background ...

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**1**answer

209 views

### Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...

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76 views

### How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...

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155 views

### Blocksum induces a unital H-space structure on the space of Fredholm operators

Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...

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**1**answer

164 views

### Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?

In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\...

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384 views

### Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...

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**1**answer

251 views

### Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...