# Tagged Questions

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### Could one recover the relative K-theory from the quotient derived category?

Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the ...

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62 views

### K-theory of ringed spaces (including henselian and formal schemes); excision and Mayer-Vietoris

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...

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72 views

### Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...

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**1**answer

151 views

### Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$

Consider the semiring
$$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$
Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)?
...

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**1**answer

186 views

### Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...

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142 views

### (Non trivial) coidempotents(Co-$K$-theory)

I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
...

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120 views

+50

### Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...

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141 views

### K-theory of ultrapowers

It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...

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86 views

### Definition of a cylinder functor in Waldhausen's K-theory

In Waldhausen's Algebraic K-Theory of Spaces, he defines a cylinder functor on a category $\mathcal C$ with cofibrations and weak equivalences (henceforth called a Waldhausen category) as the ...

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304 views

### What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...

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194 views

### Equivalent fomulations of Bott periodicity

Is there an easy way to see the equivalence of the two statements of Bott periodicity.
$$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and
$$K(X)\otimes K(S^2) \cong K(X\times S^2)$$

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89 views

### What is the Atiyah-Bott-Shapiro map for a bundle of *complex* quadratic forms?

In order to ask the question in the title more precisely, let me
recall some standard stuff introduced in [1; Atiyah, Bott, Shapiro].
Suppose $X$ is a compact CW complex and $V \to X$ is an oriented ...

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228 views

### Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...

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**1**answer

73 views

### Integration from vector bundles

Let $(E,M,p)$ be a smooth n dimensional vector bundle. Then $(TE,TM,Dp)$ is a 2n dimensional vector bundle. We restrict this bundle to $M\subset TM$. We denote this restricted bundle by $F$, as a ...

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192 views

### Equivariant algebraic K-theory of affine space

Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...

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61 views

### Relative Gersten resolution for a flat projective morphism

I am reading two papers by Daniel Grayson: "Localization for flat modules in algebraic K-theory" and "Algebraic cycles and algebraic K-theory" and I am wondering if any recent advances in K-theory ...

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**1**answer

309 views

### What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...

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**1**answer

218 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

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**1**answer

448 views

### Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$,
a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...

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133 views

### A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity

In Wegge-Olsenâ€™s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of ...

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511 views

### Applications of Atiyah-Singer using pseudodifferential operators

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...

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109 views

### The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?

Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...

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79 views

### Le Gall's equivariant KK-theory and twisted equivariant KK-theory

I am starting to study Le Gall's equivariant KK-theory and twisted equivariant KK-theory.
Does anyone knows a good reference on this subject?
By the way, I can not access to Journal of K-theory and ...

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84 views

### Shapiro's Lemma for topological K-theory of groups

Chabert, Echterhoff and Oyono-Oyono proved in [Shapiro's Lemma for topological K-theory of groups] that $K^{top}_*(X\rtimes G;A)\cong K^{top}_*(G;A)$ for any $X\rtimes G$-algebra $A$. They claimed ...

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88 views

### Comparison of products in Quillen and Waldhausen K-theory

I'm relatively new to algebraic K-theory and stumbled upon the following question. I would be very glad If someone could provide a reference to an answer or a short argument.
We are given an exact ...

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**1**answer

233 views

### Totally non parallelizable manifold

Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...

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94 views

### Is the $K$-Theory of $\mathbb{P}X$ a $\lambda$-Ring?

If we let $\mathbb{P}X$ denote the free commutative algebra generated by the spectrum $X$, then we have a weak equivalence
$$
\mathbb{P}X\simeq \bigvee_r E\Sigma_{r+}\wedge_{\Sigma_r}X^{\wedge r}.
$$
...

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78 views

### K-Exactness for groups and C*-algebras

We say that a C*-algebra $A$ is K-exact, if for any exact sequence of C*-algebras
$0\rightarrow I\rightarrow B\rightarrow B/I\rightarrow0$, the sequences
$K_i(I\otimes_{min}A)\rightarrow ...

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137 views

### Replacing commutative C*-algebras by simple ones

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$.
I am particularly ...

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81 views

### Lifting triangles in K-theory to KL-groups

Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...

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76 views

### H-flux by any other name

There are more than a few papers referring to H-flux and/or H-twist etc.
Is there anywhere a survey relating these variants?

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121 views

### On closed model categories: standard arguments and fibrantly cogenerated categories

Some not very clever questions on closed model categories.
For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when ...

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127 views

### Yoneda embeddings of stable model categories; composition with Bousfield localizations

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question ...

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157 views

### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...

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296 views

### filtration in K-theory and ordinary cohomology

I am going to ask a question, which could be a stupid one.
I am reading a paper "an index theorem in differential K-theory". The first paragraph of section 8.28 recalls a filtration of K-theory ...

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302 views

### Representation ring and induced representation

Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring ...

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155 views

### Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...

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364 views

### Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras:
$$\begin{array}{ccc}
n & Cl_n & M_n/i^{*}M_{n+1}\\
1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\
2 & \mathbb{H} & ...

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**1**answer

512 views

### What does the moduli stack of G-torsors over the multiplicative group look like?

I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the world of algebraic geometry ...

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140 views

### Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...

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113 views

### Interpretation of the product $K(X)\otimes K^{-1}(X) \to K^{-1}(X)$

We can represent every element of the group $K^{-1}(X)=\tilde{K}(SX)$ by a isomorphism of trivial vector bundles $L:\, X\times \mathbb{C}^k \to X\times \mathbb{C}^k$ because $SX$ is the union of two ...

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158 views

### $K_0$ group of graph underlying an approximately finite (AF) C* algebra

Say we have an AF C* algebra $A$ described by some Bratteli diagram $E$. If $M_\infty (A)=\displaystyle{\lim_\rightarrow M_n(A)}$ and $P(A)$ are the projections in this algebra, we know that ...

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73 views

### Homology of special linear group over local field

I am trying to compute the group
$H_1(Sl_2(\mathbb{Z}_2),M)$, where $\mathbb{Z}_2$ are $2$-adic integers and M is a module $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. I suppose that the group acts on $M$ by ...

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### What happened to online articles published in K-theory (Springer journal)?

As most people probably know, the journal "K-theory" used to be published by Springer, but was discontinued after the editorial board resigned around 2007. The editors (or many of them) started the ...

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161 views

### Surfaces and the boundary map of K-theory

In the reduced K-theory we have the exact sequence of pair
$$ \tilde{K}(S(X/A)) \to \tilde{K}(SX) \to \tilde{K}(SA) \to \tilde{K}(X/A) \to \tilde{K}(X) \to \tilde{K}(A) $$
If choose $X$ to be a ...

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**1**answer

238 views

### Quasi-isomorphisms in exact categories

I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...

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260 views

### When an exact embedding of abelian categories induces a full embedding of their derived categories?

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient ...

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156 views

### the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme

Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does ...

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358 views

### Duality between K-theory and K-homology in the non-spin^c case.

I posted this question on Math.SE (http://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times ...

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**5**answers

612 views

### Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
...