# Questions tagged [k-theory]

K-theory is the study of invariants of large matrices, in a suitable sense. It has many variations: (algebraic-k-theory), (topological-k-theory), or in the study of (operator-algebras).

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### Prove that $v=vv^*v$ for a partial isometry $v$.

Let $A$ be a $C^*$-algebra and let $v$ be a partial isometry so that $v^*v$ is a projection. Show that $v=vv^*v$. The hint in the book recommended setting $z=(1-vv^*)v$ and then computing $z^*z$. I ...
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### Path-lifting property.

What is the typical example $\pi \colon X \to B$ of a surjective morphism between topological (or $C^{\infty}$) manifolds, i.e. locally euclidean space via continuous bijections, where path lifting ...
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### Bott Periodictity for Real and Complex Algebras

In Rordam's Introduction to K-Theory for $C^*$ Algebras they prove \begin{equation} K_{n+2}(A) \cong K_{n}(A) \end{equation} using isomorphism in terms of the suspensions of $A$. I have also heard ...
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### Resolution of coherent sheaves on abelian varieties.

If $A$ is a commutative unital ring and $E$ is a finite rank projective $A$-module there is a surjective $A$-linear map $\phi: A^n \rightarrow E$, with kernel $F:=ker(\phi)$ and $F\oplus E \cong A^n$ ...
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### Intuition behind 4-fold periodicity of $L$-theory

The quadratic and the symmetric L-groups are 4-fold periodic. What is the simple argument to obtain the intuition behind the 4-fold periodicity of $L$-theory? (For example, why not have the Bott ...
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### Bott Periodicity of Class Group: Linkage to $K$-theory

I am currently reading Bott's The Stable Homotopy of the Classical Groups (1959), which was his original proof towards Bott periodicity. As a consequence of his proof, the stable homotopy group of ...
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### K theory of the wedge of circles

I am interested in finding the $K_0, K_1$ groups of $C(S^1 \vee S^1)$. We know that $K_0 (C(S^1)) = K_1(C(S^1)) = \mathbb Z$, but is this directly helpful?
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### $K_0$ of an inner automorphism

An exercise in a book I am studying says $K_0(\alpha) =id$ for every inner automorphism $\alpha$. I am not sure why this is true, I suppose it has to do with the fact that inner automorphisms look ...
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### Does every vector bundle have a 'tensor inverse'?

For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial. For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
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### Cyclic homology of $C(S^1)$?

As an exercise for myself, I am trying to figure out the cyclic homology of the algebra of functions over $S^1$, i.e. $C(S^1, \mathbb{C})$. Since the cyclic homology is defined via the universal ...
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### Restriction of $G$-equivariant sheaves to fixed locus

Let $G$ be a group acting on a Noetherian scheme $X$ over $\mathbb{C}$ and $\mathcal{E}$ be a $G$-equivariant coherent sheaf on $X$ where the support of $\mathcal{E}$ contains the fixed locus $X^G$. I ...
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### Equivalence of triangulated categories of perfect complexes

I'm reading a paper by Paul Balmer where he supposes to have a nice (let's say noetherian) scheme $X$, with an open $U\subseteq X$, and proves the existence of an equivalence of triangulated ...
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### Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
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### Example of a space whose complex K-theory is not easily computable from singular cohomology

I am looking for a counterexample to the formula $$K^n(X) \cong \prod_{i\equiv n \mod 2} H^i(X)$$ where $K^*$ denotes complex topological $K$-theory, $H^*$ singular cohomology and $X$ a compact ...
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### On stable isomorphism of separable nuclear C$^*$-algebras and Morita equivalence

I'm a new one to the theory of C$^*$-algebras, and I'm really missing something significant. According to Blackadar,"Operator algebras", page 153 Brown-Green-Rieffel theorem - For $\sigma$-...
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### Help with the setup for Atiyah's proof of Bott Periodicity.

I'm trying to understand Atiyah's proof of Bott Periodicity from his little book on K-Theory - in particular his formulation in terms of $K(P(L \oplus 1))$ where $L$ is a line bundle on a space $X$. ...
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### What is the meaning of $KO^{-1}(S^1)$?

I am interested in the KO-theory of the circle $S^1$. In particular $KO^{-1}(S^1)$. Using the suspension theorem and reduced $K$-theory I can easily show that \begin{equation} KO^{-1}(S^1) \simeq KO^{-...
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### Ideas for calculating $K_0(l_{\infty})$ and $K_1(l_{\infty})$.

Thank you for answering my question. I'm a bit new to K-theory. So I was wondering how can I calculate $K_0(l_{\infty})$ and $K_1(l_{\infty})$. I think if we have one, then by using bott periodicity ...
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### $K_1(A)$ is countable when A is separable C*-algebra

We know that when A is a separable C*-algebra then $K_0(A)$ is countable. How can I show that $K_1(A)$ is also countable?
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### $K_0(C_0(X, A))$ , when X is compact and contractible.

Let A be a $C^{*}$-algebra and $B = C_0(X, A)$ be the set of all continuous functions from a locally compact Hausdorff space $X$ to $A$, vanishing at infinity. Prove that $K_0(B) \cong K_0(A)$ and ...
### Is $St(n,R) \rightarrow St(n+1,R)$ injective?
For every (not necessarily commutative) ring R and every $n\geq 3$, one can define the n-th (unstable) Steinberg group St(n,R) as in https://ncatlab.org/nlab/show/Steinberg+group. Is the canonical map ...