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Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
1
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0answers
52 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrickā€™s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
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0answers
17 views

reference request for finiteness theorem for $K_2$

Is there an algebraic proof of the finiteness theorem for $K_2$ of number fields available? The Garland's proof (1971) heavily relies on transcendental methods.
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0answers
28 views

Finding a Monoid

I want to find the monoid $ colim_n \pi_0 (\mathcal{I} M_n(\mathbb{C})) $ which I know is isomorphic to $ \pi_0 (colim_n \mathcal{I} M_n(\mathbb{C})) $ where $ \mathcal{I}$ is the set of idempotents ...
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0answers
34 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
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3answers
152 views

Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
3
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0answers
64 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
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1answer
36 views

$P \oplus A^m\cong A^n$

Let $A$ be a commutative ring with unit and $P$ an $A$-module, I know that if $A^n\cong A^m\implies n=m$, then this number $n$ is well defined. I would like to prove that if $P\oplus A^m\cong A^n$, ...
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1answer
94 views

Relationship between topological and Quillen's K-theory

Up until now, I've taken it for granted that the topological k-theory of a space $X$ is equal to the K-theory of vector bundles on $X$. $K_0$'s of both coincide (Serre-Swan) however, is it the case ...
6
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1answer
178 views

Symmetric groups and the “field with one element”

I have heard several times that one may regard the symmetric group on $n$ letters as the general linear group in dimension $n$ over the "field with one element". In particular this heuristic would ...
1
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1answer
68 views

Question on the map from algebraic K_0 to topological K_0

Let $X$ be a scheme over a field $F$ where $F$ is either the real or the complex numbers. If $q:Y\to X$ is an algebraic vector bundle over $X$, the $F$-points $q(F):Y(F)\to X(F)$ constitute a ...
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103 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
12
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0answers
785 views

Are there open problems in Linear Algebra?

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There ...
0
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1answer
58 views

Non-finitely generated module isomorphism theorems

I'm searching for nice isomorphism theorems for non-finitely generated $R$-modules. I guess that if $A$ is an $R$-module which is not finitely generated, then there is an isomorphism between ...
2
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1answer
45 views

K-theory - dependence of algebraic structure

I want to figure out the dependence of K-class of an finitely generate projective ring and its algebraic struture. For example consider $K_0(\mathbb{C})\cong \mathbb{Z}$ and ...
2
votes
1answer
162 views

Subgroup of a virtually cyclic group

Let $G$ be a virtually cyclic group, i.e., G has an infinite cyclic subgroup $H$ of finite index. Is it true that if $H'$ is another infinite cyclic subgroup of $G$ then $H'$ must be of finite index ...
5
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2answers
159 views

Induced map on $K_{1}$-group

Let $A$ be a unital $C^{\ast}$-algebra. Any automorphism $\alpha$ of $A$ induces a map on $K_{1}(A)$ by $\alpha_{\ast}[u]=[\alpha(u)]$. Let the automorphism $\alpha$ be inner, does it follow that ...
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1answer
45 views

Rings with finitely many finitely generated free modules, up to isomorphism

If $A = \mathrm{End}(V)$, where $V$ is an infinite-dimensional vector space over some field, then it's not hard to see that $A \cong A^2 \cong \dotsb$. In particular, the map $\mathbb{Z} \to K_0(A)$ ...
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2answers
80 views

Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module? This might seem like an odd question but it matters to me. The background ...
6
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1answer
89 views

Idea behind the factorization of the matrix $\operatorname{diag}(a,a^{-1})$ in algebraic K-Theory

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} ...
5
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0answers
167 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
2
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1answer
178 views

How to compute the formal group law of K-Theory

Could anyone point me to a reference where the formal group law of (topological or motivic) K-theory is computed in as much detail as possible?
5
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0answers
88 views

Milnor $K_2$ of an inverse limit is inverse limit of Milnor $K_2$'s?

Let $\{A_n\}$ be an inverse system of rings and $\hat{A}$ be the inverse limit of this system. Let $K_2(R)$ denote Milnor $K_2$ (I will assume that the case I am interested in the Milnor $K_2$ is ...
1
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1answer
133 views

Basic question on $K$-theory

Let $h\colon A\to A'$ be a ring homomorphism between $A,A'$ which are commutative rings with $1$. Let $P,Q$ be $A$-modules. Then, are there any gaps in my following argument? ...
2
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1answer
138 views

Question on K-theory

If $R$ is a ring with 1 which satisfies $R^r=R^s$ for some $r\neq s$. Are there any explicit calculation of $K_0(R)$ for such $R$? I want to know such examples because I think that such $R$ may not ...
3
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4answers
323 views

Chromatic Filtration of Burnside Ring

I just attended a seminar on the chromatic filtration of the Burnside ring. I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration ...
12
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3answers
507 views

What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics?

Quote from Don Zagier (Mathematicians: An Outer View of the Inner World): " I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, ...
13
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1answer
726 views

Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras ...
10
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1answer
356 views

How to show directly that two elements become equal in Grothendieck group?

Consider commutative semigroup S and its Grothendieck completion group G(S).Suppose I insist on defining G(S) as free abelian group on basis $[a]$ (with $a\in S$) divided out by the relations ...