Algebraic K-theory is a tool from homological algebra that defines a sequence of functors from rings to abelian groups. It has many applications in algebraic geometry. See also (topological-k-theory).

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Quillen's K-theory and ring homomorphisms

I am a beginner in algebraic K-theory and I want to make sure that I understand the following correctly: Let $f:A \to A'$ be an isomorphism of commutative rings. Denote by $\mathcal{P}(A)$ (resp. ...
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The spectral transfinite open spaces with quintic characteristics of second kind

Context: Beginning with the formal definition of transfinite spaces together with the Picker-Hansel theorem, we obviously get a relation $$ \bigcap\xi_{|\sigma|\mapsto \theta^*} \oplus_\psi ...
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What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
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When does a smooth projective variety X have a free Grothendieck group

Let $X$ be a smooth projective variety (e.g. Grassmannians). Since $X$ is smooth, the groups $G_0(X):=K_0(CohX)$ and $K_0(X):=K_0(VectX)$, the Grothendieck groups of coherent sheaves of modules on $X$ ...
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A question on algebraic K-theory- show that $K_1(R)\cong R^\times$ if $R$ is a field.

If $R$ is a ring we write $K_1(R)$ for the abelian group $K_1({\rm category\; of\; finitely\; generated\; projective\;R-modules})$. Swan's 'Algebraic K-Theory' defines $K_1$ in terms of the usual ...
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Algebraic results using lower K-theory as a blackbox

There is an algebraic K-theory seminar at my school and we are struggling to find applications from areas other than topology. We'd like a nice statement like "If X then Y" whose proof makes ...
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Motive of Pfisterforms spectral sequence

In this famous paper http://www.math.uni-bielefeld.de/~rost/data/motive.pdf Rost constructs the motive of a Pfister-Form/Pfister-Quadric. In the last proof on page 13 he writes: "By a spectral ...
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A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’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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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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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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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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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, ...
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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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$P \oplus A^m\cong A^n$

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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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 ...
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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 ...
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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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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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Examples of Eilenberg-type Swindles

I am compiling a list of 'swindles' in the style of the Eilenberg-Mazur swindle. I've already got some swindles in K-theory, the Mazur Swindle and the proof of the Cantor–Bernstein–Schroeder theorem. ...
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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 ...
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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} ...
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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 ...
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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?
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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 ...
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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? ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...