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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What is the motivation of studying $P[A]$ in operator K-theory?

I am reading the last chapter of Murphy's $C^*$-algebras and operator theory. He defines $$P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$$ and construct the Grothendieck group ...
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$K_0$ group of Direct Sum of C*-Algebras

We know that for C*-algebras, $$ K_0(A\oplus B) \stackrel{(*)}{=} K_0(A)\times K_0(B)$$where the $\oplus$ on the L.H.S. is a direct sum of C*-algebras (derived from the notion of direct sum of ...
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The necessity of defining the stable equivalence in the construction of the Grothendieck group $K_0$

I am confused about the process of the construction of the Grothendieck group $K_0$ in Murphy's $C^*$-algebras and operator theory section 7.1. Let $A$ be a $*$-algebra and ...
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43 views

How to construct the map relating topological and algebraic $K$ - theory ?.

Let $X$ be a complex projective variety, and let $ X^{an} $ denote the topological space of complex points of $ X $ equipped with the analytic topology. Then, any algebraic vector bundle $ E \to X $ ...
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Examples of Waldhausen categories.

Waldhausen's wS construction of K-theory assigns K-groups to an arbitrary small Waldhausen category, my main goal in reading this construction was to apply it to the case of exact categories with weak ...
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classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
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20 views

Reference request: K-theory of finite fields by Q construction

I have read Quillen's Q construction and Waldhausen's S construction of K-groups of an exact category. Most references for K theory, calculate K groups for finite fields by the + construction and show ...
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Multiplicative structure on algebraic K-theory

Let $R$ be a commutative ring. Using Quillen's $+$-construction, it is relatively easy to see that the algebraic K-theory of $R$, $K_*(R)$, admits a graded commutative product $$K_i(R)\otimes K_j(R) ...
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Algebraic K-theory of the cotangent bundle

Below, always let $A$ be the coordinate ring of a smooth affine variety over $\mathbb C$. What can be said about the (non)-triviality of the module of Kahler differentials $\Omega_{A/\mathbb C}^1$? ...
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58 views

What are the pre-requisites required to understand Milnor's book on algebraic K- theory?

I want to understand Steinitz’ theorem on the structure of finitely generated modules over Dedekind domains. I also want to have some general awareness regarding what Algebraic K-theory is about. ...
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understanding a statement in Weibel's “The K-book” about bisimplicial sets

Here is a theorem from Weibel's The K-book, Chapter IV Theorem 3.6.1. Let f : X → Y be a map of bisimplicial sets. (i) If each simplicial map $X_{p,∗} → Y_{p,∗}$ is a homotopy equivalence, so is ...
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Units in Semiperfect Skew Group Rings

Let $k$ be a field and $S$ the ring $k[[x_1,\ldots, x_n]]$. Let $G$ be a finite subgroup of $GL_n(k)$ that does not contain any nontrvial pseudo-reflections and such that $|G|$ is invertible in $k$. ...
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135 views

Proof of Rim's Theorem (Milnor)

I'm currently learning Algebraic K Theory from Milnor's "An Introduction to Algebraic K Theory" and am having trouble understanding his proof of Rim's theorem using a "Mayer-Vietoris" exact sequence. ...
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What is the purpose of K-Theory?

I have recognized that there is a theory called K-Theory in mathematics is used also for applications in mathematical physics. There is existing algebraic K-Theory and topological K-Theory. Are ...
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Pullback along the Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then $F^*(\mathcal{L}) \cong ...
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29 views

Different definition of $K_0(R)$. Prove equivalence

I need to prove that the following two definitions of the zero-th $K$-theory group of a ring $R$ (with unit) are equivalent. def 1 the abelian group generated by the isomorphism classes $[P]$ of ...
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Eilenberg-MacLane rings in algebraic K theory?

Given any abelian group $G$ and a natural number $n$, is there always a ring (unital) whose algebraic K-groups are $K_i(R)=0, i\neq n$ and $K_n(R)=G$?
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The computation of $K_2(\mathbb Q)$

I have a small question on Tate's proof of the structure of the group $K_2\mathbb Q$, as found in e.g., Milnor's book "Introduction to Algebraic $K$-theory". The proof goes by showing that the map ...
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Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function ...
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K-theory of projective space

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for ...
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$K_0$ of a ring via idempotents

As it is described here, the group $K_0(R)$ of a unital ring can also be described in termes of conjucation classes of idempotents. In this text it is shown that the semigroup of isomorphism classes ...
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$Z_2$ Equivariant K-theory of $S^1$

I am interested in the $\mathbf{Z}_2$ equivariant K-theory of $S^1$, but I cannot find any good references or methods to calculate it with the action I have in mind. The action on $S^1$ is an ...
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How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here ...
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Show that $R(G)\cong K_0(F[G])$

Let $F$ be a field of characteristic $0$, $G$ a finite group and let $R(G)$ be the additive group of functions $G\to F$ generated by characters of $G$ of degree $1$. Question: How can we show ...
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The +-construction on a homology n-sphere

I am working on Weibel's K-Book and when defining higher K-Theory for a Ring via $BGL(R)^+$, I have encountered a question concerning a homology n-sphere. The statement I want to show is the ...
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71 views

What is the Grothendieck group of finitely generated $R[G]$-modules?

Let $R$ be a ring with unity, $G$ a finite group and $R[G]$ the group ring. What is the definition of the Grothendieck group of finitely generated $R[G]$-modules? How is this connected to the ...
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67 views

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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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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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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45 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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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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321 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 ...
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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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255 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 ...
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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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271 views

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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104 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 ...