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Questions tagged [algebraic-k-theory]

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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K theory of integers [on hold]

How can i calculate $K_2$ of $\mathbb{Z} $ i.e. $K_2 (\mathbb{Z})$
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What does it mean to say $K$-theory satisfy Mayer–Vietoris sequence?

I saw the statement "a $K$-theory is expected to satisfy Mayer–Vietoris property and Bott Periodicity" somewhere and I am trying to understand what it means. What does it mean to say a $K$-theory ...
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Kernel of the Adams operation on the algebraic K-theory

Adams operation $\psi^n$ acts on the algebraic $K$-theory groups of a commutative ring $R$. Its known that eigenvalues of $\psi^n$ on $\mathbb{Q}\otimes K_i(R)$ can only be non-negative powers of $n$. ...
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Computing $K$-theory elements in a $C^*$ algebra $A$

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. Now consider the matrices, the projections, $$ \left[ \begin{pmatrix} 1-p &...
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$K$ theory of $C^*$ algebra is different to algebraic $K$ Theory?

Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory? We assume $A$ is unital (I am not sure if this matters), i.e. what is an example ...
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Certain exact functor on the Grothendieck group of a module category

Let $A$ be a be a finite dimensional, associative, and unital $\mathbb{C}$-algebra. Let $\mathcal{A}$ be the category of finitely generated $A$-modules. Since $A$ is an Artinian ring, there are only ...
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K Theory of $C^*$ algebras I, Higson's notes

Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the ...
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On rings whose $K_0$ has nice properties

Let $R$ be a commutative, reduced ring. It can be seen that $K_0(R) \cong \mathbb Z$ as groups if and only if every finitely generated projective module is stably free. My question is, are there ...
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What are the inverses in the Grothendieck group of a module category?

Let $R$ be a unital commutative ring and $\mathcal{C}$ be the category of finitely generated $R$-modules. The Grothendieck group $K_{0}(\mathcal{C})$ is the free abelian group generated by the ...
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$K$ theory of vector bundle, $K(V)$, is a $K(X)$ module

This is on page 67, definition 5.6 when the author defines the Thom homomorphism: $V$ is a hermitian vector bundle over a compact space. $K(V)$ is a $K(X)$ module. How does $K(X)$ act on $K(V)$? ...
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Short exact sequence in $K_0$ of non unital rings.

Where may I find some reference for explaining the basics of $K_0$ for non unital rings? I was reading this pdf by Max Karoubi. He stated two results. Let $A$ be any $k$-algebra, where $k$ is a ...
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$K_1(\mathbb{Z})$ is covered by diagonal matrices with only $\pm 1$ entries

Let me give some definitions first: We define the inclusion $GL(n,\mathbb{Z})\to GL(n+1,\mathbb{Z})$ as $A\mapsto \begin{pmatrix}A&0\\0&1\end{pmatrix}$ We call $E(n,\mathbb{Z})$ the subgroup ...
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The transfer map from $K_2L$ to $K_2F$ for a galois extension $L/F$

In J. Tate's "Relations between $K_2$ and Galois Cohomology" Lemma 3.4 J. Tate talks about the transfer map in k-theory, and references the Milnor's "Introduction to Algebraic K-Theory", but in the ...
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Torsion Grothendieck group

What is an example of an abelian/exact/triangulated category that has Grothendieck group isomorphic to $\mathbb{Z}/k\mathbb{Z}$ for some $k$? The category of finitely generated abelian groups has ...
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Why we call monomial matrices?

Currently, I am reading Milnor’s book on algebraic K-theory, where he defined a monomial matrix over commutative ring with 1 to be a matrix of the form PD, where P is a permutation matrix and D is a ...
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Relation/difference between proof of constant rank of f.g. projective module over commutative domain

Let $M$ be a f.g projective module over a domain $R$. I am wondering how I can see openness and closedness via $K_0$ group proof. Since $M$ is f.g. projective, it is clear localization at $p\in Spec(...
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Rings and categories with zero Grothendieck group

I am interested in examples of rings (or triangulated categories) that have zero Grothendieck group but are somehow still interesting. More example, for what rings $R$ is the category of finitely-...
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1answer
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Meaning of K in Algebraic K theory

I am reading algebraic K-theory but I have doubt not in the subject but in the name. I want to ask what K stands in the word Algebraic K-theory as well as in Topological K-theory.
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Clarifying how we can identify $SK_1(R)$ with set of path components

Let $R$ be a commutative Banach algebra. Let $E_n$ denote the group generated by $n\times n$ elementary matrices. $SK_1(R)$ denotes the kernel of the induced determinant map $K_1(R) \to R^{\times}$. ...
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When $\mathbb Z$ is a direct summand of $K_0(R)$

Suppose that there is a ring homomorphism $R \to F$ where $F$ is a field. I'm trying to verify that $\mathbb Z$ is a direct summand of $K_0(R)$. We have an induced ring homomorphism $(K_0(R), \...
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Understanding a proof from algebraic k-theory

I've been reading Milnor's notes on algebraic K-theory, and have trouble understanding the last step of the proof of Lemma 3.2. Here is the set up. Let $A$ be a ring, and $P$ a finitely generated ...
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Nerve of a simplicial category

Here a simplicial category is a simplicial object in $\textbf{Cat}$ (that is, a functor from $\Delta^{op}$ to $\textbf{Cat}$). I wonder why the nerve of a simplicial category is a simplicial set? For ...
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Definition of algebraic K-theory space

Let $(C,wC)$ be a Waldhausen category. The algebraic K-theory space is the loop space of the classifying space of the simplicial pointed category $wS_*C$, i.e. of the topological realization of the ...
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1answer
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Degreewise cofibration in $S_nC$

Given a category $C$, we have $S_nC=Fun(Ar[n],C)$ and given $A,B\in Ob(S_n(C))$, (i.e. A,B are functors from $Ar[n]$ to $C$), then what does a morphism $f:A\to B$ in $S_nC$ mean by degreewise ...
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Motivation of $G_0$ group

I want to know the motivation why we want to modulo the subgroup related to the exact sequences.
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Are these two definitions stably isomorphic modules / $K_0(A)$ equivalent?

Let $A$ be a ring with $1$. I have encountered two definitions of stably isomorphic (left) $A$-modules. Def. 1. Two finitely generated $A$-modules $M,N$ are stably isomorphic if there exists $r,s\geq ...
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trying to understand the connecting homomorphism between K theory groups

the connecting homomorphism from $K_{1}(A/J)$ to $K_{0}(J)$ is defined by the composition $(j_{*})^{-1} k_{*}$ where $j$ is the inclusion of $J$ to the mapping cone $C_{\pi}$, which induce isomorphism ...
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1answer
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Algebraic $K_2$ as “universal receptacle”?

In algebraic $K$-theory, $K_0$ and $K_1$ have nice descriptions in terms of the category of finitely generated projectives. $K_0$ is motivated as the "universal receptacle" for (additive) invariants ...
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An immediate result of fundamental theorem of algebraic $K$-theory.

The fundamental theorem of algebraic $K$-theory says that $K_1(R[t,t^{-1}])\cong K_1(R)\oplus K_0(R)\oplus NK_1(R)\oplus NK_1(R)$ On the page 153 of Rosenberg's book on algebraic $K$-theory, he said ...
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Rational points on conics over fields of dimension 1

Let $K$ be a field of cohomological dimension 1 and $C$ be a smooth projective conic over $K$. Is it true that $C$ always has a $K$-point?
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Compute Thom and Euler class

If $\gamma \colon S^1 \rightarrow SO(2)$, we define $E_{\gamma}= D_{a}^2 \times \mathbb{R}^2 \sqcup D_{b}^2 \times \mathbb{R}^2 / \sim$, where $(x,v) \sim (x,\gamma(x)\cdot v)$ for $(x,v)\in S_{a}^1 \...
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Why is $BGL(R)^+$ an $H$-space?

Of course, an easy answer is to use the fact that $BGL(R)^+$ is the identity component of $K(R)$ and construct the $H$-space structure using a more convenient model for $K(R)$, such as the group ...
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1answer
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Atiyah–Jänich for $K_1$

Atiyah–Jänich's theorem says that $$ \left[X\to\mathcal{F}\left(\mathcal{H}\right)\right] = K_0\left(X\right) $$where $\mathcal{H}$ is any separable complex Hilbert space, $\mathcal{F}\left(\mathcal{H}...
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Koszul complexes of twisting sheaves

I am considering the following problem: For the projective space $X=\mathbb P^n$ is there an exact sequence of the form $0\rightarrow \binom{n+1}{n+1} \mathcal O(-n-1)\rightarrow \binom{n+1}{n}\...
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Grothendieck group of $\mathbb P^n$

I am trying to prove that the Grothendieck group $K_0(\mathbb P^n)$ is generated by $\{\mathcal O, \mathcal O(1),...,\mathcal O(n)\}$. I already showed that this set generates a full sublattice of $...
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1answer
153 views

Establishing a right-exact sequence in K-theory

I am confused about problem II.6.10 b) in Hartshorne's Algebraic Geometry. The question goes: For a scheme $X$ let $K(X)$ be the Grothendieck group of coherent sheaves on $X$, i.e. the free abelian ...
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1answer
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How does group cohomology relate to algebraic k-theory?

In the wikapedia article on group cohomology (https://en.wikipedia.org/wiki/Group_cohomology#Algebraic_K-theory_and_homology_of_linear_groups) , there is a short section on how it relates to group ...
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What are the sheaves in the $G$-equivariant resolutions seen as vector bundles?

$\newcommand{\F}{\mathcal{F}}$ Let $\mathcal{F}$ be a $G$-equivariant sheaf over a smooth projective $G$-variety $X$, generated by global sections $s_1,...s_n$. One can construct the first locally ...
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1answer
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If $P$ is a fg projective $\mathbb{Z}[G]$-module, what is $P\otimes\mathbb{R}$?

Let $G$ be a finite group, and $P$ a fg projective $\mathbb{Z}[G]$-module. By a theorem of Swan, $P\otimes\mathbb{Q}\cong\mathbb{Q}[G]^n$ for some $n$. Is the same true for $\mathbb{R}$? Clearly, $(P\...
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2answers
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Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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1answer
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Showing an isomorphism of $R[G]$-modules using the regular representation

Let $R$ be a commutative ring, and $G$ a finite group. Now suppose $M,N$ are finitely generated $R[G]$-modules such that $M\cong_R N$ (let's say they have $R$-rank$=n$). To show $M\cong_{R[G]}N$, is ...
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Example of Waldhausen category with quotient maps not closed under composition

Let $\mathscr{C}$ be a Waldhausen category, i.e. a category pointed by $0$, with cofibrations and weak equivalences. Recall a map $B \to B/A$ in $\mathscr{C}$ is called a quotient map if it is the ...
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1answer
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Some properties of $K_i(\mathcal C)$

I'm learning the abstract definition of the $i$-th $K$-group $K_i(\mathcal C)$ for an exact category $\mathcal C$. The definition is quite complicate, basically you start from an exact category $\...
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1answer
51 views

$M$ and $N$ equal in $K_0$ $\Rightarrow$ $\exists P$ such that $M\oplus P\cong N\oplus P$

Let $\mathcal P(A)$ be the category of finitely generated projective $A$-modules ($A$ is a ring with unity). Then consider the free group $F$ over the isomorphism classes of $\mathcal P(A)$. I will ...
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1answer
105 views

Some basic facts about algebraic $K$-theory

I'm definitely not an expert of $K$-theory but I need to know a couple of results in order to complete a computation. I tried o find these things on the standard books but without success. I'm sorry ...
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1answer
78 views

Correspondence between the Algebraic $K_1$ and the topological $K_1$

By Serre Swan theorem we have a nice correspondence between Topological $K_0$ and Algebraic $K_0$ when we consider the ring to be continuous functions on a topological space. I am wondering if there ...
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K-theory of the Grassmannians

For complex projective space its K-theory is $\mathbb{Z}[H]/\langle H-1\rangle^{n-1}$. How does this generalise to the Grassmannian case?
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1answer
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Writing out projection classes on $C(S^2)$

I am very new to K-theory, either topological or algebraic. Please bear with me if this sounds naively stupid. After reading a bit of Wegge-Olsen, I have got the idea of how to calculate $K_0(C(S^2))$...
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2answers
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Rings with isomorphic center not necessarily Morita equivalent

a friend asked me this question; after a bit of searching, I'm still unable to answer it. Question Are there two rings with isomorphic centers which are not Morita equivalent? Our first method ...
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1answer
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Ring theoretic properties of the rings $K_0$.

Let $A$ be a commutative ring, and $K_0(A)$ its algebraic K-theory ring. Are there are any notable results asserting ring-theoretic properties (being Noetherian, reduced, Krull dimension, etc..) of $...