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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Showing $K_0(F)=\mathbb{Z}$

In Morel's "$\mathbb{A}^1$-algebraic topology over a field", Milnor-Witt K-theory is defined to be the graded ring $K_*^{MW}(F)$ generated by the symbols, $[u]$ of degree $1$ and $\eta$ of ...
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What does $p+q \leq 1$ means in the context of $C^*$-algebra?

Let $A$ be a $C^*$-algebra, and let $p,q \in A$. The book I'm following(An Introduction to K-theory for $C^*$-algebra) has this notation $p+q \leq 1$. What does this mean? I looked through the book, ...
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Is a topological space that is homotopy equivalent to a CW-complex locally path-connected?

While describing Quillen’s $ + $-construction in his book Algebraic K-Theory, V. Srinivas assumes that his topological spaces are equivalent to CW-complexes and are path-connected. As universal ...
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What does the computation of K_0(R) tell me about the finitely-generated projective module?

Just like the question, supposing I calculated two different PID's, if they were PID's, I would calculate out Z--assuming the question only just stated that they were, that is. That's an example, but ...
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Calculating K_0(Z/m)?

I'm reading Jonathan Rosenberg's book Algebraic K-theory and Its Applications, and I finished skimming the first chapter, (I also did do a deep read in between some parts, just to be honest). However, ...
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How to compute the G-theory groups of $k[x,y]/(xy)$ for any field $k$

I am learning algebraic $K$-theory and is stuck on the following problem. How to compute the groups $G_n(A)$? Here $A$ is the ring $k[x,y]/(xy)$ and $k$ is a field. So far I have tried to localize $A$ ...
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Direct Proof that Artinian Rings have Stable Range 1

Is there a direct proof that any (right) Artinian ring has stable range 1? More precisely, let $R$ be a right Artinian ring and $a,b\in R$ be such that $aR+bR=R$. Can we prove that $(a-bt)R=R$ for ...
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Mayer-Vietoris Exact Sequence for Whitehead groups and projective module groups

I came across the claim that there is an exact sequence $K_1\Lambda\to K_1\Lambda_2\oplus K_1\Lambda_2\to K_1\Lambda'\to K_0\Lambda\to K_0\Lambda_1\oplus K_0\Lambda_2\to K_0\Lambda'$ while reading ...
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Why is the fundamental group of the Volodin space $X(R)$ the Steinberg group $St(R)$?

The Volodin space $X(R)$ is defined in A.A. Suslin's "On the Equivalence of K-Theories" (https://www.tandfonline.com/doi/abs/10.1080/00927878108822666) as the union of classifying spaces $\...
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Understanding the definition of vector bundle

I'm currently studying from Husemaller Vector Bundles and I'm having some problems understanding the definition given and the conventions used by the author. I think that the book gives a definition ...
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Presentation of Grothendieck-Witt group $GW(\mathbb{F})$ in terms of generators and relations.

Let $\mathbb{F}$ be a field, which for the sake of this discussion, is such that char $\mathbb{F} \neq 2$. By Corollary 9.4 in Scharlau's Quadratic and Hermitian Forms, the Grothendieck-Witt group $GW(...
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Lambda operations in K-theory

In Weibel's K-Book, before defining the lambda-operations on higher (Quillen) K-theory, he states that "Although many constructions of $\lambda$-operations have been proposed in more exotic ...
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Etale $K$-theory of finite fields.

Do etale $K$-theory of finite fields coincide with algebraic $K$-theory? what is the easiest way to see this (I think etale and algebraic $K$-theory coincide above a certain value like cohomological ...
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Chow group of a specific scheme related to the localization of the affine line at two points.

Given a scheme $X$, let $S=\varprojlim\limits_{U\subset X\times \mathbb{A}^1}U$ where $U$ is a Zariski open that contains both of $X\times\{1\}$ and $X\times\{0\}$. What does the Chow group $CH^i(S)$ ...
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Counterexamples to the Grothendieck group of a commutative monoid respecting tensor products

$\newcommand{\Q}{\mathbb{Q}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$Recall the definitions of the tensor product $\otimes_{\mathbb{N}}$ of commutative monoids (see also this note by ...
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Show the spectral sequence of a fibre sequence is a spectral sequence of algebras

I'm reading Notes on the K-theory of Finite Fields by Steve Mitchell. Here is a lemma from this paper. We want to show that the maps in the spectral sequence of the fibre sequence is $\mathbb{Z}/l$ -...
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Where and by whom is the stable rank first defined?

To the best of my knowledge, Bass defined the notion of stable range. Then somewhere it was shown that the condition on unimodular sequences defining the stable range holds for $N\geq n$ if it holds ...
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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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A reference question on the Dedekind zeta function and other L-functions.

On page 501 in the following ICM1983 conference proceeding https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1983.1/ICM1983.1.ocr.pdf you will find a conjecture on the L-function $L(X,s)$ of an ...
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Note or textbook about algebraic k-theory

I'm reading the Weibel's K-book. It's difficult for me. I have read some $K_0$ with the hlep of Bass' book. But Bass' book is so old. So, are there any more detailed books or notes to recommend. I ...
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Does isomorphism at the level of residue fields imply isomorphism of etale $K$-theory?

I am wondering whether something like the following argument is correct or not? If a morphism of regular schemes $f:X\rightarrow Y$ induces an isomorphism on residue fields and also induces a ...
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Bott inverted algebraic $K$-theory vs algebraic $K$-theory.

Bott inverted mod $l$ algebraic $K$-theory coincides with mod $l$ etale $K$-theory under mild conditions. The Bott element is an element in $K_2$ that is inverted with respect to the graded product ...
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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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When existence of transfer map of algebraic $K$-theory implies rational injection.

Given a map of smooth projective varieties $f:X\rightarrow Y$ over fields, there is a projection formula in algebraic $K$-theory given by $f_*(\alpha.f^*(\beta))=\beta.f_*(\alpha)$. I was wondering ...
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$K$-theory of formal power series.

I was wondering whether there is a calculation of algebraic $K$-groups of the formal power series $\mathbb{F}_p[[x]]$?
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$K$-theory of union of smooth curves.

Is there a method that one can calculate algebraic $K$ theory of a number of smooth curves with some singularity points corresponding to the intersection points of two different curves, given we know ...
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$\mathcal{K}$-homology sheaf on affine space.

The $n$-th $\mathcal{K}$-homology sheaf on $X$ is defined as the sheafification of the presheaf $U\mapsto K_n(U)$. Here $K_n(U)$ is the $n$-th algebraic $K$-group. I was just wondering whether it is ...
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$G_1$ of a nodal projective curve.

Let $C$ be the projective nodal curve over the field $k$ i.e. the projective line with two points identified. I'm having trouble calculating the $G$-theory of $C$, especially $G_1$. Writing the ...
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K Theory: Book Recommendations

Good people! So I've been hoping to get into K Theory for a while now, and the book that I have been trying to use (and failing) has been Charles Weibel's book by that very title. The book itself isn'...
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Elementary computations for hermitian and non-hermitian algebraic K-theory

As the title states, I am looking for some examples of non trivial albeit elementary calculations of algebraic hermitian and non-hermitian $K$-theory groups. Concretely, I would be interested in ...
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A contradiction involved extending vector bundles from punctured affine plane.

Consider the punctured affine plane $\mathbb{A}^2\setminus \{(0,0)\}$ over a field. Every vector bundle on this is trivial. Especially every vector bundle on this extends to a vector bundle on $\...
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On a characterization of certain $D(R,I)$-projective modules

Let $R$ be a ring with unit and $I$ a two sided ideal. We define the double ring $D(R,I)$ as the kernel pair for $q : R \to R/I$, i.e. as $$ D(R,I) = \{(x,y) \in R^n \times R^n : x = y \pmod{I} \} $$ ...
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An example in algebraic $K$-theory

I'm reading Carlsson's Derived Representation Theory and the Algebraic $K$-theory of Fields, and in section 4.2 he gives an example which I'm not fully understanding. The example is as follows: $k$ is ...
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The infinite loop space associated to the spectrum associated to a special $\Gamma$-space

I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim. On the top of ...
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Relating $G_0(R/fR)$ and $\oplus_{P\in V(f)} G_0(R/P)$?

For a Commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism ...
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When the classes of two finitely generated modules are equal in the Grothendieck group

Let $R$ be a Commutative Noetherian ring. Let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism ...
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Does $K_0$ of a regular (affine) scheme always surjects onto $K_0$ of any open subscheme?

Let $X$ be a Noetherian separated regular scheme and let $U$ be an open subscheme. Then does there always exist an exact sequence $K_0(X)\to K_0(U)\to 0$ ? If not, then is it at least true when $X=\...
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Is the nerve of a symmetric monoidal category a K-theory space?

It's an amazing fact that if the $C$ is a symmetric monoidal category so that its components form a group, then $NC$ is an infinite loop space. Now if we have a Waldhausen category $D$, a category ...
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$K$-group of category of bounded chain complexes of free modules with finite length homologies

For a Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free modules with ...
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Induced map on the $K_0$ of punctured spectrum of completion

Let $(R,\mathfrak m)$ be a local Gorenstein ring (may also assume excellent) of dimension at least $2$, and let $(\hat R,\hat {\mathfrak m})$ be the $\mathfrak m$-adic completion. Let $U:=\text{Spec}(...
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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 ...
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On transfer and base change map on Grothendieck groups induced from injective ring homomorphism

For a commutative Noetherian ring $R$, let $G_0(R)$ denote the Grothendieck group (an abelian group) of the abelian category of finitely generated $R$-modules (Note that I'm Not talking about $K_0(R)$ ...
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The generator of $K_0(C(\partial(]0,1[^2)))$ and $K_1(C(\partial(]0,1[^2)))$

Let $C=[0,1]^2 \subseteq \mathbb{C}$ and $\partial C$ the boundary of $C$. I'm looking for the $K_0(C(\partial C))$ and $K_1(C(\partial C))$ and its generator.
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Proving $K_1(\mathcal{T})=0$ (is trivial)

Let $\mathcal{T}$ the toeplitz algebra and we define the short exact sequence? where $C(\mathbb{T})=\{z\in \mathbb{C}/ |z|\leq 1\}$: $$ 0 \rightarrow \mathcal{K} \rightarrow \mathcal{T }\rightarrow C(...
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Grothendieck group of local affine- surfaces with rational singularities

Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $ \pi: Y \to X=\operatorname {Spec}(R)$ be a resolution ...
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When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme

Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ ...
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Is there a categorical way to see that $K_1(A)\twoheadrightarrow K_1(A/I)$ for a nilpotent ideal $I\subset A$?

Let $A$ ba a (not necessarily commutative) unital ring. The $K_1$ group of $A$ is defined as $K_1(A)=\pi_1((\mathrm{Proj}(A)^\simeq)^\mathrm{gp})$. Here $\mathrm{Proj}^\simeq$ is the core of the ...
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Excision sequence for $K_0$ in algebraic $K$-theory

I am looking at Exercise I.2.3 in Weibel's $K$-book. Here is the statement: If $I$ is an ideal of a ring $R$, we form the augmented ring $R\oplus I$ and let $K_0(R,I)$ denote the kernel of $K_0(R\...
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2 votes
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Recovering components in the projective bundle formula

Projective bundle formula is telling us that for a projective bundle $Y$ over $X$, where the fibers are $\mathbb{P}^n$, we have $G(Y)\cong G(X)^{n+1}$. Here $G$ is the $K$-theory of coherent sheaves. ...
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What are the K groups of $X_n$, the wedge sum of $n$ circles at a single point?

If $X_n$ is the bouquet of n circles, then what is $K_0(X_n)$ and $K_1(X_n)$? I am super confused on how to approach this problem, since I've only seen $K_0$, $K_1$ computed for algebras, but I don't ...
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