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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24 views

Bott Periodictity for Real and Complex Algebras

In Rordam's Introduction to K-Theory for $C^*$ Algebras they prove \begin{equation} K_{n+2}(A) \cong K_{n}(A) \end{equation} using isomorphism in terms of the suspensions of $A$. I have also heard ...
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28 views

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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70 views

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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26 views

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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75 views

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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51 views

$K$-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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49 views

$K$-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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37 views

$\mathcal{K}$-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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Simply connected and rationally connected varieties in char $p$.

Consider a smooth projective variety in char $p$ with trivial etale fundamental group. You can assume it is also rationally connected. I believe there are a lot of examples for these types of ...
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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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175 views

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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40 views

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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60 views

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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100 views

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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37 views

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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28 views

Units of a ring extension

In the paper Gersten Conjecture on Milnor K theory by Moritz Kerz, he says: Let $A \subset A'$ be a local extension of semilocal factorial rings and $ f \neq 0$ in $A- A^{*}$ be such that $ A/fA \cong ...
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36 views

A problem on $K_0$

Let $i: R \hookrightarrow R[[t]]$. If $R$ is left regular, then $K_0i$ is an isomorphism ? It's well known that this holds for $R[t]$.
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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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1answer
72 views

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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61 views

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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60 views

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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39 views

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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38 views

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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109 views

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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94 views

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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35 views

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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40 views

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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Help with a problem in K-theory/C- algebras.

For this problem you may assume the fact that $K_1(C_0(D)) = 0$. Let $n > 1$, let $\omega = e^{\frac{2πi}{n}}$, and let $E_n$ be the space obtained from $D$ by identifying $z$ and $\omega z$ for ...
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46 views

What are the K groups of $X_n$?

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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374 views

17 definitions of algebraic K-theory of a ring. Which should I take?

There are multiple definitions of algebraic K-theory, but I have trouble differentiating between them. Could someone help me out? Let $R$ be a commutative ring. I would like to define $K_n(R)$, and ...
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1answer
148 views

Compatibility of tensor product and external tensor product in algebraic K-theory

Let $X$ be a smooth, compact, quasi-projective, complex algebraic variety and let $K(X)=K_0(X)=K_0(\mathrm{Coh}(X))$ be the Grothendieck group of coherent sheaves on $X$. There are several notions of ...
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195 views

Properties of injective modules

I am reading A course in Homological Algebra by Hilton and Stammbach. In the first chapter they showed that a $\Lambda$-module is projective iff it is a direct summand of a free module. They then ...
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235 views

Show that $K_2(F)$ is a direct summand of $K_2F(t)$

I have a question regarding Example 6.1.2 (page 252) from the book "The K-Book" by Charles Weibel. Here is the statement: Example 6.1.2: Let $F(t)$ be a rational function field in one ...
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1answer
89 views

$a\otimes(-a) = a \otimes (1-a) + a^{-1}\otimes(1-a^{-1})$???

J. Browkin writes in his article "K-Theory, Cyclotomic Equations, and Clausen's Function" (which appears on Chapter 11 of Lewin's book "Structural properties of polylogarithms") that for any field $F$ ...
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1answer
54 views

Identification of $M_n(\tilde{SA})$,where $\tilde{SA} $ is the unitalization of suspension $C^*$-algebra

In the section 11.1 of Rordam's book, there is a remark : If $A$ is a unital $C^*$-algebra, denote $SA$ by the suspension of $A$, $\tilde{SA} $ is the unitalization of $SA$. We can identify $M_n(\...
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Geometric Resolutions of C* -Algebras

I am reading "K-theory FOR OPERATOR ALGEBRAS" Bruce Blackadar, Proposition 23.5.1. Let B be a separable $C^{*}$-algebra. Then there is a separable commutative $C^{∗}$-algebra F, whose spectrum ...
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1answer
20 views

Admissible layers Q-construction

I have a question which seems to be extremely trivial but for some reason I don't get it and am very confused about it. In the paper by Quillen, "Higher Algebraic K-theory I" page 94 top. Quillen ...
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1answer
136 views

Grothendieck group of equivalent categories

Let $\Gamma : \mathcal A \to \mathcal B$ be an equivalence of categories where $\mathcal A$ is an exact category, $\mathcal B$ is an additive full subcategory of the category of $R$-modules for some ...
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59 views

Additivity in algebraic K-theory — what does it truly mean?

--- Question --- I have seen several definitions of 'additivity' in algebraic K-theory. In all cases, I can more or less see that there is something additive going on. But I have difficulty seeing ...
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106 views

Projective modules of rank $n$ over a ring with $\operatorname{Pic}(R) = 0$

Let $R$ a ring with trivial Picard group, so every rank $1$ projective module is free. What does that tell me about the structure of projective modules of rank $n$? For starters, if $P$ is any ...