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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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Concretely determining homotopy cofibers, an example from algebraic K-theory

For context this questions is "caused" by the proof by Thomason of the Gillet-Waldhausen theorem (which is proposition 1.11.7 in this paper). Let $\mathcal{C}$ be an exact category, we have ...
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quotient group of Idele group is its own Pontryagin dual with respect this pairing

In the J. Tate's paper "Relations Between K2 and Galois Cohomology" Lemma 5.2, it says that given $\alpha_1,\cdots,\alpha_r\in $Br$_lF$, i.e. $\alpha_i$ is killed by $l$, where $l$ is prime ...
I Am Fish's user avatar
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Plus construction on Simplicial Sets?

Write $\mathsf{sSet}$ for the category of simplicial sets and $\mathsf{Top}$ for the category of topological spaces. I would like to know if there a functor $\mathsf{sSet}\to\mathsf{sSet}$ that ...
wind's user avatar
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Waldhausen's $S.$ construction as an adjoint

I am studying K-theory with Weibel's K-book and have just read the definition of the $S.$ construction for Waldhausen categories. I also recently watched a series of talks by Thomas Nikolaus which ...
DevVorb's user avatar
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rank of an $\mathcal{O}_X$-module being constant

Given a finitely generated projective module $M$ over a ring $R$ with exactly two idempotents $0,1$ ($X=\operatorname{Spec} R$ is connected). We have a coherent $\mathcal{O}_X$-module $\widetilde{M}$ ...
Mizutsuki's user avatar
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Subtleties in commuting colimits

For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
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Algebraic K-theory of compositum of fields $K_2(F.M)$

Let $F$ and $E$ be two subfields of a fixed field $K$, then the compositum $F.E$ is just $F(E)=E(F)$. Let $K_2$ be the second Milnor algebraic K-group. I ma asking if we can express (or lift ) ...
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Plus construction and classifying space

Suppose $G$ is a perfect group, and let us consider $BG$ and its plus construction (We consider $BG$, the classifying space, by the nerve construction). By following Hatcher’s book (Proposition 4.40, ...
May's user avatar
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Plus construction is functorial

Given a nice map (cellular map) $f : X\rightarrow Y$ between CW complexes $X$ and $Y$, how is $f^{+}$ defined between their plus constructions? Going through my old notes, I've learnt that the plus ...
May's user avatar
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On Bloch's higher chow.

I stumbled across the statement that for an integral scheme $X$ and an elements $f_1, \ldots, f_n \in {\cal O}_X^{\times}$, there is a symbol map, viz., $$ {\mathrm S} \colon \{ f_1,\ldots,f_n \} \...
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2 votes
1 answer
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Non-isomorphic AF-algebras

I am studying the Elliot's Classification Theorem of AF-algebras (see Theorem 7.3.4 in An Introduction to $K$-theory for $C^*$-algebras by Rørdam, Larsen and Laustsen). I am trying to understand the ...
Jose M Barrientos's user avatar
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Question 3.1.2 in Allen Hatcher, Algebraic Topology(comparing multiplication by n map in $H$ and $G$).

Here is the question I am trying to understand its solution: Show that the maps $G \xrightarrow{n} G$ and $H \xrightarrow{n} H$ multiplying each element by the integer $n$ induce multiplication by $n$ ...
Emptymind's user avatar
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Question 3.1.1 in AT

Here is the question I am trying to understand its solution: Show that $\operatorname{Ext}(H, G)$ is a contravariant functor of $H$ for fixed $G$, and a covariant functor of $G$ for fixed $H.$ Here is ...
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Isomorphism of projective modules modulo nilpotent ideals.

Let $R$ be a commutative ring, $I$ be an ideal of $R$ with $I^2 = 0$ and let $M, N$ be projective $R$-modules. Then $M/IM$ and $N/IN$ are projective $R/IR$-modules and we get a map $\mathrm{Hom}_R(M, ...
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On the norm map in Milnor K-theory for $K_2$

I am currently studying Milnor $K$-theory and came across his definition of a transfer map which works if we have an inclusion of rings. However, in Quillen's $K$-theory, he defines a transfer map for ...
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Give an example of a commutative ring with unity $R$, such that $K_0(R) \ncong K_0(R[t])$

In Weibel's $K$-book I have read that it is Quillen's classical result that if $R$ is a regular Noetherian ring then $K_0(R) \cong K_0(R[t])$. So out of curiosity I have tried and failed quite a lot ...
Divya's user avatar
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stable range of stably free modules

This is part of exercise 1.1.5 of the K-book: Notation: we say $R$ has stable range at most $n$ if every unimodular row $(r_0,\ldots, r_n)$ induces a unimodular row $(r_1',\ldots, r_n')$ with $r_i'=...
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What is the Grothendieck (K_0) group of the ring of Laurent polynomials?

I am curious to know what is the Grothendieck group of the ring of Laurent polynomials over a field. I am a beginner in the study of Algebraic K-theory. I have learned that the Grothendieck group of ...
Promit Mukherjee's user avatar
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Pushforward of the Segre embedding in K-theory

Fix $n$, $m\ge 1$, and let $d=\binom{m+n}{m}$ and $N=mn+m+n$. Consider the Segre embedding $\sigma:\mathbb{P}^m\times \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$, which has degree $d$. I'm trying to ...
Alvaro Martinez's user avatar
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When is the representation ring of neutral component of reductive group flat?

Let $G$ be a reductive complex algebraic group, not necessarily connected. Let $G^\circ$ be the connected component of the identity. When is it true that $R(G^\circ)$ is flat over each connected ...
Stefan  Dawydiak's user avatar
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The definition of (special) $\lambda$-rings by using the ring $W(K)$ of big Witt vectors.

I am reading C. Weibel's 'The K-book', the definition of (special) $\lambda$-rings (in Page 102, Definition 4.3.1) is A pre-$\lambda$-ring $(K,+,\cdot)$ is called a $\lambda$-ring, if the following ...
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The multiplication '$*$' of the ring $W(K)=1+tK[[t]]$ in C.Weibel's 'The K-book'

I am reading C. Weibel's 'The K-book', and in page 101, example 4.3, there is a construction of the ring $(W(K)=1+tK[[t]],\cdot,*)$. I could not understand how can we construct the multiplication '$*$'...
Frank's user avatar
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Minimal polynomial of $\mathcal{O}(1)\otimes -$-operator

This is somewhat of a follow-up to this question. Take $X= \mathbb{P}^1, G = \operatorname{GL}(2)$. Compute the minimal polynomial of the operator $\mathcal{O}(1)\otimes -$ on $K_G(X)$. How does one ...
fish_monster's user avatar
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Pushforward of a line bundle on $\mathbb{P}^1$ to a point

In his notes on K-theory, A. Okounkov states the following exercise: The group $\operatorname{GL}(2)$ acts naturally on $\mathbb{P}^1$ and on line bundles $\mathcal{O}(k)$ over it. Push forward these ...
fish_monster's user avatar
1 vote
1 answer
58 views

$T$-equivariant resolution of coherent sheaf

Let $X := \{x_1x_2= 0\} \subset \mathbb{C}^2$ and let $\mathcal{F} := \mathcal{O}_0$ where $0\in X$ is the origin. Let $T = \left\{\begin{pmatrix} t_1 & 0 \\ 0 & t_2\end{pmatrix}\right\}\...
fish_monster's user avatar
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Quillen +-construction

Loday in the article available here http://www.numdam.org/item/?id=ASENS_1976_4_9_3_309_0, described the Quillen +-construction in the proof of the Theorem 1.1.1. I don't understand his last equality, ...
newuser's user avatar
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How to get into K-Theory and have least interaction with Algebraic Topology

I am doing PhD in Leavitt Path Algebras (LPAs). While going through some recent works on LPAs I found this paper titled K-Theory of Leavitt path algebras by Ara etal. I tried reading it, but failed ...
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Why is there a homotopy equivalence between $s_\bullet \mathcal{C}$ and $iS_\bullet \mathcal{C}$?

I'm reading Waldhausen's 1985 Algebraic K-Theory of Spaces and have not been able to follow his argument for Corollary 2 of Lemma 1.4.1. In Waldhausen's notation. $s_\bullet\mathcal{C}$ is the ...
Tanner Carawan's user avatar
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Are algebraic maps from the n-dimensional torus to the special unitary group of large enough degree null-homotopic?

Let $S^n\subset \mathbb{R}^{n+1}$ be the unit sphere, and $T^n=(S^1)^n$ the $n$-torus. Loday proved in 1 that every algebraic map $T^n\to S^n$ is null-homotopic. In particular, since $SU(2)\simeq S^3$,...
Vincent Nesme's user avatar
2 votes
0 answers
68 views

When the norm map $K_2(L) \rightarrow K_2(K)$ is Zero

Given a number field $K$ which contains at least one primitive root of unity. Let $L/K$ be a finite extension of $K$. Let $N_{L/K}: K_2(L) \rightarrow K_2(K)$ be the norm map of Milnor or Quillen ...
Hajar hajar's user avatar
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0 answers
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The $K$ Book exercise $II\;6.4$

The exercise is Let $\mathcal{A}$ be a small abelian category. If $[A_1] = [A_2]$ in $K_0(\mathcal{A})$, show that there are short exact sequences in $\mathcal{A}$ $0 \rightarrow C'\rightarrow C_1 \...
Divya's user avatar
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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, ...
Phil's user avatar
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1 vote
1 answer
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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 ...
Transcendental's user avatar
6 votes
1 answer
101 views

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 ...
Tipping Octopus's user avatar
1 vote
0 answers
53 views

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 ...
kid111's user avatar
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1 answer
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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 $\...
Joshua Graham's user avatar
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1 answer
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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 ...
jacopoburelli's user avatar
4 votes
1 answer
193 views

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(...
Sunny Sood's user avatar
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1 answer
132 views

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 ...
Captain Lama's user avatar
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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 ...
user127776's user avatar
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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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2 votes
1 answer
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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 ...
Emily's user avatar
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1 vote
1 answer
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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 $\mathbb{Z}/l$ -...
Flying pencil's user avatar
2 votes
2 answers
110 views

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 ...
Daniel W.'s user avatar
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1 answer
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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 ...
wonderich's user avatar
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1 vote
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159 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 ...
hm2020's user avatar
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1 answer
393 views

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 ...
Cille's user avatar
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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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2 votes
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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 ...
user127776's user avatar
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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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