# Questions tagged [algebraic-k-theory]

The algebraic-k-theory tag has no usage guidance.

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### Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?

Quillen famously constructed a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^q-1$. Does this lift to the ...

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557 views

### Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?

It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...

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134 views

### Group $C^*$ vs group von-Neumann algebras

Let $\Gamma$ be a countable (discrete) group (in what follows, make additional assumptions as you wish). Let $C^*_r(\Gamma)$ and $W^*_r(\Gamma)$ be the reduced $C^*$-algebra respectively the reduced ...

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266 views

### K-theory of finite diagram categories

Suppose $I$ is a finite $\infty$-category and $F:I\rightarrow\text{fCW}$ is a functor that takes values in finite CW complexes. For each $X\in I$, let $[F(X)]$ denote the class of $F(X)$ in $K_0(\text{...

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518 views

### Reference for the algebro-geometric proof of Matsumoto theorem

Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...

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294 views

### Stable Cohomotopy as $K \mathbb{F}_1$

Various classical results suggest that stable cohomotopy may usefully be regarded as being the algebraic K-theory over the "field with one element" $\mathbb{F}_1$:
$$
K \mathbb{F}_1 \;\simeq\; \...

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206 views

### Examples for a conjecture of Beilinson

Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in ...

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183 views

### Are these two constructions of $K_0(A)$ isomorphic?

The following question is extracted from this question on MSE, which got no answer so far, probably because it was a bit hidden by another question which a posteriori was totally obvious.
Let $A$ be ...

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160 views

### Parsing the definition of center of an algebra in a higher-categorical setting

I'm having trouble parsing a definition in Lurie's "Rotation Invariance in Algebraic $K$-Theory". The definition os for the notion of center of an associative algebra object, and occurs in Remark 2.1....

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229 views

### Finite generation of rational algebraic k-theory

Parshin's conjecture states that higher algebraic k-theory of smooth projective varieties over finite fields are rationally trivial. This has been shown for curves. Quillen showed that the K-groups in ...

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69 views

### Is there Thom isomorphism for equivariant K groups in algebraic geometry, not necessarily complex number field?

In Chriss and Ginzburg's fantastic book 'representation theory and complex geometry', they use the following Thom Isomorphism:
$\pi:E\rightarrow X$, is a G-equivariant affine linear bundle, then $\pi^...

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493 views

### Andrei Suslin's works

Andrei Suslin, a well known mathematician, died 10 July 2018. (https://en.wikipedia.org/wiki/Deaths_in_2018) I believe it may be appropriate to give an overview of his work on this site. Personally, ...

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708 views

### Is algebraic $K$-theory a motivic spectrum?

I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...

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134 views

### Presentation of special linear group over localizations of the integers

I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\...

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150 views

### Using the universal property of K-theory

A paper of Blumberg, Gepner and Tabuada gives a universal property of K-theory: from their abstract "connective algebraic K-theory is the universal
additive invariant, i.e., the universal functor with ...

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541 views

### Entering to the K-Theory Realm

I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation
and interaction with the field of Algebraic Topology. I mainly had
concentrated on ...

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130 views

### Meaning of notations in Rost's cycle modules

In chapter 2 of markus Rost's "Chow Groups with Coefficients", I encountered the notations $c_{\kappa(v)\vert F}$ and $r_{\kappa(v)\vert F}$ in formula (2.1.0) and in the homotopy property for $\...

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244 views

### Galois descent in motivic cohomology

Let $X_N$ denote the Fermat curve defined over $\mathbb{Q}$ by the equation $x^N+y^N-z^N=0$ and let $X_{N,\mathbb{Q}(\mu_N)}$ be the base change. Let $G$ be the Galois group of $\mathbb{Q}(\mu_N)/\...

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113 views

### Extending vector bundles over a regular divisor in a regular affine scheme

This is more-or-less question (3) on page 170 of Quillen's "Projective Modules over Polynomial Rings" (link):
Let $A$ be a regular Noetherian ring and let $f \in A$ be an element of $A$ such that $...

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280 views

### Hilbert 90 for higher K-groups

For a field $F$, Let $K_n(F)$ be the Quillen's $n$-th K-group of $F$.Then $K_0(F)\cong \mathbb{Z}$, $K_1(F)\cong F^\times$.
For a finite Galois extension $L/K$, $K_n(L)$ are Galois modules.
Then $\...

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359 views

### Ring structures on algebraic K-theory spectrum, and its non-connective counterpart

I have a few naive questions on the algebraic K-theory spectrum construction, but whose answers I couldn't figure out using the internet. I'm mostly interested in the case of a commutative ring, but I ...

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244 views

### Positive cones in K-groups

Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just ...

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2k views

### Homology of $\mathrm{PGL}_2(F)$

Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...

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94 views

### Is the Milnor boundary map, a natural transformation?

Consider the Milnor $K_n$-functors for discrete valuiation fields. For any discrete valuation field $F$ we can associate an abelian group $K_n(F)$ and the construction is given thanks a universal ...

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240 views

### Direct proof of the equivalence of symmetric monoidal $K$-theory and exact sequence $K$-theory?

When all exact sequences split in $C$, we have $\Omega B C \simeq K(C):=\Omega Q(C)$. Heuristically, this is because the space of upper-triangular matrices is contractible. Can this be made precise? I ...

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73 views

### reduction of torsion modules

Let $G$ be a profinite group.
Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb ...

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246 views

### Difference between algebraic and etale K-theory

Due to the Quillen-Lichtenbaum conjecture (now proven by Rost, Voevodsky, and Weibel), the map $K_\ast(X,\mathbb{Z}/n)\rightarrow K_\ast^{et}(X,\mathbb{Z}/n)$ from algebraic K-theory to etale K-theory ...

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### Higher order algebraic and topological k group [duplicate]

Is the higher order algebraic k group has some direct analogy with the corresponding topological k group ? since the 0th,1th algebraic k group are direct algebraic version of topological k group,but ...

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194 views

### Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain

Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M_n(R)$-modules. Is it true that
$A\oplus M_n(R)\cong B\oplus M_n(R)\:\:\Rightarrow\:\:A\cong B$?
Here, the isomorphism is ...

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280 views

### Poincaré duality for motivic cohomology

Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...

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195 views

### K-theory of the infinite dimensional projective space

What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "...

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855 views

### $K$-theory backwards

Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?
That is, ...

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311 views

### Motivic $\mathbf{Z}(1)$

I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...

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### Fundamental class in equivariant K-theory

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.
The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...

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81 views

### Defining structure maps of spectra by lifting from the homotopy category

Voevodsky's original definition of the algebraic $K$-theory spectrum, $KGL$, was given as follows:
The component spaces were fibrant replacements of the infinite Grassmannian $BGL$. The structure ...

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36 views

### When is genus the same as stable equivalence?

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...

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237 views

### On the existence of a norm map for Milnor K-theory for a finite extension $A \to B$ which is free of finite rank

I have one technical question on norm maps on Milnor K-theory.
When $K \subset L$ is a finite extension of fields, we know (by Bass-Tate and Kato) that there exists a norm map $N_{L/K} : K^M_n (L) \...

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497 views

### K-theory of an elliptic curve

Given an elliptic curve $E$ over $\mathbb{Q}$, I have read somewhere (But I can't remember exactly where) that the Beilinson conjecture asserts that: The rank of the albelian group $K_{2}(E)$ (the ...

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61 views

### What is $K_1(\mathrm{Var}_\Bbbk)$? [duplicate]

Ok, this is a very naive question, and not seriously motivated. But I was just wondering: did anybody define any (interesting) higher K-theory Grothendieck group of varieties $K_n(\mathrm{Var}_{\Bbbk})...

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785 views

### Spectral sequences in $K$-theory

There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.
For a field $k$, let $X$ be smooth variety $X$ ...

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131 views

### Non-vanishing of the Borel classes in the cohomology of $SL_n(\mathbb Z)$

The stable real cohomology of $SL_n(\mathbb Z)$ was computed by Borel: it is given by $\mathbb R[z_i\mid i=5,9,13,\cdots]$ with $z_i$ in degree $i$. One may wonder whether the pull back of the stable ...

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465 views

### K-theory and homology of groups

It is known that for any ring $R$,
$$K_{1}(R)=H_{1}(GL_{\infty}(R), \mathbb{Z})$$
$$ K_{2}(R)= H_{2}(E_{\infty}(R),\mathbb{Z})$$
$$ K_{3}(R)= H_{3}(St_{\infty}(R),\mathbb{Z})$$
where $GL_{\infty}= ...

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555 views

### Local property of split exact sequence

In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal?
Thanks!

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231 views

### Bass and Quillen K-theory

What is the difference of Bass and Quillen K-theory groups of a ring $R$.
More concretely, what is $K^{Bass}_{i}(R)$? does it equal to $K^{Quillen}_{i}(R)$ if $R$ is a regular ring ?
Does it make ...

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264 views

### About Riemann-Roch without denominators

The Riemann-Roch without denominators can be expressed as follows:
Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ ...

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107 views

### Bounded self-adjoint perturbation of a p-summable spectral triple

I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book 'Elements of Noncommutative Geometry' by Gracia-Bondía,Várilly and Figueroa.Now,after ...

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317 views

### Reference to Chern classes in algebraic k theory

I am reading P. Schneider's paper, Introduction to the Beilinson conjectures. Section 4 in this paper is something very formal about Chern classes. Personally I find some terminologies in the paper a ...

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### For $T$ the $2\times 2$ triangular matrices over $R$, can we write $GL_2(T)=U(T)E_2(T)$?

Let $R$ be a commutative ring with identity, and let $T = T_2(R)$ be the ring of $2\times 2$ upper triangular matrices over $R$. Is it true that the following identity holds?
$$GL_2(T)=U(T)E_2(T)$$
...

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157 views

### Non-existence of nontrivial finite group extension of any simply-connected Lie group

Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that
there does not exist any group $G$ (with no topology) ...

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### Does the localization sequence (for exact categories) in algebraic K-theory come from a homotopy fibration sequence?

In Quillen's paper, he states localization, as:
Let $B$ be a Serre subcategory of $A$, let $A/B$ be the quotient abelian category, and let $e: B \to A$ and $s: A \to A/B$ denote the canonical ...