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7
votes
0answers
202 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 ...
5
votes
0answers
99 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 ...
5
votes
1answer
253 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{...
12
votes
1answer
497 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 ...
13
votes
1answer
287 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\; \...
9
votes
0answers
195 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 ...
5
votes
1answer
177 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 ...
6
votes
1answer
153 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....
7
votes
0answers
221 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 ...
3
votes
0answers
67 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^...
9
votes
0answers
470 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, ...
21
votes
1answer
663 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 ...
6
votes
1answer
128 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}\...
8
votes
0answers
141 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 ...
7
votes
1answer
494 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 ...
-2
votes
1answer
128 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 $\...
6
votes
2answers
239 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)/\...
3
votes
0answers
112 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 $...
11
votes
1answer
275 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 $\...
5
votes
2answers
330 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 ...
9
votes
1answer
234 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 ...
31
votes
0answers
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 ...
2
votes
0answers
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 ...
11
votes
0answers
235 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 ...
2
votes
1answer
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 ...
6
votes
0answers
232 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 ...
0
votes
0answers
45 views

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 ...
2
votes
1answer
187 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 ...
5
votes
0answers
270 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\...
6
votes
0answers
190 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 "...
12
votes
2answers
851 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, ...
6
votes
0answers
310 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 ...
5
votes
0answers
112 views

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 ...
3
votes
0answers
80 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 ...
1
vote
0answers
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 ...
7
votes
0answers
229 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) \...
10
votes
1answer
484 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 ...
1
vote
0answers
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})...
21
votes
1answer
774 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 $X$ ...
7
votes
0answers
117 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 ...
16
votes
0answers
451 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}= ...
5
votes
2answers
530 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!
3
votes
1answer
224 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 ...
8
votes
1answer
258 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 $E$ ...
4
votes
0answers
106 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 ...
11
votes
1answer
307 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 ...
3
votes
1answer
212 views

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? $$GL_2(T)=U(T)E_2(T)$$ ...
0
votes
0answers
154 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) ...
3
votes
0answers
96 views

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 ...
1
vote
0answers
134 views

Basic question about relation between algebraic k-theory and group cohomology

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 ...