Questions tagged [algebraic-k-theory]

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13 votes
1 answer
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"a sign that one should be computing K-theory"

Allen Knutson said here in comments below the question that I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it. I know one ...
Praphulla Koushik's user avatar
3 votes
0 answers
159 views

$G_0(X) \cong G_0(X_{red})$ where X is a noetherian scheme

Let $\textbf {X}$ be a noetherian scheme, $\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$. We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$. Now I ...
user avatar
10 votes
1 answer
576 views

Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
Liam Keenan's user avatar
11 votes
1 answer
690 views

Status of the extended form of the Lichtenbaum conjecture

The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$. ...
skupers's user avatar
  • 7,923
9 votes
1 answer
398 views

$K_3(\mathbb{Z})$ and $\pi ^S_3$

This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the ...
abx's user avatar
  • 37.1k
1 vote
0 answers
172 views

Injective envelope in the category of left exact functors

Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
Danimenru's user avatar
4 votes
1 answer
542 views

Question about an implication of Thomason's étale descent spectral sequence

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads $$H^p_{\acute{e}t}(X, \...
xir's user avatar
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5 votes
0 answers
140 views

Failure of devissage vs link topology in algebraic K-theory

This is somehow related to (or maybe a simplified version of) an earlier question (see here) regarding Gersten complexes for singular varieties. The Gersten complexes arise from the coniveau spectral ...
Matthias Wendt's user avatar
7 votes
0 answers
754 views

Serre presentations over $\mathbb{Z}$

Given a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a classical result of J.-P. Serre asserts that the complex semisimple Lie algebra $\mathfrak g=\mathfrak g(A)$ corresponding to $A$ admits a presentation ...
Timothée Marquis's user avatar
6 votes
1 answer
413 views

Equivalence between categories of coherent sheaf of codimension p

Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \...
Sunny's user avatar
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3 votes
0 answers
258 views

Homotopy equivalence of $K$-theory and $G$-theory

Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
user127776's user avatar
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2 votes
0 answers
308 views

Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers: 1) Is there any example of a smooth projective variety $X$ over a ...
user127776's user avatar
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7 votes
0 answers
271 views

Adequate equivalence relations and algebraic $K$-theory

I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
user127776's user avatar
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10 votes
1 answer
318 views

About $K$-rectification of increasing tableaux

Let $T$ be a standard Young tableaux on $[n]$. Denote the RSK algorithm $\text{RSK}(w)=(P(T),Q(T))$ for $w\in\mathfrak{S}_n$, where $P(T)$ is the Schencted insertion tableaux. For $1\leq i\leq j\leq n$...
Sylvester W. Zhang's user avatar
4 votes
0 answers
138 views

On the Beilinson's conjecture regarding the proper flat integral models

I had a question which seems to be straightforward but I wasn't able to figure it out. In page 13 of this paper a conjecture of beilison is mentioned that if $\mathcal{X}_{\mathbb{Z}}$ is a proper ...
user127776's user avatar
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7 votes
1 answer
517 views

Motivation for Suslin’s Rigidity Conjecture

Suslin Rigidity conjecture states that motivic cohomology $$ H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n)) $$ of the field $F$ coincides with motivic cohomology for the subfield of ...
Daniil Rudenko's user avatar
8 votes
1 answer
607 views

Original reference for Adams Riemann-Roch theorem

Let $f\colon Y\to X$ be a proper morphism between smooth quasiprojective $k$-algebraic varieties. Denote by $\psi^j$ the $j$-th Adams operation on the Grothendieck group of vector bundles and $\theta^...
Tintin's user avatar
  • 2,741
6 votes
0 answers
92 views

Elliptic deformation of the second Chern class

Second Chern class $$c_2 \in H^4(BGL,\mathbb{Q}(2))$$ admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
Daniil Rudenko's user avatar
9 votes
1 answer
680 views

Bass' conjecture implies the Parshin's conjecture

In the appendix of this paper. It is proved that Bass' conjecture for $K_n$ implies the rational Beilinson-Soulé conjecture for $K_n$. Then at the end the author claims that the same method can be ...
user127776's user avatar
  • 5,831
4 votes
0 answers
251 views

Effects of the first algebraic K-theory on the higher algebraic K-theory

Is there any counterexamples known to the following statement? ($A$ a regular noetherian integral domain of finite Krull dimension) If $A^{\times}$ is finitely generated then $K_n(A)$ is finitely ...
user127776's user avatar
  • 5,831
5 votes
1 answer
227 views

Kernel of the determinant morphism from the first algebraic K-theory

If $A$ is the coordinate ring of a smooth variety over a finite field is it known whether the kernel of the determinant map $K_1(A)\rightarrow A^{\times}$ is torsion or not?
user127776's user avatar
  • 5,831
6 votes
0 answers
217 views

Adams operation on Q-construction of fields

Let $F$ be a field that we want to compute its rational algebraic $K$-theory using the Quillen's $Q$-construction. Let $QF$ be the $Q$ construction of the category of finite dimensional vector spaces ...
user127776's user avatar
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3 votes
0 answers
108 views

Adams operation on the rational homology

The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...
user127776's user avatar
  • 5,831
4 votes
0 answers
462 views

Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...
rori's user avatar
  • 257
5 votes
0 answers
349 views

making the group completion in homology sense unique via the plus construction

A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology. In the paper they prove that when $X$ is a ...
Hari Rau-Murthy's user avatar
9 votes
1 answer
504 views

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 $\psi^\ell-1$ for $\ell$ a generator ...
xir's user avatar
  • 1,964
14 votes
1 answer
1k 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 ...
user40276's user avatar
  • 2,199
6 votes
0 answers
225 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 ...
Matthias Ludewig's user avatar
5 votes
1 answer
321 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{...
John Berman's user avatar
13 votes
1 answer
669 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 ...
anna abasheva's user avatar
13 votes
1 answer
456 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\; \...
Urs Schreiber's user avatar
9 votes
0 answers
346 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 ...
user127776's user avatar
  • 5,831
5 votes
1 answer
241 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 ...
GreginGre's user avatar
  • 1,661
6 votes
1 answer
373 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....
Brian Shin's user avatar
9 votes
0 answers
373 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 ...
user127776's user avatar
  • 5,831
4 votes
0 answers
103 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^...
Bin Wang's user avatar
  • 193
9 votes
0 answers
686 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, ...
Alex Gavrilov's user avatar
21 votes
1 answer
2k 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 ...
Tim Campion's user avatar
  • 60.6k
10 votes
2 answers
413 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}\...
ahulpke's user avatar
  • 1,200
10 votes
0 answers
402 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 ...
Jakob's user avatar
  • 1,986
14 votes
1 answer
2k 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 ...
B.K-Theory's user avatar
-2 votes
1 answer
207 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 $\...
masa M's user avatar
  • 141
6 votes
2 answers
518 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)/\...
Nicola Nesa's user avatar
3 votes
0 answers
198 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 $...
Minseon Shin's user avatar
  • 1,987
11 votes
1 answer
405 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 $\...
J.Li's user avatar
  • 781
6 votes
2 answers
841 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 ...
xir's user avatar
  • 1,964
9 votes
1 answer
321 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 ...
Grisha Papayanov's user avatar
37 votes
0 answers
5k 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 ...
Peter Scholze's user avatar
3 votes
0 answers
120 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 ...
Dubious's user avatar
  • 1,237
11 votes
0 answers
264 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 ...
Tim Campion's user avatar
  • 60.6k

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