Questions tagged [algebraic-k-theory]

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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
23 votes
0 answers
615 views

Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
მამუკა ჯიბლაძე's user avatar
19 votes
0 answers
883 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
Dmitri Pavlov's user avatar
19 votes
0 answers
865 views

Epsilon factors - a la Beilinson - What is it?

I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or "...
Sasha's user avatar
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16 votes
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582 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}= ...
Ofra's user avatar
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15 votes
0 answers
363 views

Dennis trace map for stable $\infty$-category, naively

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...
Simon Henry's user avatar
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12 votes
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364 views

Can Quillen-Lichtenbaum recover Borel's computation?

Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
skupers's user avatar
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Global version of Gabber's rigidity theorem

I had a question regarding Gabber's rigidity. Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
user127776's user avatar
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12 votes
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Goodwillie's notes from MSRI Lecture Series

Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
Juan Villeta-Garcia's user avatar
12 votes
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Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
JBorger's user avatar
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11 votes
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614 views

Does Merkurjev's argument help Voevodsky's program?

In the talk Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract) Voevodsky mentioned that he was ...
David Roberts's user avatar
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11 votes
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258 views

Criteria for a map of rings to induce an equivalence on K-theory?

Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...
Reuben Stern's user avatar
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
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321 views

$K$-theory spectrum of the category of finite groups

(I asked some people this question in person and got the answer "no", but wanted to see if the Internet had more to say)$ \newcommand{\FinGrp}{\mathbf{FinGrp}} $ Way back in my first group theory ...
Yuri Sulyma's user avatar
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747 views

Rosenberg's proof of Bass-Heller-Swan

I'm reading the proof the Bass-Heller-Swan Theorem in Rosenberg's book Algebraic K-Theory and Applications (Theorem 3.2.22), which asserts $$K_1(R[t,t^{-1}]) \cong K_0(R) \oplus K_1(R) \oplus NK^+_1(R)...
Martin Brandenburg's user avatar
10 votes
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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
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152 views

Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...
Matthias Wendt's user avatar
10 votes
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332 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
sphere's user avatar
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Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first: Given an integral ...
Marty's user avatar
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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
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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
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9 votes
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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
9 votes
0 answers
594 views

Relative Chow groups

Most cohomologies have the notion of cohomology with support on a closed subspace, and also cohomology with compact support. In general, for any morphism $f\colon Y\to Z$ the inverse image fits into a ...
Tintin's user avatar
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9 votes
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939 views

Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...
Anandam Banerjee's user avatar
9 votes
0 answers
724 views

When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...
Dmitry Vaintrob's user avatar
9 votes
0 answers
411 views

Geometrizing the Third Cohomology of a Complex Lie Group

If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\...
Daniel Litt's user avatar
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8 votes
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111 views

The homotopy inverse on Quillen's $S^{-1}S$ construction

Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by ...
Georg Lehner's user avatar
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8 votes
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408 views

$K$-theory of $D$-modules

I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects: According to this page 23, a different $t$-structure on $D^b(\text{...
user127776's user avatar
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8 votes
0 answers
373 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) \...
Jinhyun Park's user avatar
8 votes
0 answers
414 views

Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
Luc Guyot's user avatar
  • 7,353
8 votes
0 answers
299 views

How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
Andrei Smolensky's user avatar
7 votes
0 answers
201 views

Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?

In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
Connor Malin's user avatar
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7 votes
0 answers
247 views

Homotopy invariant analogues of localizing invariants

Given a localizing invariant, $E$, valued in spectra, by following the recipe prescribed in 3.13 of https://arxiv.org/abs/1808.05559, we can define a homotopy-invariant version of $E$ on $H\mathbb{Z}$-...
Liam Keenan'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
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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7 votes
0 answers
784 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 ...
John Berman's user avatar
7 votes
0 answers
174 views

Conceptual proof of a theorem of Bloch on $K_2$ of Artinian $\mathbb Q$-algebras

Recall the following theorem of S. Bloch from his paper ($K_2$ of Artinian $\mathbb Q$-algebras, with applications to algebraic cycles, 1975): For any local $\mathbb Q$-algebra $B$ and an augmented $B$...
guest's user avatar
  • 528
7 votes
0 answers
275 views

Is there a derived geometric interpretation of morse functions?

Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\...
54321user's user avatar
  • 1,706
7 votes
0 answers
201 views

Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
Dmitrii Korshunov's user avatar
7 votes
0 answers
320 views

Split exact categories arising naturally

If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...
Tom Harris's user avatar
7 votes
0 answers
182 views

Torsion in Whitehead group

Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
W. Politarczyk's user avatar
7 votes
1 answer
506 views

When is $GL_m(R)$ generated by elementary and diagonal matrices?

Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary ...
Sam Williams's user avatar
6 votes
0 answers
139 views

$K_0$ of arithmetic surfaces

In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
Daniel Schäppi's user avatar
6 votes
0 answers
360 views

Subgroup of algebraic $K$-theory generated by split vector bundles

Is there any description of a subgroup of the algebraic $K$-groups of a curve that its generators lie in the subcategory that its objects are direct sums of $\mathcal{O}(n)$'s (for possibly different $...
user127776's user avatar
  • 5,831
6 votes
0 answers
176 views

Abelian localisation for K theory?

Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like $$\text{id}...
Pulcinella's user avatar
  • 5,506
6 votes
0 answers
100 views

$K_0$ of configuration of hyperplanes

Let $\ell_n$ where $n\geq 3$ be the configuration of $n$ lines in a plane, such that $n-1$ of them pass through a single point and the last one does not and it intersects rest of the $n-1$ lines. I'm ...
user127776's user avatar
  • 5,831
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
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
  • 5,831
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
6 votes
0 answers
365 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 "...
Mikhail Bondarko's user avatar