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104 votes
10 answers
24k views

Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
57 votes
2 answers
7k views

What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
David White's user avatar
  • 30.3k
38 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
37 votes
6 answers
6k views

Why is Milnor K-theory not ad hoc?

When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is ...
user717's user avatar
  • 5,243
37 votes
1 answer
3k views

Morava on Shafarevich conjecture

$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture. The Shafarevich Conjecture states: ...
Romeo's user avatar
  • 2,734
34 votes
1 answer
2k views

Is every ''group-completion'' map an acyclic map?

I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the Quillen plus construction. I have seen outstanding experts being confused ...
Johannes Ebert's user avatar
31 votes
1 answer
2k views

A modern interpretation of Quillen's computation of the K theory of finite fields

In his beautiful paper On the cohomology and K theory of the general linear group over a finite field, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory ...
Dmitry Vaintrob's user avatar
26 votes
3 answers
3k views

Does Milnor K-Theory arise from Waldhausen K-Theory

Quillens higher K-groups of rings can be realized as πnK(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from ...
user2146's user avatar
  • 1,273
26 votes
1 answer
1k views

Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$? A motivation is ...
YCor's user avatar
  • 63.9k
24 votes
3 answers
4k views

Plus construction considerations.

In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction: Recall that $K_1(R) = GL(R)/E(...
Joshua Seaton's user avatar
23 votes
0 answers
647 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
2 answers
702 views

Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$? Here, by "trivial examples" ...
user avatar
18 votes
5 answers
4k views

"Must read" papers in algebraic K-theory?

I'm mainly interested (graduate student) in surgery theory and geometric topology. If I have a chance to suggest "must read" papers in geometric topology for beginner, I'm very glad to suggest "...
17 votes
2 answers
2k views

Who first noticed that the Hilbert symbol is a Steinberg symbol ?

Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula $$ \prod_v(a,b)_v=1 $$ for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ ...
Chandan Singh Dalawat's user avatar
16 votes
2 answers
2k views

Can anyone explain to me what is an assembly map?

Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide ...
Xiaolei Wu's user avatar
  • 1,598
16 votes
3 answers
797 views

For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?

By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $...
Timothée Marquis's user avatar
16 votes
0 answers
603 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
  • 1,613
15 votes
1 answer
454 views

Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to K_3^{\operatorname{...
Matthias Wendt's user avatar
15 votes
0 answers
402 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
  • 42.4k
14 votes
2 answers
2k views

Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...
Stefan Witzel's user avatar
14 votes
1 answer
3k 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
14 votes
2 answers
734 views

A looping of algebraic K-theory

Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K_0(\mathcal{C})$. ...
Pavel Safronov's user avatar
14 votes
2 answers
2k views

Motivic cohomology and cohomology of Milnor K-theory sheaf

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$). ...
David Loeffler's user avatar
14 votes
1 answer
800 views

Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...
John Pardon's user avatar
  • 18.7k
14 votes
1 answer
1k views

Simplest example of failure of finite Galois descent in algebraic $K$-theory?

Let $E \to F$ be a $G$-Galois extension of fields. What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois ...
Akhil Mathew's user avatar
  • 25.6k
13 votes
1 answer
941 views

"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
13 votes
3 answers
552 views

f.g. modules vs. f.g. projective modules

In algebraic K-theory one defines $K_0(R)$ as the result of application of the Grothendieck construction to the semigroup of isomorphism classes of left f.g. projective $R$-modules. But we can also ...
W. Politarczyk's user avatar
13 votes
1 answer
700 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 ...
cll's user avatar
  • 2,305
12 votes
1 answer
2k views

Why does K-theory need schemes to be Noetherian?

The definition of K-theory of a scheme $X$ is defined as $G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$. But usually the schemes are required to be (at least locally) Noetherian, and ...
Li Guanyu's user avatar
  • 449
12 votes
2 answers
794 views

What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?

Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...
Dmitry Vaintrob's user avatar
12 votes
2 answers
1k 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, ...
Tim Campion's user avatar
  • 63.9k
12 votes
2 answers
657 views

Maps between K-groups induced by rings homomorphism

Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free ...
Hailong Dao's user avatar
  • 30.5k
12 votes
1 answer
312 views

Group ring with infinite stable rank

In searching for a counterexample in homological stability, I came across the following question: Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ ...
user124543's user avatar
12 votes
1 answer
795 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
  • 8,167
12 votes
1 answer
458 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
  • 433
12 votes
0 answers
410 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
  • 8,167
11 votes
3 answers
3k views

Is there a simple relationship between K-theory and Galois theory?

I can (barely) understand the definition of the higher algebraic K-groups a la the plus construction right now (I have some past familiarity with K-theory for C*-algebras and can recall the rudiments ...
Steve Huntsman's user avatar
11 votes
2 answers
864 views

Solving polynomial equations in spectra?

Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
Tim Campion's user avatar
  • 63.9k
11 votes
2 answers
1k views

Good reference for topological Hochschild homology

I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC). I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
Sunny's user avatar
  • 629
11 votes
1 answer
432 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
  • 38k
11 votes
1 answer
846 views

$K_0$ of a non-separated scheme

This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K_0(X)$ be the Grothendieck group of coherent ...
Ariyan Javanpeykar's user avatar
11 votes
1 answer
1k 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 ...
symmetry 's user avatar
11 votes
1 answer
429 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
  • 1,053
11 votes
1 answer
490 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 ...
Wenzhe's user avatar
  • 2,971
11 votes
0 answers
265 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
  • 63.9k
11 votes
0 answers
779 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
3 answers
3k views

The localisation long exact sequence in K-theory over an arbitrary base

If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory ... Kn+1(Dx) --> Kn(k) --...
Peter McNamara's user avatar
10 votes
2 answers
1k views

When do non-exact functors induce morphisms on $K$-theory?

Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...
Akhil Mathew's user avatar
  • 25.6k
10 votes
2 answers
2k views

When do the $\gamma$-filtration and codimension filtration of K-theory agree?

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by ...
David Loeffler's user avatar