All Questions
Tagged with kt.k-theory-and-homology algebraic-k-theory
134 questions
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 ...
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 ...
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 ...
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: ...
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 ...
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 ...
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 ...
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 ...
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(...
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 ...
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" ...
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 $(\ ,\ ...
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 ...
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 $...
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}= ...
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{...
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 ...
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 ...
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 ...
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 $K_0(\mathcal{C})$.
...
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 $\mathbf{Q}_p$).
...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 $\mathbb{Z}[G]$ ...
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 $O_K$.
...
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 ...
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}...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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 ...
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)...
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) --...
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 $F: \mathcal{A} \to \mathcal{B}$ ...
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
...