The algebraic-k-theory tag has no usage guidance.

**64**

votes

**10**answers

11k 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 ...

**43**

votes

**6**answers

4k views

### Which of Quillen's Papers Should I read?

I just heard that Dan Quillen passed on. I am not familiar with his work
and want to celebrate his life by reading some of his papers. Which one(s?)
should I read?
I am an algebraic geometry who is ...

**39**

votes

**2**answers

2k 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 ...

**34**

votes

**6**answers

4k 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 ...

**30**

votes

**3**answers

2k views

### How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...

**26**

votes

**1**answer

2k views

### Why is Riemann-Roch for stacks so hard?

First some indication that it really is a difficult problem: Both Vistoli and Gillet in their classics on intersection theory on stacks remark that their should be a Riemann-Roch theorem for proper ...

**24**

votes

**1**answer

862 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. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...

**22**

votes

**3**answers

2k 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 ...

**21**

votes

**4**answers

2k views

### Motivation/interpretation for Quillen's Q-construction?

This question has been on my mind for a while. As I understand it, the Q-construction was the first definition for higher algebraic K-theory. Some details can be found here.
...

**21**

votes

**1**answer

1k views

### Morava on Shafarevich

Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: $Gal(\bar Q / Q_{cycl})$ is free. ...

**20**

votes

**4**answers

2k views

### Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...

**19**

votes

**1**answer

457 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 ...

**18**

votes

**1**answer

1k views

### What is a path in K-theory space?

In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement:
In brief, the current view is ...

**18**

votes

**1**answer

416 views

### Can topological cyclic homology compute Picard groups?

Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism
$$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$
where $Pic(\mathcal{O}_K)$ is the ...

**16**

votes

**1**answer

234 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" ...

**15**

votes

**3**answers

2k views

### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

**15**

votes

**7**answers

2k views

### universal cover of SL2(R): does it admit central extensions?

Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective action of ...

**15**

votes

**1**answer

435 views

### When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...

**15**

votes

**2**answers

901 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 $(\ ,\ ...

**15**

votes

**1**answer

422 views

### Is there any publication of Bombieri about the standard conjectures on algebraic cycles?

In "Standard conjectures of algebraic cycles" Grothendieck says:
"... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and ...

**14**

votes

**2**answers

2k views

### What's about “quantum modular forms”?

Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling!
Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...

**14**

votes

**1**answer

419 views

### Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...

**14**

votes

**1**answer

472 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**14**

votes

**1**answer

617 views

### Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?

In very short:
When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...

**13**

votes

**3**answers

1k views

### Can we decompose Diff(MxN)?

If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there anything we can say ...

**13**

votes

**3**answers

2k views

### Is Higher K-functor the derived functor of K0?

It might be a stupid question. I wonder whether the derived functor of functor K0 is Quillen Higher K-functor?
If not, is there any relationship between derived functor of K0(or satellites of ...

**13**

votes

**2**answers

1k 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 ...

**12**

votes

**3**answers

847 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) = ...

**12**

votes

**2**answers

1k views

### Why was it reasonable to ask what the higher K-groups are?

To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the ...

**12**

votes

**1**answer

784 views

### Rationalised K-theory of number fields

Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...

**12**

votes

**3**answers

990 views

### Stable Homology of arithmetic groups.

Suppose that F/Q is a number field.
Using automorphic forms, Borel computed the (R-) stable cohomology of SL_n(O_F), and as a ...

**12**

votes

**2**answers

459 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

**1**answer

215 views

### The multiplication on $THH$ of finite fields

Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...

**12**

votes

**0**answers

292 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 ...

**11**

votes

**4**answers

2k 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 ...

**11**

votes

**2**answers

585 views

### Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...

**11**

votes

**3**answers

928 views

### What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...

**11**

votes

**1**answer

496 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 ...

**11**

votes

**2**answers

317 views

### Bass's paper “Symplectic groups and modules”, used in proof of the congruence subgroup property for Sp

Let $R$ be the ring of integers in a number field. While studying the congruence subgroup property for $\text{Sp}_{2g}(R)$ in
Bass, H.; Milnor, J.; Serre, J.-P.
Solution of the congruence subgroup ...

**11**

votes

**1**answer

2k views

### Values of zeta at odd positive integers and Borel's computations

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$.
I always assumed this was well known. More precisely I thought this result ...

**11**

votes

**1**answer

375 views

### Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences

A nice property of $\mathbb P^n$ is:
Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.
(for example, any 2 curves in $\mathbb ...

**10**

votes

**3**answers

2k 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 ...

**10**

votes

**2**answers

1k views

### Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?

I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...

**10**

votes

**2**answers

696 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 ...

**10**

votes

**1**answer

978 views

### Explicit description of boundary map in algebraic K-theory

Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory,
$$K(k) \rightarrow K(A) \rightarrow K(F).$$
In Remark 5.17 of his ...

**10**

votes

**2**answers

413 views

### What is the Q-construction, metaphysically?

An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the ...

**10**

votes

**1**answer

401 views

### Eilenberg-Mazur swindle for higher K groups

The Eilenberg-Mazur swindle shows that the Grothendieck group of an additive category with countable coproducts is trivial. The strategy is to observe that any "Euler characteristic" $\chi$ on such a ...

**10**

votes

**1**answer

564 views

### K-Theory space of finite abelian groups

Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a ...

**10**

votes

**2**answers

315 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 $K_0(\mathcal{C})$.
...

**10**

votes

**1**answer

768 views

### When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims.
On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...