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

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### 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 ...

**10**

votes

**1**answer

581 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 ...

**2**

votes

**1**answer

181 views

### Cube of cofibrations II

Let $\mathcal{C}$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(\mathcal{C})$ the category with cofibrations consisting of sequences of ...

**6**

votes

**1**answer

267 views

### About Tate's computation of $K_2^{\rm M}(\mathbb Q)$

For any field $F$, there is a natural group homomorphism $K_n^{\rm M}(F) \to K_n(F)$ from Milnor's $K$-theory to Quillen's $K$-theory. If $n=2$, it is an isomorphism, by Matsumoto's theorem. It is a ...

**8**

votes

**2**answers

371 views

### whitehead group of product of groups

I am wondering is there a formula for the whitehead group of product of groups. In other words, if we know the whitehead group of two groups, are we able to calculate the whitehead group of their ...

**4**

votes

**0**answers

267 views

### where to learn K-group of coherent sheaves modulo numerical equivalence?

I am trying to emerge from my complete ignorance about intersection theory.
I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From ...

**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 ...

**2**

votes

**0**answers

918 views

### Online Number Theory Video?

Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses http://www.math.upenn.edu/~ted/noframes.html, ...

**7**

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**1**answer

370 views

### The K-theoretic Farrell-Jones conjecture for cat(0) groups

Is the fibered K-theoretic farrell-jones conjecture true for cat(0)-groups?

**6**

votes

**1**answer

331 views

### When is a cube of cofibrations are “lattice”?

Let $C$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(C)$ the category with cofibrations consisting of sequences of $n$ cofibrations ...

**10**

votes

**1**answer

415 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 ...

**4**

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**0**answers

279 views

### Vanishing of Nisnevich cohomology of K-theory over a one-dimensional local ring.

Let $A$ be a one-dimensional, Noetherian, local ring, and let $\mathcal{K}_n$ denote the sheafification of $K_n$ (degree $n$ $K$-theory) on the Nisnevich site of $X:=\mbox{Spec }A$. Then is it true ...

**4**

votes

**1**answer

295 views

### A Reference on Multicategories with “Internal Hom”

The multicategory of Waldhausen categories is "enriched over itself": the Hom-set of $k$-exact functors can be given a Waldhausen category structure by letting the morphisms be natural ...

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**0**answers

285 views

### A modified version of K-theory for manifolds ?

If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...

**8**

votes

**1**answer

415 views

### Nonnegative additive functions on coherent sheaves

Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...

**3**

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**0**answers

351 views

### K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...

**5**

votes

**1**answer

357 views

### K-theory and regular rings

Let $R$ be a noetherian (commutative) ring. It is a well-known fact that for $R$ regular, $K$-theory of (finitely generated) projective modules and $K$-theory of arbitrary (f.g.) modules agree. Does ...

**15**

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**1**answer

453 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 ...

**9**

votes

**2**answers

433 views

### Genus of smooth varieties with small Chow group

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = ...

**9**

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**1**answer

704 views

### Current status of a conjecture of Bloch

In the seminal paper $K_2$ and algebraic cycles, Bloch make the following conjecture :
Suppose $A$ is a local Noetherian integral domain with quotient field $F$
$K_2(A)$ → $K_2(F)$ is ...

**2**

votes

**1**answer

247 views

### Gersten for homotopy invariant K-theory of non-singular varieties.

Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?
If $X$ is ...

**8**

votes

**2**answers

358 views

### Algorithm to calculate $Wh(G)$ for finitely presented group $G$?

Let $G$ be a finitely presented group.
Are there any algorithm to calculate whitehead group $G$, $Wh(G)$ in terms of presentation of $G$?

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vote

**1**answer

587 views

### Nerves of simplicial objects in categories/Waldhausen's S-construction

Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?
To give this some context, I'd ...

**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 ...

**11**

votes

**1**answer

875 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 ...

**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

806 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 ...

**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 ...

**8**

votes

**1**answer

669 views

### Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.
When is
$$
...

**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 ...

**5**

votes

**2**answers

374 views

### Uses of the Chern--Connes Pairing?

The backbone of Connes' approach to noncommutative geometry is the Chern--Connes pairing. By discovering the cyclic homology of an algebra and then pairing it the $K$-theory of that algebra, Connes ...

**4**

votes

**1**answer

236 views

### A detail in the construction of the coarse index of a Dirac operator in “Roe: An Index Theorem on Open Manifold, I”

Hi,
I'm currently wreading "Roe: An Index Theorem on Open Manifolds, I, J. Differential Geometry 27 (1988), p. 87-113" and there is a detail in the construction of the coarse index of a Dirac ...

**10**

votes

**1**answer

1k 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 ...

**4**

votes

**1**answer

276 views

### A question on equivariant K-theory

Let $G$ be a connected linear algebraic group over $\mathbb{C}$ and $X$ a $G$-variety. Let $K_G^0(X)$ be the Grothedieck group of coherent $G$-equivariant sheaves on $X$, and $K_G^i(X)$ for $i>0$ ...

**6**

votes

**1**answer

1k views

### Atiyah class for non-locally free sheaf

Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$.
The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the ...

**12**

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 ...

**6**

votes

**2**answers

468 views

### Why does the map $BG\to A(*)$ fail to split?

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is
Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus ...

**4**

votes

**1**answer

292 views

### Adams graded parts of rational K-theory of a number field.

Let $F$ be a number field and $r_{1}$ and $r_{2}$ the numbers of real and pairs of complex embeddings respectively of $F$. Then Borel computed that for $n\geq 2$
$$
K_{n}(F)_{\mathbb{Q}}\simeq
...

**4**

votes

**1**answer

380 views

### Is there a clean definition of the residue map in Milnor K-theory?

If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...

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**0**answers

443 views

### Algebraic description of double vector bundles.

It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...

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votes

**1**answer

1k views

### Geometric Realization of a Simplicial Category

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, ...

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**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 ...

**3**

votes

**1**answer

307 views

### Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...

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**0**answers

277 views

### homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...

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**0**answers

448 views

### Inverse Galois Problem…and parallelizable vector fields?

Usual approaches to the inverse Galois problem start with realizations of a group $G$ over a larger field, and then try to specialize to ${\Bbb Q}$.
One could also start by building suitable objects ...

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votes

**1**answer

308 views

### Algebraic K-groups and braids

This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?

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votes

**3**answers

757 views

### Group rings of infinite products of groups

Given a infinite family of groups $(G_i)$ for $i\in I$. Is there a ring theoretic construction, that produces $R[\prod_{i\in I} G_i]$ using only the rings $(R[G_i])_{i\in I}$ ?
For the case of a ...

**11**

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 ...

**2**

votes

**1**answer

1k views

### Z/48 and Moonshine Beyond the Monster

I am interested in pursuing an understanding of K-theory. Primarily, the
$K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the ...

**11**

votes

**1**answer

387 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 ...