The algebraic-k-theory tag has no wiki summary.

**2**

votes

**1**answer

221 views

### Path components of a monoidal category form a monoid?

In Grayson's 'Higher Algebraic K-theory II', leading up to the categorical generalisation of the plus construction, he considers $\pi_0(S) = \pi_0(BS)$, where $S$ is a (small, symmetric) monoidal ...

**8**

votes

**1**answer

597 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 $\mathbf{Q}_p$).
...

**12**

votes

**3**answers

834 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) = ...

**4**

votes

**0**answers

292 views

### K-Theory and Tame Symbol

This might be a little bit spesific but here it goes. While reading a paper (Brauer-Manin pairing...) by Yamazaki, I encountered this definition.
Let $V$ be a variety. $y$ be a one dimensional point ...

**2**

votes

**1**answer

456 views

### Any abelian category as filtered colimit of categories of projective modules

Recently I have heard somewhere that any (edit: small) abelian category can be expressed as the colimit of categories of projective modules over some rings. The remark was that this is "basically just ...

**9**

votes

**0**answers

292 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 ...

**2**

votes

**1**answer

145 views

### For which local $R$ its K-theory mod l is isomorphic to the one of its residue field?

It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong ...

**6**

votes

**1**answer

167 views

### rationalized K-Theory of the group ring of finite cyclic groups

I am interested in calculating the rationalized algebraic K-Theory groups of the group ring of $\mathbb Z/n$, that is $K_i(\mathbb Z[\mathbb Z/n])\otimes \mathbb{Q}$ for any natural number $n\geq 2$. ...

**5**

votes

**2**answers

384 views

### Normal subgroups of $SL_2$ of a polynomial ring

What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction ...

**7**

votes

**1**answer

368 views

### Waldhausen Additivity in a More General Context

The following arose when I was thinking about a talk at the Midwest Topology Seminar:
Background
I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object ...

**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 ...

**5**

votes

**2**answers

497 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
...

**15**

votes

**2**answers

898 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 $(\ ,\ ...

**9**

votes

**2**answers

377 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 ...

**4**

votes

**1**answer

408 views

### K-theory, monoidal vs. exact

My question is somewhat related to this one. However I think it adds something new to the table so I decided to post it sperately.
There is a construction of K-theory for symmetric monoidal ...

**1**

vote

**1**answer

114 views

### Triviality of SK_0(Lambda) for Lambda an order in a group algebra over a $p$-adic field

CR refers to Methods of Representation Theory by Charles Curtis and Irving Reiner.
Let $F$ be a finite extension of $\mathbb{Q}_p$ with valuation ring $\mathcal{O}_F$.
Let $G$ be a finite group and ...

**2**

votes

**0**answers

164 views

### Are connected categories with pullbacks weakly contractible?

Quillen's Theorem A says that a functor between (small) categories $f:I\rightarrow J$ induces a weak equivalence of the nerves if for each $j\in J$ the comma category $f/j$ is weakly contractible. In ...

**5**

votes

**1**answer

349 views

### Milnor patching for schemes

Let $R_1,R_2$, and $S$ be commutative rings with maps $R_1,R_2 \to S$ and form the fiber product $R = R_1 \times_S R_2$. A well-known theorem of Milnor says that under certain assumptions the category ...

**2**

votes

**3**answers

637 views

### Classify matrices up to similarity over arbitrary (commutative) ring.

One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the ...

**2**

votes

**2**answers

499 views

### pull backs (and tensor product) in algebraic K-theory

In the context of algebraic (equivariant) K-theory (more specifically, in the context of Chriss and Ginzburg's book representation theory and complex geometry) I would like to know if I have the ...

**9**

votes

**0**answers

395 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 ...

**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 ...

**2**

votes

**1**answer

179 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

249 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 ...

**5**

votes

**2**answers

343 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

248 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

841 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**

votes

**1**answer

353 views

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

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

**5**

votes

**1**answer

323 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

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 ...

**4**

votes

**0**answers

252 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

280 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 ...

**0**

votes

**0**answers

271 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

409 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**

votes

**0**answers

337 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

342 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**

votes

**1**answer

431 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

410 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**

votes

**1**answer

682 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

243 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

351 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 $G$?

**1**

vote

**1**answer

563 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 ...

**10**

votes

**1**answer

757 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

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 ...

**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

633 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 ...