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

**11**

votes

**2**answers

325 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 ...

**1**

vote

**0**answers

162 views

### Computation of Quillen K-groups for mod R

The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the ...

**6**

votes

**0**answers

131 views

### Torsion in Whitehead group

Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?

**2**

votes

**0**answers

188 views

### Certain central extensions of simply connected simple algebraic groups

An offbeat question involving Milnor's $K_2$ has come up recently. Start with an algebraically closed field $F$ (perhaps required to be of characteristic 0). Let $G$ be a connected, simply connected ...

**3**

votes

**1**answer

341 views

### Finite tor dimension in Quillen's paper

Quillen gives the following projection formula his foundational paper on higher algebraic k-theory. (For simplicity, I assume all schemes are noetherian.)
Let $f: X \rightarrow Y$ be a proper map of ...

**2**

votes

**1**answer

147 views

### Can this reduced version of algebraic K-theory be identified with this direct limit?

Let $X$ be a connected smooth scheme over a field and let $VB(X)$ be the exact category of vector bundles (i.e. locally free $\mathcal{O}_X$-module whose rank is finite at every point) over $X$.
Let ...

**9**

votes

**1**answer

449 views

### Quasi-isomorphisms in exact categories

I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...

**5**

votes

**2**answers

437 views

### Is the algebraic Grothendieck group of a weighted projective space finitely generated ?

This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !

**3**

votes

**1**answer

226 views

### the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme

Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does ...

**11**

votes

**4**answers

1k views

### Algebraic K-theory and Homotopy Sheaves

Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...

**8**

votes

**2**answers

632 views

### Waldhausen $K$-theory for $G$-spaces

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?
Let $G$ be a finite group. We know that ...

**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 ...

**10**

votes

**2**answers

424 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 ...

**5**

votes

**2**answers

420 views

### K-theory of monoidal categories

I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions.
Consider an exact monoidal ...

**0**

votes

**0**answers

187 views

### A question about higher K-theory

Suppose $\mathcal{A,B,C}$ are additive categories, $\mathcal B$ is a subcategory of $\mathcal C$. Now let $F,G: \mathcal A\rightarrow\mathcal B$ be two additive functors. Suppose $F,G$ are naturally ...

**1**

vote

**1**answer

124 views

### homology of $B S^{-1} S$ computation in the proof that $+ = Q$

Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that
$BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$
In proving this, in Srinivas' algebraic ...

**3**

votes

**1**answer

504 views

### Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space

For an ideal $I \subset R$ with relative K-groups $K_i(R,I)$ we have an exact sequence
$K_2(R) \to K_2(R,I) \to K_2(R/I) \to K_1(R) \to K_1(R,I) \to K_1(R/I)$
$\to K_0(R) \to K_0(R,I) \to K_0(R/I)$
...

**8**

votes

**2**answers

310 views

### Categorical description of the second K-group

Let $\mathcal{P}$ be a (small) exact category. Without delving into any homotopy theory, we can provide characterisations of $K_0(\mathcal{P})$ and $K_1(\mathcal{P})$ as plain categorical ...

**3**

votes

**2**answers

328 views

### Morphisms between $K_0$

I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map
$$
f: \operatorname{K_0}(A) \to \operatorname{K_0}(B)
$$
is it ...

**1**

vote

**1**answer

219 views

### Simplicial sets from bisimplicial sets, and their realisations.

From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T_{pp} $, $\underline p \mapsto |\underline q \mapsto T_{pq} |$, and $\underline q \mapsto |\underline p ...

**2**

votes

**1**answer

224 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 ...

**9**

votes

**1**answer

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

**13**

votes

**3**answers

919 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) = ...

**6**

votes

**1**answer

386 views

### K-Theory and Tame Symbol

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

**2**

votes

**1**answer

484 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

303 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

148 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 ...

**7**

votes

**1**answer

193 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

397 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

376 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 ...

**25**

votes

**1**answer

899 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

544 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

938 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

379 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

443 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

176 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

368 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

646 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

597 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

415 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

575 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

180 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

264 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

364 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

262 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

891 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

364 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

325 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 ...