The tag has no wiki summary.

learn more… | top users | synonyms

8
votes
0answers
174 views

Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first: Given an integral ...
1
vote
1answer
202 views

Additivity theorem for algebraic L-theory?

There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?
8
votes
1answer
386 views

Algebraic K-theory of odd-dimensional spheres

Let $A(X)$ denote the Waldhausen's algebraic K-theory of a space $X$, and let $n$ be odd. Are the rational homotopy groups of $A(S^n)$ known? Is the group $\pi_{2k}(A(S^n))$ finite for all ...
2
votes
1answer
60 views

Action of GL(2,O_k) on 1d subspaces of (O_k)^2

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k ...
11
votes
2answers
298 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
0answers
144 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
0answers
121 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
0answers
165 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
1answer
269 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
1answer
133 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 ...
7
votes
1answer
284 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 ...
3
votes
1answer
288 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
1answer
177 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 ...
8
votes
4answers
759 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
2answers
499 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 ...
34
votes
2answers
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
2answers
376 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 ...
4
votes
2answers
361 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
0answers
173 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
1answer
115 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
1answer
452 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
2answers
287 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
2answers
326 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
1answer
202 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
1answer
216 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
1answer
498 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 $\mathbf{Q}_p$). ...
10
votes
3answers
723 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
0answers
228 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
1answer
390 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
0answers
270 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
1answer
141 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
1answer
146 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
2answers
373 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 ...
6
votes
1answer
317 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 ...
21
votes
1answer
750 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
2answers
406 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 ...
14
votes
2answers
807 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
2answers
362 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 ...
3
votes
1answer
349 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
1answer
103 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
0answers
155 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
1answer
299 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
3answers
613 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
2answers
407 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
0answers
362 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
1answer
534 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
1answer
174 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
1answer
233 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
2answers
310 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
0answers
226 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 ...