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

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### Faltings-Riemann-Roch Theorem

I found the famous Faltings book ``Lectures on arithmetic Riemann-Roch theorem".
In the book, very analytic techniques such as Garding inequality or heat kernel are explained. I have no idea where ...

**4**

votes

**1**answer

208 views

### Codimension zero embeddings and diffeomorphism groups

Let $V$ be a smooth manifold obtained by attaching the ``open collar'' $[0,1)\times \partial N$ to a compact smooth manifold $N$ along the boundary. Let $\mathrm{Emb}(N, V)$ be the space of embeddings ...

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132 views

### Comparison of products in Quillen and Waldhausen K-theory

I'm relatively new to algebraic K-theory and stumbled upon the following question. I would be very glad If someone could provide a reference to an answer or a short argument.
We are given an exact ...

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**1**answer

238 views

### Can any suspension spectrum be realized as Waldhausen K-theory?

If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the ...

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65 views

### Do GE rings have matrix completion?

If $R$ is a ring, $E_n(R)$ is the subgroup of the group $GL_n(R)$ generated by matrices obtained from the multiplicative identity matrix by replacing an off-diagonal entry by some $r \in R$. The ...

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202 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 ...

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**1**answer

240 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?

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**1**answer

415 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 ...

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**1**answer

61 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 ...

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337 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 ...

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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 ...

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133 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?

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191 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 ...

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**1**answer

350 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 ...

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**1**answer

150 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 ...

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**1**answer

483 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 ...

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453 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" !

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**1**answer

240 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 ...

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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 ...

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641 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 ...

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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 ...

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435 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 ...

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429 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 ...

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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 ...

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125 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 ...

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**1**answer

514 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
$K_2(R) \to K_2(R,I) \to K_2(R/I) \to K_1(R) \to K_1(R,I) \to K_1(R/I)$
$\to K_0(R) \to K_0(R,I) \to K_0(R/I)$
...

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316 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 ...

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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 ...

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221 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 ...

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228 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 ...

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

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957 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) = ...

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411 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 ...

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502 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 ...

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304 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 ...

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150 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 ...

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199 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 $n\geq 2$. ...

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404 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 ...

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**1**answer

377 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 ...

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**1**answer

918 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 ...

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575 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
...

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957 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 $(\ ,\ ...

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381 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 ...

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**1**answer

450 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 ...

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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 ...

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### 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 ...

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379 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 ...

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### 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 ...

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### 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 ...

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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 ...