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

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### Split exact categories arising naturally

If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...

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### What is the Atiyah-Bott-Shapiro map for a bundle of *complex* quadratic forms?

In order to ask the question in the title more precisely, let me
recall some standard stuff introduced in [1; Atiyah, Bott, Shapiro].
Suppose $X$ is a compact CW complex and $V \to X$ is an oriented ...

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### Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...

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### Riemann-Roch without denominators?

The Riemann-Roch Theorem, the Grothendieck-Riemann-Roch Theorem , the Grothendieck-Hirzebruch-Riemann-Roch Theorem , all of them are well explained at Wikipedia .I would like to understand the meaning ...

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### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...

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### K-theory and completion

I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...

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### K-Theory and completion [duplicate]

I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the ...

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### Equivariant algebraic K-theory of affine space

Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...

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### Relative Gersten resolution for a flat projective morphism

I am reading two papers by Daniel Grayson: "Localization for flat modules in algebraic K-theory" and "Algebraic cycles and algebraic K-theory" and I am wondering if any recent advances in K-theory ...

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### Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...

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### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

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### on the Zariski sheafification of Quillen's K-theory

Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$.
The Bloch-Quillen formula says that ...

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### an elementary question on K-theory

Sorry for asking such an elementary question.
1) What is Quillen's $K_1$ of a (nice) scheme $X$?
If $X=Spec(k)$, I guess one gets $k^\times$, is that correct? What about the case of a curve $C$ ...

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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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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 $n\geq 2$. ...