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

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### Counterexamples to Kunneth formula in algebraic K-theory

Let $X$ be a smooth projective variety with an action of linear algebraic group $G$. Theorem 5.6.1 in Criss/Ginzburg (Representation Theory and Complex Geometry) lists a bunch of equivalent ...

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### Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...

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### 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 ...

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### Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...

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### When do non-exact functors induce morphisms on $K$-theory?

Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...

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### How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...

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### Chow ring of an algebraic group for another equivalence relation than rational

For $G$ a split algebraic group of arbitrary Dynkin typ, the Chow ring with rational equivalence and $\mathbb{Z}/p\mathbb{Z}$, for $p$ some torsion prime of $G$, is well known and will be denoted as
...

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### Simplest example of failure of finite Galois descent in algebraic $K$-theory?

Let $E \to F$ be a $G$-Galois extension of fields.
What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois ...

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### Geometric contractibility of noetherian rings

Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if
All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial.
There exist a non-trivial locally free sheaf of ...

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### Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture?
Is there some book (or expository article) about this conjecture?
Is there any connection between this conjecture and ...

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### Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...

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### K theory and derived categories

Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra".
As next step, I moved to ...

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### Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...

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### On non-vanishing of Milnor K-groups for infinite fields

It is well-known that for $n \geq 2$ and a finite field $k$, the Milnor $K$-group $K_n ^M (k)$ vanishes. I don't know who proved this first, but if curious, you may look at somewhere in Srinivas's ...

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### Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?

Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...

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### Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The ...

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### Milnor-Witt K-theory for finite fields

Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?

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### Is this a model for $K$-theory of a triangulated category?

The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...

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### Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...

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### Waldhausen's regular coherent groups: torsionfree non-examples and behaviour under taking products?

Waldhausen defined a group $G$ to be regular coherent, if for all regular noetherian rings $R$ the group algebra $RG$ is regular coherent. (see Waldhausen - Algebraic $K$-Theory of generalized free ...

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### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction),
as defined by Grayson in Higher algebraic K-theory: II (page 219),
takes as an ...

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### Algebraic cobordism (of a point) outside the geometric diagonal

This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2*,*}(\...

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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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### Reference: Relative cohomology of a morphism

Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence
$$
\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots
$$
where the ...

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### The saturation of Murray von Neumann relation

Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version:
Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $...

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### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

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### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann ...

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### research articles in topology/geometry [closed]

There is a saying "Do you read the masters?"
I want to read some basic papers in Topology/geometry...
I can not clearly state what is basic as of now...
My back ground includes course in
...

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### 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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### Plugging $1-x$ into Schur polynomials

I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...

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### Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...

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### Algebraic K theory, Karoubi completion and splitting

Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...

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### When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...

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### Waldhausen and Segal's delooping machinery

I was recently thinking about the proof of a theorem where Waldhausen compared the Segal's delooping machinery with his, in the case when the cofibration is splittable (sec.1.8 in 'Algebraic $K$-...

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### Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ?
I ...

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### Pullback along Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...

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### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset X$...

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### Goodwillie's notes from MSRI Lecture Series

Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...

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### $K_0$ an $KH_0$ of a normal crossing variety

Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...

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### 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 ...

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### An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...

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### Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...

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### Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?

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### What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...

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### Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it.
For those who have the book (or want to look it up via google books etc.), it's the ...

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### Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...

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### Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...

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### Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...

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### When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...

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### generalization of result on K_1 of $SL(n,R)$

Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te_{i,j}$ is equal to $SL(n,R)$.
My question is as follows: Instead of $SL(n,R)$ I look ...