Questions tagged [algebraic-k-theory]

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"Burnside ring" of the natural numbers and algebraic K-theory

The construction of the Burnside ring $A(G)$ of a group $G$ (usually, but not always, finite) is given by taking the Grothendieck group of the commutative semi-ring of isomorphism classes of finite $G$...
John Klein's user avatar
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0 answers
170 views

How to compute the $G$-theory groups of $k[x,y,z,w]/(xy-zw,yz-w^2,xw-z^2)$ for any field $k$

I am trying to compute the $G$-theory groups of the ring $R=k[x,y,z,w]/(xy-zw,yz-w^2,xw-z^2)$ for any field $k$. This is my progress so far: Note that $R/y$ is isomorphic to $k[x,z,w]/(zw,w^2,xw-z^2)$,...
Boris's user avatar
  • 501
7 votes
1 answer
339 views

How can I detect the homology image of a unipotent group in the general linear group?

Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements. Denote $\text{GL}_n(F)$ the general linear group in the usual sense and $U_n(F)$ the unipotent ...
XYC's user avatar
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9 votes
1 answer
413 views

Abelianization of $\mathrm{GL}_2(R)$

$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
qqqqqqw's user avatar
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4 votes
1 answer
263 views

The third homology stability of general linear groups over finite fields

Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\...
XYC's user avatar
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1 vote
0 answers
200 views

Motivic cohomology commutes with field extension

$\DeclareMathOperator\Cor{Cor}$Let $X$ be a smooth scheme over $k$ and $k \subset F$ a field extension. Let $X_F$ be the field extension of $X$. Then there is a map $$\varinjlim_{k\subset E \subset F} ...
XT Chen's user avatar
  • 1,064
3 votes
1 answer
217 views

(Nonconnective algebraic) K-theory cohomology = K-theory of cohomology?

Situation Suppose that we have a commutative ring (or an $E_{\infty}$-ring) $R$ and a homotopy type $X$. Then we get a canonical morphism $$ f \colon K(R ^ {\Sigma ^ \infty X_+}) \to K(R) ^ {\Sigma ^...
B. W.'s user avatar
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4 votes
0 answers
214 views

How to to understand the homology groups $H_*(\Omega_0^\infty S^\infty)$?

The original statement of the Barratt--Priddy theorem says there is an isomorphism of homology groups $$H_*(\Sigma_\infty)\cong H_*(\Omega_0^\infty S^\infty),$$ where $\Omega_0^\infty S^\infty$ is the ...
Chase's user avatar
  • 93
5 votes
1 answer
210 views

Computation of the torsion of K-groups related to elliptic curves

Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$. The $K_2$ group of $F$ may be described by elements in $F^\times ⊗_\mathbb{Z} F^\times$ quotiented by the ...
LeechLattice's user avatar
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8 votes
1 answer
516 views

Importance of third homology of $\operatorname{SL}_{2}$ over a field

$\DeclareMathOperator\SL{SL}$I am reading some papers about the third homology of linear groups. In particular for the $\SL_{2}$ over a field. Why is it important to study these homologies? I have ...
Liddo's user avatar
  • 259
4 votes
1 answer
159 views

Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$

I am trying to understand the assembly map $$\pi_i ((BC_2)_+ \wedge A( \ast )) \rightarrow A_i( BC_2 ) $$ in low degrees for the space $BC_2 = \mathbb R P^\infty$ in Waldhausen $A$-theory. I know we ...
Georg Lehner's user avatar
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2 votes
0 answers
99 views

Generalisations of Volodin's construction of algebraic K-theory

In a previous question I asked about uses of Volodin's construction of the algebraic K-theory of rings. Some of these are striking and it made me wonder whether those proofs can be extended. This ...
user124543's user avatar
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0 answers
173 views

Higher Chow group of complex field

It is well-known fact that there is an isomorphism $$ K_i(\mathbb{C})\simeq \left\{ \begin{array}{ll} \mathbb{Q}/\mathbb{Z} & \text{if } i:odd \\ 0 & \text{if }i:even \end{array} \right. $$ My ...
OOOOOO's user avatar
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5 votes
0 answers
219 views

Two Hattori-Stallings trace questions

$\DeclareMathOperator\THH{THH}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\map{map}\DeclareMathOperator\tr{tr}\DeclareMathOperator\HH{HH}\DeclareMathOperator\fib{fib}\DeclareMathOperator\id{id}\...
Maxime Ramzi's user avatar
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12 votes
0 answers
364 views

Can Quillen-Lichtenbaum recover Borel's computation?

Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
skupers's user avatar
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3 votes
0 answers
174 views

Finite generation of algebraic $K$-theory with finite coefficients

Given a smooth connected complex quasi-projective variety $X$, is it possible that $K_i(X, \mathbb{Z}/l)$ to be infinitely generated for $i>0$? I think Quillen-Lichtenbaum implies that above the ...
user127776's user avatar
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2 votes
0 answers
177 views

Existence of numerically trivial classes in the algebraic $K$-theory of a threefold with nontrivial Chern characters

This is a follow-up question to my previous post. Let $X$ be a complex smooth projective variety of dimension $d$. Let $K(X)$ denote the Grothendieck group of coherent sheaves on $X$. There is an ...
Yuhang Chen's user avatar
  • 1,099
62 votes
2 answers
4k views

Thomason's "open letter" to the mathematical community

In 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter explained ...
John Klein's user avatar
  • 18.6k
2 votes
0 answers
151 views

construction of $K_0$-group and Karoubian completion

Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most old fashioned way as the Grothendieck group of the set of isomorphism classes of its finitely generated projective $R$ modules, ...
user267839's user avatar
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2 votes
0 answers
189 views

pseudo-abelian category / Karoubian category in K-theory

A pseudo-abelian category or Karoubian category $\mathcal{C}$ is a preaditive category such that every idempotent morphism $i: A \to A$ in $\mathcal{C}$ has a kernel and consequently a cokernel as ...
user267839's user avatar
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3 votes
0 answers
104 views

Periodicity of algebraic $K$-theory in high enough degrees with finite coefficients

Given this it seems that higher algebraic $K$-theory and the etale one coincide in high enough degrees. The etale $K$-theory with finite coefficients is also Bott inverted $K$-theory, so it should be ...
user127776's user avatar
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4 votes
1 answer
234 views

Etale $K$ theory coincides with algebraic one in high enough degrees

I have seen the claim that Beilinson Lichtenbaum implies that higher algebraic $K$ groups coincides with etale ones integrally in high enough degrees. Is this statement accurate? What conditions are ...
user127776's user avatar
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2 votes
0 answers
103 views

When mod $l$ algebraic $K$-groups inject into the mod $l$ etale algebraic $K$-group?

I was wondering whether in general it is known that for an invertible prime $l$, the mod-$l$ algebraic $K$-group of a regular Noetherian scheme $X$ injects into the mod-$l$ etale $K$-groups? I just ...
user127776's user avatar
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2 votes
1 answer
184 views

When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?

This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ...
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3 votes
1 answer
259 views

Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups

This is a question about the answer in this other post: Symplectic group over integers and finite fields. In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
user avatar
12 votes
1 answer
384 views

Uses of Volodin's construction of algebraic K-theory

There is a construction of the algebraic K-theory groups $K_i(R)$ of a ring $R$ by Volodin. He gave an explicit construction of the plus-construction $BGL(R)^+$ as the quotient of the bar construction ...
user124543's user avatar
6 votes
1 answer
435 views

Algebraic K-theory and intersection theory (Bloch's formula)

It seems to be a well known fact that algebraic K-theory can be used to understand intersection theory, at least for varieties (or stacks!) over a field. A first glimpse of this result seems to be ...
Thiago's user avatar
  • 221
3 votes
0 answers
188 views

Generalization of conjectures involving Beilinson regulators

I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
user127776's user avatar
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12 votes
1 answer
306 views

Group ring with infinite stable rank

In searching for a counterexample in homological stability, I came across the following question: Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ ...
user124543's user avatar
3 votes
1 answer
274 views

Algebraic K-theory of a category containing all perfect complexes

Let $R$ be a ring and let $\mathcal{C}$ be the category of perfect $R$-complexes. Suppose that $$S=\bigoplus_{i=1}^{\infty}R$$ Let us define $\mathcal{D}$ the smallest thick category generated by $S$. ...
LGO's user avatar
  • 169
2 votes
0 answers
162 views

Understanding Sha through $K_2$

Consider the following setup, to keep things easy: let $F$ be a number field with ring of integers $A$. Let $E$ be an elliptic curve over $F$ with Neron model $N$ over $A$. Let $Sha(E)$ be the ...
Marty's user avatar
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4 votes
0 answers
286 views

Is there algebraic $K$-theory of a group independent of the base ring?

Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...
David Corwin's user avatar
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0 votes
0 answers
158 views

Pullback of algebraic $K$-theory along the surjection of abelian varieties

Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...
user127776's user avatar
  • 5,831
1 vote
0 answers
105 views

Divisible elements in the cohomology of Milnor $K$-theory

As a consequence of the strong Tate conjecture over finite fields one can deduce (see here proposition 8.20) for every smooth variety $X$ over a finite field: $$H_{cont}^i(X,\mathbb{Q}_l(n))=H_{Zar}^{...
user127776's user avatar
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4 votes
0 answers
132 views

Borel-Moore variant of the Lichtenbaum conjecture

A conjecture of Lichtenbaum expects that for a smooth proper variety $X$ over a finite field, the etale motivic cohomology groups $H^i(X_{et}, \mathbb{Z}(n))$ are finite for $i\neq 2n, 2n+2$, finitely ...
user127776's user avatar
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2 votes
0 answers
64 views

Recovering Milnor K-theory of a field extension and $l$-divisible elements in the Milnor $K$-theory

I have a couple questions regarding Milnor K-theory. Given a field $k$ of char $p$, let $k'$ be an Artin-Schreier field extension of $k$. Let's say we know all $K_i^M(k)$, can way recover $K_i^M(k')$?...
user127776's user avatar
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2 votes
0 answers
106 views

$l$-adic rigidity for Milnor $K$-theory

Given a local henselian ring $A$ with the maximal ideal $m$, does the quotient map $A\mapsto A/m$ induce isomorphisms on $l$-adic Milnor $K$-theories? ($K_n^M(R)\otimes \mathbb{Z}_l$, where $l$ is an ...
user127776's user avatar
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5 votes
0 answers
166 views

Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra

By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...
user127776's user avatar
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3 votes
0 answers
141 views

Comparing $K$-cohomology groups and weight filtration on the $K$-groups

The second page of the Quillen-Brown-Gersten is in the following form: $$E_2^{p,q}=H^{p}(X, \mathcal{K}_{-q})\Rightarrow K_{-q-p}(X)$$ Here $\mathcal{K}_n$ is sheafification of the $U\mapsto K_n(U)$ ...
user127776's user avatar
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2 votes
1 answer
118 views

Does the inclusion functor induce an injection in this case?

Notations : $R$ is a commutative ring with unity. $P(R)$ is the category of finitely generated projective $R-$ modules, $Ch^{b}(P(R))$ is the the category of bounded chain complexes on $P(R)$ and $C^q(...
user avatar
8 votes
0 answers
408 views

$K$-theory of $D$-modules

I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects: According to this page 23, a different $t$-structure on $D^b(\text{...
user127776's user avatar
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2 votes
0 answers
177 views

Lefschetz type theorems/conjectures for algebraic $K$-theory

Lefschetz hyperplane theorem, compares the homology/cohomology of a projective variety with a hyperplane section of it and claims they are isomorphic in certain ranges. There are Lefschetz type ...
user127776's user avatar
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2 votes
0 answers
243 views

Zariski descent of algebraic $K$-theory on formal schemes

This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. ...
user127776's user avatar
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12 votes
0 answers
562 views

Global version of Gabber's rigidity theorem

I had a question regarding Gabber's rigidity. Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
user127776's user avatar
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1 vote
0 answers
161 views

Descent of rational algebraic $K$-theory with respect to a special type of blow up

An abstract blow-up square is the following square: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\...
user127776's user avatar
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23 votes
1 answer
835 views

Do Schlichting's and Balmer's definitions of higher Witt groups of a scheme agree when 2 is inverted?

My question is whether the construction of higher Witt groups of a scheme in Schlichting's Hermitian K-theory of Exact Categories agrees with the definition in Balmer's chapter in the Handbook of K-...
Nati PT's user avatar
  • 333
11 votes
2 answers
817 views

Solving polynomial equations in spectra?

Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general ...
Tim Campion's user avatar
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3 votes
0 answers
113 views

$K$-theory with respect to two different choices of quasi-isomorphisms

This question is related to another question asked here. Let's assume we have an exact category $C$ that consists of specific vector bundles on a variety. Furthermore assume $C$ is idempotent complete ...
user127776's user avatar
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10 votes
2 answers
580 views

Stable rank one and corners of $C^\ast$-algebras

Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we ...
Julien's user avatar
  • 650
3 votes
0 answers
178 views

Cofinality theorem for derived categories

For a projective variety $X$ and an ample line bundle $L$ on it, we consider the family of line bundles $L^{\otimes i}$ for $i\in \mathbb{Z}$. Let $\mathfrak{C}$ be the category generated by the ...
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