Questions tagged [algebraic-k-theory]
The algebraic-k-theory tag has no usage guidance.
498
questions
13
votes
2
answers
523
views
"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 $G$...
0
votes
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 $k[x,z,w]/(zw,w^2,xw-z^2)$,...
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 ...
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, ...
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}))\...
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
$$\varinjlim_{k\subset E \subset F} ...
3
votes
1
answer
217
views
(Nonconnective algebraic) K-theory cohomology = K-theory of cohomology?
Situation
Suppose that we have
a commutative ring (or an $E_{\infty}$-ring) $R$ and
a homotopy type $X$.
Then we get a canonical morphism
$$
f \colon K(R ^ {\Sigma ^ \infty X_+}) \to K(R) ^ {\Sigma ^...
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 ...
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 ...
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 ...
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 ...
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 ...
0
votes
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 ...
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}\...
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}...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\mathbb{Z}[G]$ ...
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 $S$.
...
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 ...
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 ...
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 ...
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:
$$H_{cont}^i(X,\mathbb{Q}_l(n))=H_{Zar}^{...
4
votes
0
answers
132
views
Borel-Moore variant of the Lichtenbaum conjecture
A conjecture of Lichtenbaum expects that for a smooth proper variety $X$ over a finite field, the etale motivic cohomology groups $H^i(X_{et}, \mathbb{Z}(n))$ are finite for $i\neq 2n, 2n+2$, finitely ...
2
votes
0
answers
64
views
Recovering Milnor K-theory of a field extension and $l$-divisible elements in the Milnor $K$-theory
I have a couple questions regarding Milnor K-theory.
Given a field $k$ of char $p$, let $k'$ be an Artin-Schreier field extension of $k$. Let's say we know all $K_i^M(k)$, can way recover $K_i^M(k')$?...
2
votes
0
answers
106
views
$l$-adic rigidity for Milnor $K$-theory
Given a local henselian ring $A$ with the maximal ideal $m$, does the quotient map $A\mapsto A/m$ induce isomorphisms on $l$-adic Milnor $K$-theories? ($K_n^M(R)\otimes \mathbb{Z}_l$, where $l$ is an ...
5
votes
0
answers
166
views
Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra
By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...
3
votes
0
answers
141
views
Comparing $K$-cohomology groups and weight filtration on the $K$-groups
The second page of the Quillen-Brown-Gersten is in the following form:
$$E_2^{p,q}=H^{p}(X, \mathcal{K}_{-q})\Rightarrow K_{-q-p}(X)$$
Here $\mathcal{K}_n$ is sheafification of the $U\mapsto K_n(U)$ ...
2
votes
1
answer
118
views
Does the inclusion functor induce an injection in this case?
Notations :
$R$ is a commutative ring with unity. $P(R)$ is the category of finitely generated projective $R-$ modules, $Ch^{b}(P(R))$ is the the category of bounded chain complexes on $P(R)$ and $C^q(...
8
votes
0
answers
408
views
$K$-theory of $D$-modules
I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects:
According to this page 23, a different $t$-structure on $D^b(\text{...
2
votes
0
answers
177
views
Lefschetz type theorems/conjectures for algebraic $K$-theory
Lefschetz hyperplane theorem, compares the homology/cohomology of a projective variety with a hyperplane section of it and claims they are isomorphic in certain ranges. There are Lefschetz type ...
2
votes
0
answers
243
views
Zariski descent of algebraic $K$-theory on formal schemes
This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. ...
12
votes
0
answers
562
views
Global version of Gabber's rigidity theorem
I had a question regarding Gabber's rigidity.
Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
1
vote
0
answers
161
views
Descent of rational algebraic $K$-theory with respect to a special type of blow up
An abstract blow-up square is the following square:
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\...
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-...
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 ...
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 ...
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 ...
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 ...