Questions tagged [algebraic-k-theory]

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2 votes
1 answer
36 views

Induced map in k-theory by an involution

Let $T$ be a ring with involution $s:T\rightarrow T$. And let $$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$. suppose ...
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2 votes
0 answers
52 views

Derived category of a exact categories with (unusual) weak equivalences

Every exact category $\mathcal{E}$ has an attached derived category (for simplicity I will just refer to the bounded one) $D^b(\mathcal{E})$. The construction is for example explained in A. Neeman, ...
1 vote
0 answers
85 views

How to compute the G-theory groups of a blow-up of Noetherian schemes

Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...
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2 votes
0 answers
100 views

Does the category of stable infinity categories form a "subtractive Waldhausen" category?

In "The $K$-theory spectrum of varieties", Jonathan Campbell introduces the notion of a subtractive Waldhausen category, a slight generalization of the notion of Waldhausen category that ...
3 votes
2 answers
321 views

Involution map, and induced morphism in K-theory

Let $T$ be a ring with involution $s:T\rightarrow T$. And let $$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$. I was ...
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4 votes
1 answer
139 views

Higher Chow cycles

Recall that the higher Chow groups $CH^k(X,m)$ are defined as the homology of the complex $Z^k(X,\bullet)$, where $Z^k(X,m)$ is the subgroup of codimension $k$ cycles of $X\times \Delta^m$ which meet ...
4 votes
0 answers
108 views

When is the degree $(2,2)$ motivic cohomology generated by products of units?

The motivic coniveau spectral sequence tells us that for a scheme $X/k$, its cohomology $H^2(X,\mathbb{Z}(2))$ is the kernel of the tame symbol $K_2^M(k(X))\to \oplus_{Y} K_1^M(k(Y))$ where $Y$ runs ...
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1 vote
0 answers
154 views

Proof of Geisser-Levine

I am trying to understand the proof of the Geisser-Levine theorem (Thm 8.4 here ) which claims that for a smooth variety $X$ over a perfect field of characteristic $p$ we have an isomorphism $$H^s(X, ...
4 votes
0 answers
113 views

Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields

Let $F$ be a finite field with many elements, say more than 7 for example, and $X$ be the corresponding projective plane $\mathbb{P}^2(F)$. For a set of points in $X$, if any three of them are ...
3 votes
1 answer
179 views

$K_1(\mathbb{Z}_4)$ and $K_1(\mathbb{Z_4}[t])$

I am an amateur in $K$-theory, I have just started reading from "The K-book" by Charles Weibel. I have only read the definition of $K_1$ which is stated as a quotient of $GL(R)$. The union ...
1 vote
1 answer
156 views

Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...
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13 votes
2 answers
475 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 $G$...
  • 17.7k
0 votes
0 answers
64 views

How to compute $G_0(kG)$ for any field $k$

$\newcommand{\red}{\mathrm{red}}$I am trying to compute $G_0(kG)$ for any field $k$, where $G$ is the set of all monomials $x^iy^j$ such that the line through the origin and $(i,j)$ has slope between ...
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0 votes
0 answers
142 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 $k[x,z,w]/(zw,w^2,xw-z^2)$,...
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7 votes
1 answer
304 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
369 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, ...
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4 votes
1 answer
236 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
181 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} ...
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2 votes
1 answer
147 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 ^...
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4 votes
0 answers
176 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 ...
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5 votes
1 answer
171 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 ...
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8 votes
1 answer
454 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 ...
  • 249
4 votes
1 answer
140 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 ...
  • 1,583
2 votes
0 answers
93 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
153 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 ...
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4 votes
0 answers
165 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}\...
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11 votes
0 answers
299 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}...
  • 7,500
3 votes
0 answers
147 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 ...
  • 5,333
2 votes
0 answers
139 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 ...
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41 votes
0 answers
2k views

Thomason's "open letter" to the mathematical community

In the 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 ...
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2 votes
0 answers
128 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
152 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
96 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 ...
  • 5,333
4 votes
1 answer
200 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 ...
  • 5,333
2 votes
0 answers
102 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 ...
  • 5,333
2 votes
1 answer
129 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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2 votes
1 answer
187 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 ...
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12 votes
1 answer
345 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 ...
5 votes
1 answer
284 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 ...
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3 votes
0 answers
154 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 ...
  • 5,333
12 votes
1 answer
274 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
234 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 $S$. ...
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2 votes
0 answers
135 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 ...
  • 12.5k
4 votes
0 answers
254 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 ...
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0 votes
0 answers
117 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 ...
  • 5,333
1 vote
0 answers
91 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}^{...
  • 5,333
4 votes
0 answers
120 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 ...
  • 5,333
2 votes
0 answers
57 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 $K_i^M(k')$?...
  • 5,333
2 votes
0 answers
99 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 ...
  • 5,333
5 votes
0 answers
157 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 ...
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