Questions tagged [algebraic-k-theory]

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Plus construction on Simplicial Sets?

I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here. Write $\mathsf{sSet}$ for the category of simplicial sets and $...
wind's user avatar
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1 vote
0 answers
24 views

Coordinate transformation for 3-dimensional simplicial cone in $\mathbb{R}^3$

Let $k$ be an algebraically closed field and let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$.Let $X$ be the affine toric variety over $k$ associated to the cone $\sigma$, i.e. set $X$...
Boris's user avatar
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7 votes
0 answers
109 views

The homotopy inverse on Quillen's $S^{-1}S$ construction

Suppose $S$ is a symmetric monoidal groupoid. Take Quillen and Grayson's $S^{-1}S$-construction, which is a symmetric monoidal category with objects given by pairs $(m,n)$ and maps given by ...
Georg Lehner's user avatar
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2 votes
0 answers
114 views

Can K$_3$ of finite fields be related to Teichmüller cocycles?

This is sort of a blind shot, but... For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$. To simplify matters, let $R$ be a finite field $\mathbb ...
მამუკა ჯიბლაძე's user avatar
0 votes
1 answer
161 views

Does going-down theorem hold for local homomorphism of finite flat dimension?

Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down theorem hold for $f$? If yes, then by Theorem 15.1 in Matsumura’s ...
Boris's user avatar
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4 votes
1 answer
146 views

Does local homomorphism of finite flat dimension preserve Krull dimension?

Let $f:A\rightarrow B$ be a local homomorphism of Noetherian local rings, such that the $A$-module $B$ has finite flat dimension. Is it true that the Krull dimensions of $A$ and $B$ agree? If yes, ...
Boris's user avatar
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0 votes
0 answers
112 views

Is the BGQ spectral sequence functorial with respect to morphisms of finite Tor-dimension?

It is well known that the BGQ (Brown-Gersten-Quillen) spectral sequence for the G-theory of a Noetherian scheme of finite Krull-dimension is contravariant with respect to flat morphisms. My question ...
Boris's user avatar
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4 votes
1 answer
156 views

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
atinag's user avatar
  • 43
1 vote
0 answers
268 views

Presentation of Chevalley groups over Bezout domains

Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 ...
Timothée Marquis's user avatar
1 vote
0 answers
41 views

Extension of a cylinder functor on C to the S_n C

I was looking at Waldhausen's definition of a cylinder functor and reading his proof that a cylinder functor on $C$ induces cylinder functors on $S_n C$ for all $n$. It seems to me that he is using ...
Tanner Carawan's user avatar
0 votes
0 answers
76 views

How to compute the higher G-theory of the weighted projective space $\mathbb{P}(1,1,m)$ using Mayer-Vietoris sequence?

Let $k$ be an algebraically closed field of characteristic zero. Let $m$ be a positive integer and let $X$ be the weighted projective space $\mathbb{P}(1,1,m)$ over the field $k$.I am trying to ...
Boris's user avatar
  • 501
1 vote
0 answers
113 views

Computing $G$-theory for a 3-dimensional affine simplicial toric variety

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be the cone in $\mathbb{R}^3$ generated by $e_1,2e_1+e_2,e_1+2e_2+3e_3$. Then it is easy to check that $\sigma$ is a 3-...
Boris's user avatar
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2 votes
1 answer
153 views

$K_0((k[x]/(x^2))[y])$

Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
user443060's user avatar
4 votes
1 answer
172 views

When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$

Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
user443060's user avatar
4 votes
0 answers
85 views

How to describe a concrete generator of $\widetilde{K_0(\mathbb{Z}[C_{23}])} \cong \widetilde{K_0(\mathbb{Z}[\zeta_{23}])}$

Milnor (page 29, see below) gives an explicit proof that the zeroth $K$-theory of the group ring $\mathbb{Z}[C_p]$, where $C_p$ is the cyclic group of order $p$ with $p$ a prime agrees with $K_0$ of $\...
Georg Lehner's user avatar
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1 vote
1 answer
199 views

Norm/transfer functoriality of Bloch map on $K$-theory

I've seen and used the following map from the algebraic $K$-theory to the differential forms on a scheme $X$: $$ K_n(X) \to H^0(X,\Omega^n_X)$$ sending $K_1(X)\ni f\mapsto d\log f$, and extending to a ...
xir's user avatar
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2 votes
0 answers
167 views

$\mathbb{A}^1$-invariance and cdh descent

It is known that cdh-sheafification of algebraic $K$-theory coincides with homotopy $K$-theory. Although I haven't gone through the details of the proof, I was wondering whether there is a general set ...
user127776's user avatar
  • 5,821
5 votes
1 answer
298 views

Galois action on algebraic K-theory of finite fields

This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F_q$...
Andreas Thom's user avatar
  • 25.3k
5 votes
0 answers
97 views

Size of minimal generating set of ideal over Laurent polynomial ring

Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
William Thomas's user avatar
1 vote
2 answers
280 views

How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition?

I am starting to learn the $K$-theory of triangulated categories and is stuck with the following. Let $\mathcal{T}$ be a triangulated category having a semi-orthogonal decomposition $\langle \mathcal{...
Boris's user avatar
  • 501
0 votes
1 answer
163 views

How to compute the $G$-theory of the weighted projective space $\mathbb{P}(1,1,2)$?

Let $k$ be an algebraically closed field of characteristic zero. Let $\Sigma$ be the fan in $\mathbb{R}^2$ consisting of three cones, cone generated by $e_1,e_2$,cone generated by $e_2,-e_1-2e_2$ and ...
Boris's user avatar
  • 501
4 votes
1 answer
377 views

Can higher G-theory of Noetherian schemes be computed by derived categories?

Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K_0({\mathcal A})\cong K_0(D^{b}(\mathcal A))$. When we set $\mathcal A$ to ...
Boris's user avatar
  • 501
4 votes
0 answers
169 views

Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism

Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
wlad's user avatar
  • 4,792
1 vote
0 answers
115 views

Quiver representations and the standard matrix decompositions

Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form: $$M = A D B$$ where $D$...
wlad's user avatar
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1 vote
1 answer
133 views

Relation between $G_0(X)$ and $\mathrm{Cl}(X)$ for a normal variety $X$

Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$. I am trying to show that there is a surjective group homomorphism $G_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by ...
Boris's user avatar
  • 501
5 votes
0 answers
145 views

Questions about the $K$-theory of the algebraic standard Podleś sphere

Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
Branimir Ćaćić's user avatar
8 votes
1 answer
565 views

Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero?

I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by $$\...
Colin Aitken's user avatar
2 votes
0 answers
165 views

How to compute the $G$-theory of this simplicial toric surface?

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-...
Boris's user avatar
  • 501
2 votes
2 answers
289 views

How to compute the $G$-theory of the variety $\mathbb{P}^1\times\mathbb{P}^1$?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be the fiber product of two copies of $\mathbb{P}^1_k$ over the affine scheme $\operatorname{Spec}(k)$.I am trying to compute ...
Boris's user avatar
  • 501
3 votes
1 answer
129 views

How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?

$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...
Boris's user avatar
  • 501
3 votes
0 answers
102 views

Algebraic K-theory of a scheme with group action of a semidirect product

Given $A$ and $H$ linear algebraic groups over a field $k$ and a homomorphism $\phi\colon H \rightarrow \mathrm{Aut}(A)$ we can form the semi-direct product $G = A \rtimes_{\phi} H$. Suppose that $G$ ...
Simon Cooper's user avatar
2 votes
1 answer
319 views

How to compute the transfer maps for G-theory of Noetherian schemes

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $X$ be $\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of $X$ along the maximal ideal $I$ of $R$ generated by $x,xy,xy^2,xy^3$.I ...
Boris's user avatar
  • 501
1 vote
0 answers
138 views

$K_1(k[x]/(x^2))$ for a field $k$

$\DeclareMathOperator\GL{GL}$The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $\GL(R)$, where $\GL(R)$ is the union of the sequence $R^{ \times} = \GL_1(R)...
user443060's user avatar
5 votes
0 answers
150 views

Grothendieck group of coconnective dg-algebra

Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?
LGO's user avatar
  • 169
2 votes
1 answer
303 views

Grothendieck group of triangulated categories

Let $A$ be a full triangulated subcategory of $B$, $u:A\rightarrow B$ the corresponding embedding. Let $f:B\rightarrow A$ be a triangulated functor satisfying: $f\circ u = id$ Let $b \in B $, if $f(b)...
LGO's user avatar
  • 169
2 votes
0 answers
157 views

On the relative class number of a cyclotomic extension

Let $\Bbb Z[\zeta_p]$ denote the cyclotomic integers where $p$ is a prime and let $h_1 = h_1(p)$ denote its relative class number. Question: Is it known whether there are infinitely many primes $p$ ...
John Klein's user avatar
  • 18.6k
2 votes
1 answer
78 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 ...
LGO's user avatar
  • 169
2 votes
0 answers
75 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, ...
JeeheBo5's user avatar
1 vote
0 answers
110 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}...
Boris's user avatar
  • 501
2 votes
0 answers
130 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 ...
Reuben Stern's user avatar
3 votes
2 answers
360 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 ...
LGO's user avatar
  • 169
4 votes
1 answer
199 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 ...
Monsieur Periné's user avatar
4 votes
0 answers
116 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 ...
xir's user avatar
  • 1,964
1 vote
0 answers
258 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, ...
curious math guy's user avatar
4 votes
0 answers
125 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 ...
XYC's user avatar
  • 389
3 votes
1 answer
207 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 ...
user443060's user avatar
1 vote
1 answer
187 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 ...
Roxana's user avatar
  • 519
13 votes
2 answers
521 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 $G$...
John Klein's user avatar
  • 18.6k
0 votes
0 answers
168 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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