# Questions tagged [algebraic-k-theory]

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

490
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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...

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

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 ...

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### 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 $K_0(R)$ ...

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### 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 $\...

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### 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 ...

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### $\mathbb{A}^1$-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 ...

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### 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 $\mathbb F_q$...

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### 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 ...

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### 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{...

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154
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### 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 ...

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### 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 ...

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### 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 \...

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### 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 $D$...

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### 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 ...

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### 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}^\...

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### 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
$$\...

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### 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,-...

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### 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 ...

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### 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,...

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### 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 $G$ ...

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### 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 ...

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### $K_1(k[x]/(x^2))$ 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)...

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### Grothendieck group of coconnective dg-algebra

Is there an example of an $E_{2}$-coconnective differential graded algebra $A$ (with unit) such that $K_{0}(A)=0$ ?

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### 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)...

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### 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 $p$ ...

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### 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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### 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, ...

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### 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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### 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 ...

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### 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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### 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 ...

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### 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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### 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, ...

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### 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 ...

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### $K_1(\mathbb{Z}_4)$ 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 ...

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### 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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### "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 $G$...

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### 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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### 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 ...

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### 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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### 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}))\...

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### 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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### (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 ^...