Questions tagged [algebraic-k-theory]

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8
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0answers
390 views

Roadmap for Quillen 1

Question Suppose you grasped and enjoyed reading Quillen's "Higher Algebraic K-theory I". Now, if you could go back in time to when you started studying algebraic topology and create a reading list / ...
11
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1answer
291 views

Are projective modules over a certain localised Laurent polynomial ring free?

Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[...
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0answers
96 views

Homotopy invariant analogues of localizing invariants

Given a localizing invariant, $E$, valued in spectra, by following the recipe prescribed in 3.13 of https://arxiv.org/abs/1808.05559, we can define a homotopy-invariant version of $E$ on $H\mathbb{Z}$-...
10
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366 views

Does Merkurjev's argument help Voevodsky's program?

In the talk Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract) Voevodsky mentioned that he was ...
5
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1answer
301 views

Topological Hochschild homology using equivariant orthogonal spectra

In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of ...
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78 views

Coherent sheaf with big enough support is non-zero in K-theory

Let $X$ be a noetherian, separated, integral scheme of dimension $d < \infty$ and $\mathcal F \in \mbox{Coh}(X)$ coherent sheaf on $X$ with support $\operatorname{Supp} \mathcal F = X$. Is it true ...
5
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1answer
200 views

Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$

Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...
1
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124 views

Cofibration of non-equivariant spectra in Hesselholt-Madsen

In the Hesselholt-Madsen paper "On the K-Theory of finite algebras over Witt vectors of perfect fields", Proposition $2.1$ claims a cofibration sequence for non-equivariant $S^1$ spectra as follows. ...
11
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0answers
197 views

Criteria for a map of rings to induce an equivalence on K-theory?

Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...
3
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0answers
118 views

Stably deforming vector bundles

Let $X$ be a smooth projective variety. $V_1$ and $V_2$ are two vector bundles on $X\times \mathbb{A}^1$ such that $V_1|_{X\times \{0\}}\cong V_2|_{X\times \{0\}}$ and $V_1|_{X\times \{1\}}\cong V_2|_{...
12
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0answers
211 views

Dennis trace map for stable $\infty$-category, naively

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...
3
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1answer
110 views

Spherical objects and K-theory

My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the ...
4
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139 views

Beilinson regulator: a road map

I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...
4
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0answers
58 views

Filtrations of motivic spectral sequences

I had a general question about motivic spectral sequences. In order to derive them we first begin with a filtration of the algebraic $K$-theory spectra. Something like this $\cdots \rightarrow W^2(X)\...
2
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1answer
85 views

Grayson filtration and weight filtration

I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can ...
5
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1answer
176 views

Exact subcategory with trivial Grothendieck group: what are the consequences and examples

Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K_{0}(C)=0$, what can be said about the higher groups $K_{n}(...
2
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3answers
349 views

Motivation for Karoubi envelope/ idempotent completion

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...
5
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0answers
321 views

Modern context for hypercohomology spectra

In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...
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86 views

Multiplicative structure of the K-theory of Severi-Brauer varieties

There is a well-known result by Quillen stating that if $X_A$ is the Severi-Brauer variety of a central simple algebra $A$ of degree $d$ over a field $k$, then its (Quillen) K-theory decomposes as $$...
4
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1answer
187 views

Reference for computation of $K_8(\mathbb{Z})$

In an Oberwolfach report from 2016 [1, page 2] it is said that $K_8(\mathbb{Z})$ has recently been computed. Does anyone know a reference for the computation? [1] https://orbilu.uni.lu/bitstream/...
5
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1answer
133 views

Relation in Brauer group coming from trace form

Let $L/K$ be a cubic (or, more generally, odd-order) extension of fields of characteristic $0$. To every element $a \in L^\times$ we can associate the quadratic form \begin{align*} q_a : L &\to K \...
11
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1answer
1k views

Why does K-theory need schemes to be Noetherian?

The definition of K-theory of a scheme $X$ is defined as $G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$. But usually the schemes are required to be (at least locally) Noetherian, and ...
5
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0answers
258 views

Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)

I'm studying Stefan Bauer's paper The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...
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0answers
66 views

Direct sum $K$-theory localization

Direct sum $K$-theory of an exact category is defined as the homotopy groups of the $Q$-construction of the category of vector bundles, coherent sheaves and etc with short exact sequences claimed to ...
4
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0answers
192 views

K-theoretic derivation of Bézout theorem

In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says: "When the ground field $k = \mathbb C$, Bézout’...
11
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1answer
686 views

“a sign that one should be computing K-theory”

Allen Knutson said here in comments below the question that I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it. I know one ...
3
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0answers
146 views

$G_0(X) \cong G_0(X_{red})$ where X is a noetherian scheme

Let $\textbf {X}$ be a noetherian scheme, $\textbf {M(X)}$ be the categroy of coherent sheaf over the scheme $\textbf {X}$. We denote $ \textbf {K$_0$(M(X))}$ to be $ \textbf {G$_0$(X)}$. Now I ...
9
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1answer
379 views

Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
9
votes
1answer
231 views

Status of the extended form of the Lichtenbaum conjecture

The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$. ...
8
votes
1answer
285 views

$K_3(\mathbb{Z})$ and $\pi ^S_3$

This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the ...
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0answers
118 views

Injective envelope in the category of left exact functors

Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
4
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1answer
278 views

Question about an implication of Thomason's étale descent spectral sequence

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads $$H^p_{\acute{e}t}(X, \...
5
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0answers
90 views

Failure of devissage vs link topology in algebraic K-theory

This is somehow related to (or maybe a simplified version of) an earlier question (see here) regarding Gersten complexes for singular varieties. The Gersten complexes arise from the coniveau spectral ...
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0answers
690 views

Serre presentations over $\mathbb{Z}$

Given a Cartan matrix $A=(a_{ij})_{i,j\in I}$, a classical result of J.-P. Serre asserts that the complex semisimple Lie algebra $\mathfrak g=\mathfrak g(A)$ corresponding to $A$ admits a presentation ...
6
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1answer
216 views

Equivalence between categories of coherent sheaf of codimension p

Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \...
3
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0answers
207 views

Homotopy equivalence of $K$-theory and $G$-theory

Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
2
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0answers
192 views

Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers: 1) Is there any example of a smooth projective variety $X$ over a ...
7
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0answers
210 views

Adequate equivalence relations and algebraic $K$-theory

I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
10
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1answer
237 views

About $K$-rectification of increasing tableaux

Let $T$ be a standard Young tableaux on $[n]$. Denote the RSK algorithm $\text{RSK}(w)=(P(T),Q(T))$ for $w\in\mathfrak{S}_n$, where $P(T)$ is the Schencted insertion tableaux. For $1\leq i\leq j\leq ...
4
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0answers
122 views

On the Beilinson's conjecture regarding the proper flat integral models

I had a question which seems to be straightforward but I wasn't able to figure it out. In page 13 of this paper a conjecture of beilison is mentioned that if $\mathcal{X}_{\mathbb{Z}}$ is a proper ...
7
votes
1answer
368 views

Motivation for Suslin’s Rigidity Conjecture

Suslin Rigidity conjecture states that motivic cohomology $$ H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n)) $$ of the field $F$ coincides with motivic cohomology for the subfield of ...
8
votes
1answer
439 views

Original reference for Adams Riemann-Roch theorem

Let $f\colon Y\to X$ be a proper morphism between smooth quasiprojective $k$-algebraic varieties. Denote by $\psi^j$ the $j$-th Adams operation on the Grothendieck group of vector bundles and $\theta^...
6
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0answers
62 views

Elliptic deformation of the second Chern class

Second Chern class $$c_2 \in H^4(BGL,\mathbb{Q}(2))$$ admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
9
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1answer
381 views

Bass' conjecture implies the Parshin's conjecture

In the appendix of this paper. It is proved that Bass' conjecture for $K_n$ implies the rational Beilinson-Soulé conjecture for $K_n$. Then at the end the author claims that the same method can be ...
4
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0answers
239 views

Effects of the first algebraic K-theory on the higher algebraic K-theory

Is there any counterexamples known to the following statement? ($A$ a regular noetherian integral domain of finite Krull dimension) If $A^{\times}$ is finitely generated then $K_n(A)$ is finitely ...
5
votes
1answer
202 views

Kernel of the determinant morphism from the first algebraic K-theory

If $A$ is the coordinate ring of a smooth variety over a finite field is it known whether the kernel of the determinant map $K_1(A)\rightarrow A^{\times}$ is torsion or not?
6
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0answers
177 views

Adams operation on Q-construction of fields

Let $F$ be a field that we want to compute its rational algebraic $K$-theory using the Quillen's $Q$-construction. Let $QF$ be the $Q$ construction of the category of finite dimensional vector spaces ...
3
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0answers
94 views

Adams operation on the rational homology

The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...
4
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0answers
400 views

Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...
5
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0answers
222 views

making the group completion in homology sense unique via the plus construction

A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology. In the paper they prove that when $X$ is a ...

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