Questions tagged [algebraic-k-theory]
The algebraic-k-theory tag has no usage guidance.
503
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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-...
- 333
11
votes
2
answers
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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 ...
- 61.6k
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votes
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$K$-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 ...
- 5,851
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votes
2
answers
585
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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 ...
- 650
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votes
0
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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 ...
- 5,851
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1
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A noneffective descent datum: isomorphism not satisfying the cocycle condition
Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times_{S} S'$ and $S''' := S' \times_{S} S' \times_{S} S'$ with ...
- 1,987
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votes
1
answer
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$K_0(\mathsf{Nil}(R))$ when $R$ is a field
$\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel. There I found a very interesting category $\Nil(R)$, which consists of ...
user139827
6
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2
answers
801
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Idempotent completions in K-theory
I have a reference request on following comment I found in
nLab article on Karoubian categories & envelopes. It states:
The Karoubian envelope is also used in the construction of the
category of ...
- 6,000
12
votes
1
answer
479
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Homological stability and Waldhausen A-theory
$\DeclareMathOperator{\Diff}{Diff}$
From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational ...
- 5,201
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votes
0
answers
503
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$K$-theory of formal completion
Let $X_Z$ be the formal completion of $X$ along $Z$. Let's assume we are working in $char=p$. How does $K_i(Z)$ compare to $K_i(X_Z)$? You can assume everything is smooth and $Z$ is a prime divisor in ...
- 5,851
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259
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Algebraic $K$-theory of curves
The Quillen's proof of finite generation of algebraic $K$-groups of curves over finite fields has always been a mystery to me. I never understood why working with the Harder-Narasimhan filtration in ...
- 5,851
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votes
0
answers
360
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Subgroup of algebraic $K$-theory generated by split vector bundles
Is there any description of a subgroup of the algebraic $K$-groups of a curve that its generators lie in the subcategory that its objects are direct sums of $\mathcal{O}(n)$'s (for possibly different $...
- 5,851
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votes
1
answer
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Quillen, Merkurjev and Suslin results about K2 of a conic
Let $X$ be a conic without rational points over a field $F$ and $Q$ its associated quaternion algebra. The paper
https://www.math.ucla.edu/~merkurev/papers/residue.pdf
presents a proof of the ...
4
votes
1
answer
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Homotopy invariance of $K_0$
It is well-known that algebraic $K$-theory is $\mathbb{A}^1$-invariant for regular Noetherian schemes. The way this is proved is usually to prove that $K$-theory of coherent sheaves i.e. $G$-theory ...
- 5,851
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1
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What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$?
Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed ...
- 1,479
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votes
1
answer
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Constructing analog of loop space of algebraic $K$-theory at the level of varieties
I have a somewhat open-ended and vague question regarding algebraic $K$-groups. According to the fundamental theorem of algebraic $K$-theory for a regular and Noetherian ring $R$, we have $K_i(R[x,x^{-...
- 5,851
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votes
1
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Finite group such that $K_{-1} (\mathbb Z G)$ has non-trivial torsion
According to Carters Lower K-theory of finite groups for a finite group $G$ we have
$$ K_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z_2^s $$
where $s$ is the sum over all irreducible ...
- 1,993
2
votes
0
answers
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Homotopy invariant $K$-theory spectrum version vs space version
Let $K^H$ be the homotopy $K$-theory spectrum. It is defined as a colimit of the form $|\mathbb{K}(X\times \Delta^{\bullet})|$. Here $\Delta^{\bullet}$ is the co-simplicial scheme defined by the ...
- 5,851
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vote
1
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$0$-th Galois cohomology with topological Milnor K-groups coefficients
In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...
- 479
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0
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Are Milnor K-groups algebraic groups?
Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is,
$$
K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I,
$...
- 479
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0
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151
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G Theory Localization Sequence without "quasiseparated"
Let $U \subseteq X$ be an open and $Z := X \setminus U$ its closed complement. I want a sequence
$$G_0(Z) \to G_0(X) \to G_0(U) \to 0.$$
However $X, U$ are not quasiseparated and perhaps not even ...
- 1,084
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votes
0
answers
176
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Abelian localisation for K theory?
Let $X$ be a scheme acted upon by $\mathbf{G}_m$ and $K(X)=K_0(\text{Perf}X)$ the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like
$$\text{id}...
- 5,555
1
vote
0
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132
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Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
- 5,851
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1
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181
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A question on $SK_1$ of rings
Let $B$ be a commutative ring with unity and $B/nil(B):=B_{red}$, where $nil(B)$ is the nilradical of $B$. Is $SK_1(B)=SK_1(B_{red}) ?$ In particular, is it true when $B$ is an affine algebra over an ...
6
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answers
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$K_0$ of configuration of hyperplanes
Let $\ell_n$ where $n\geq 3$ be the configuration of $n$ lines in a plane, such that $n-1$ of them pass through a single point and the last one does not and it intersects rest of the $n-1$ lines. I'm ...
- 5,851
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vote
0
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$\mathbb{A}^1$-invariance of $K$-theory involving automorphisms
For a variety $X$ let's $K_0(X,\mathbb{G}_m^t)$ denote the Grothendieck group generated by vector bundles with $t$ commuting automorphisms on $X$. Subject to the relations coming from short exact ...
- 5,851
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2
answers
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Good reference for topological Hochschild homology
I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC).
I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
- 609
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votes
0
answers
123
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$K$-group of category of bounded chain complexes of Projective modules with finite length homologies
For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
- 1,199
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Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?
In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
- 5,201
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1
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Rational homotopy invariance of algebraic $K$-theory
Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K(...
- 18.6k
1
vote
1
answer
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Do limits in Waldhausen categories commute with ordinary limits?
Disclaimer : I asked this question on MSE, I have no answer and I think it's better to ask it here.
Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category.
On one ...
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1
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Induced map in K-theory by a "trivial" bimodule
Let $R$ be a ring (not necessary commutative) and let $P_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. ...
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1
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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 / ...
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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}[...
- 908
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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}$-...
- 532
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615
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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 ...
- 33.9k
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votes
1
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564
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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 ...
- 255
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0
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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 ...
- 1,229
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votes
1
answer
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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) ...
- 1,229
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0
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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 ...
- 221
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0
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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|_{...
- 5,851
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0
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363
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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 ...
- 40.2k
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votes
1
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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 ...
- 511
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0
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305
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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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0
answers
109
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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)\...
- 5,851
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votes
1
answer
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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 ...
- 5,851
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votes
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433
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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}(...
- 153
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votes
3
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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 ...
- 6,000
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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 ...
- 255
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0
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160
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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
$$...
- 262