Questions tagged [algebraic-k-theory]

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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, ...
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115 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 ...
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83 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 ...
4
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1answer
152 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 ...
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99 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 ...
2
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1answer
102 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 ...
2
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1answer
161 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 ...
10
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1answer
263 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
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1answer
231 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 ...
11
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1answer
222 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
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1answer
195 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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110 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 ...
4
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241 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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97 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 ...
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85 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}^{...
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46 views

$\mathsf{Nil}_0(R)$ is a $p$-group when $R$ is a $\mathbb{Z}_p$-algebra

$\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel, there the category $\Nil(R)$ is defined to have objects as pairs like $(P ...
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111 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 ...
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52 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 $K_i^M(k')$?...
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97 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 ...
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151 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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111 views

Algebraic K-theory of enveloping algebras and PBW-algebras - a reference request

Let for simplicity $k$ be a field of characteristic zero, let $A$ be a finitely generated $k$-algebra which is regular and let $\alpha: L\rightarrow \operatorname{Der}_k(A)$ be a Lie-Rinehart algebra (...
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108 views

Comparing $K$-cohomology groups and weight filtration on the $K$-groups

The second page of the Quillen-Brown-Gersten is in the following form: $$E_2^{p,q}=H^{p}(X, \mathcal{K}_{-q})\Rightarrow K_{-q-p}(X)$$ Here $\mathcal{K}_n$ is sheafification of the $U\mapsto K_n(U)$ ...
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1answer
81 views

Does the inclusion functor induce an injection in this case?

Notations : $R$ is a commutative ring with unity. $P(R)$ is the category of finitely generated projective $R-$ modules, $Ch^{b}(P(R))$ is the the category of bounded chain complexes on $P(R)$ and $C^q(...
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278 views

$K$-theory of $D$-modules

I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects: According to this page 23, a different $t$-structure on $D^b(\text{...
2
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120 views

Lefschetz type theorems/conjectures for algebraic $K$-theory

Lefschetz hyperplane theorem, compares the homology/cohomology of a projective variety with a hyperplane section of it and claims they are isomorphic in certain ranges. There are Lefschetz type ...
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162 views

Zariski descent of algebraic $K$-theory on formal schemes

This question is highly related to some other questions that I've previously asked, especially to this one. In this problem we have a scheme $X$ and a closed subscheme $Z$ the formal completion $X_Z$. ...
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421 views

Global version of Gabber's rigidity theorem

I had a question regarding Gabber's rigidity. Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
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121 views

Descent of rational algebraic $K$-theory with respect to a special type of blow up

An abstract blow-up square is the following square: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\...
23
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1answer
622 views

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-...
11
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2answers
667 views

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 ...
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106 views

$K$-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 ...
8
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2answers
269 views

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 ...
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166 views

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 ...
4
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1answer
217 views

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 ...
3
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1answer
162 views

$K_0(\mathsf{Nil}(R))$ 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 ...
6
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2answers
528 views

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 ...
12
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1answer
381 views

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 ...
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400 views

$K$-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 ...
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0answers
166 views

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
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0answers
330 views

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 $...
9
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1answer
224 views

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
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1answer
240 views

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 ...
4
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1answer
185 views

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 ...
2
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1answer
87 views

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^{-...
8
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1answer
207 views

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 ...
2
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0answers
116 views

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 ...
1
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1answer
94 views

$0$-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 ...
2
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0answers
189 views

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, $...
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0answers
119 views

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 ...
6
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0answers
163 views

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}...

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