The algebraic-k-theory tag has no wiki summary.

**17**

votes

**1**answer

316 views

### Can topological cyclic homology compute Picard groups?

Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism
$$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$
where $Pic(\mathcal{O}_K)$ is the ...

**0**

votes

**0**answers

74 views

### $K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Under what conditions does the following exact sequence split?
$1\to E(R)\to Gl(R)\to K_1(R)\to 1$.

**3**

votes

**0**answers

126 views

### Equivalence of various definitions of arithmetic Chow groups

If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits_in_iZ_i, g)$ where $Z := \sum\limits_in_iZ_i$ ...

**2**

votes

**1**answer

203 views

### Map from algebraic K-theory to topological K-theory

Suppose that $A$ is a Banach algebra with unit. We can consider $GL(A)$ as a topological group in either the discrete topology or the topology that it inherits from the norm topology of $A$, and the ...

**1**

vote

**1**answer

181 views

### What is $K_2(\mathbb{Z}[x,x^{-1}])$?

The question is as in the title: is $K_2(\mathbb{Z}[x,x^{-1}])$ known?

**5**

votes

**0**answers

119 views

### Gersten Conjecture for Milnor K-theory

The Gersten conjecture for Milnor K-theory, saying that the Gersten complex $$0\rightarrow \mathcal{K}^M_X\rightarrow \oplus_{x\in X^0}i_{x*}(K^M_n(x))\rightarrow \oplus_{x\in ...

**10**

votes

**2**answers

658 views

### Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...

**0**

votes

**0**answers

83 views

### Central extensions of SL2(R) by U(1) ?

Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you

**1**

vote

**0**answers

134 views

### map in K-theory

I have a very stupid question on a map I have seen in K-theory.
The situation is as follows: $X$ is a smooth variety over a field $k$ and $\iota: Z \to X$ is the inclusion of a smooth closed ...

**5**

votes

**0**answers

196 views

### Beyond Bloch-Kato conjecture [repost from math.stackexchange]

I have just asked this question on math.stackexchange, and would like to repost it here:
The norm residue isomorphism theorem establishes that the norm residue map between Milnor K-theory of a field ...

**1**

vote

**0**answers

141 views

### The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...

**4**

votes

**0**answers

112 views

### residue and regulator

Let $C$ be a curve defined over $\mathbb{Q}$. The regulator is a map
$$
reg: K_2(C)_{\mathbb{Q}} \longrightarrow H^1(C(\mathbb{C}), \mathbb{R}).
$$ Here $K_2(C)_{\mathbb{Q}}$ is the K-group tensor ...

**3**

votes

**1**answer

99 views

### Stable rank of finite rings

Has any finite ring (not necessarily commutative) always stable rank 2 ? How do you prove that or does it follow from something ? May be this question is trivial but I'm not familiar with K-theory.

**10**

votes

**1**answer

464 views

### Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...

**13**

votes

**1**answer

462 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**6**

votes

**1**answer

353 views

### K-theory of the h-cobordism category

I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...

**-1**

votes

**1**answer

341 views

### Coherent sheaves on Proj

Roughly speaking , the question is : when a f.g. graded module induces a trivial coherent sheave on Proj ? More precisely, let S be a (complex) graded polynomial algebra, where the variables have ...

**5**

votes

**1**answer

280 views

### Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...

**3**

votes

**0**answers

71 views

### The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...

**0**

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**0**answers

93 views

### Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to ...

**1**

vote

**0**answers

72 views

### Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...

**1**

vote

**0**answers

156 views

### Surjectivity of the algebraic K-functor

Let $R \to S$ be a surjective morphism of commutative rings. For a fixed integer $q$, is there any known condition under which the resulting morphism of the K-groups, $K_q(R) \to K_q(S)$ is ...

**1**

vote

**1**answer

86 views

### Equivalence of definitions of the Milnor $K$-groups

In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
...

**9**

votes

**2**answers

420 views

### Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...

**4**

votes

**0**answers

112 views

### The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...

**1**

vote

**1**answer

300 views

### Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture?
Is there some book (or expository article) about this conjecture?
Is there any connection between this conjecture and ...

**6**

votes

**0**answers

78 views

### Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to ...

**6**

votes

**1**answer

334 views

### Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...

**2**

votes

**0**answers

167 views

### What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...

**5**

votes

**0**answers

138 views

### Strengthening of Suslin's rigidity argument?

To fix the situation, let $k$ be an algebraically closed field, and let $C$ be a smooth projective curve over $k$. Suslin's rigidity argument implies in particular that any class in ...

**7**

votes

**1**answer

179 views

### Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of
transfers for Bloch groups and scissors congruence groups/pre-Bloch
groups.
To fix notation and recall definitions:
From the ...

**0**

votes

**1**answer

229 views

### Articles about Weil cohomology theory by Grothendieck and Artin

In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...

**2**

votes

**1**answer

295 views

### Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves?

In "Standard conjectures of algebraic cycles" nLab says:
"... They were also followed by “Beilinson’s dream” on motivic (complexes of) sheaves which comprise so called standard conjectures of ...

**4**

votes

**1**answer

255 views

### why Borel's computation implies Beilinson-Soulé?

Let $k$ be a field of characteristic zero and $DM(k)_{\mathrm{Q}}$ Voevodsky's category of motives over $k$ with rational coefficients. The Beilinson-Soulé conjecture says
$$
...

**3**

votes

**3**answers

634 views

### Algebraic K-theory can be seen as a generalization of Linear algebra? [closed]

Algebraic K-theory can be seen as a generalization of Linear algebra?
If yes, how so?

**4**

votes

**1**answer

495 views

### A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...

**15**

votes

**1**answer

405 views

### Is there any publication of Bombieri about the standard conjectures on algebraic cycles?

In "Standard conjectures of algebraic cycles" Grothendieck says:
"... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and ...

**1**

vote

**0**answers

219 views

### Where can I find the article of A. Borel: “Values of zeta-functions at integers, cohomology and polylogarithms”? [closed]

Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.

**1**

vote

**1**answer

392 views

### Where can I find the article of Kleiman: “Algebraic cycles and Weil conjectures. Dix exposés sur la cohomologie des schémas”?

Where on the Internet can I find the article of Kleiman: "Algebraic cycles and Weil conjectures. Dix exposés sur la cohomologie des schémas" ?

**6**

votes

**1**answer

247 views

### Is this spectrum the algebraic K-theory spectrum of something?

Given a spectrum, is there any kind of machinery that can tell you whether it is the K-theory spectrum of some recognizable category? For example, could TMF be realized in this way? In this case we ...

**8**

votes

**2**answers

468 views

### is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. ...

**13**

votes

**1**answer

399 views

### Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...

**12**

votes

**2**answers

444 views

### What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?

Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...

**11**

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**0**answers

263 views

### Epsilon factors - a la Beilinson - What is it?

I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or ...

**9**

votes

**2**answers

291 views

### A looping of algebraic K-theory

Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K_0(\mathcal{C})$.
...

**15**

votes

**1**answer

222 views

### Besides F_q, for which rings R is K_i(R) completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...

**2**

votes

**0**answers

161 views

### (Non trivial) coidempotents(Co-$K$-theory)

I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
...

**8**

votes

**1**answer

354 views

### Over which fields (of positive characteristic) is the Beilinson-Soulé vanishing conjecture known to hold?

Let $k$ be a field, and denote by $K_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.
The Beilinson-Soulé (BS) vanishing conjecture predicts that
$$
...

**5**

votes

**1**answer

219 views

### Definition of a cylinder functor in Waldhausen's K-theory

In Waldhausen's Algebraic K-Theory of Spaces, he defines a cylinder functor on a category $\mathcal C$ with cofibrations and weak equivalences (henceforth called a Waldhausen category) as the ...

**6**

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**0**answers

137 views

### Split exact categories arising naturally

If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...