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

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### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...

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**1**answer

220 views

### Pullback along Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...

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96 views

### $K_0$ an $KH_0$ of a normal crossing variety

Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...

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votes

**2**answers

140 views

### Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...

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**1**answer

479 views

### Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?

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400 views

### What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...

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139 views

### Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it.
For those who have the book (or want to look it up via google books etc.), it's the ...

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120 views

### Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...

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186 views

### An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...

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175 views

### Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...

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79 views

### Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...

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124 views

### When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...

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133 views

### generalization of result on K_1 of $SL(n,R)$

Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te_{i,j}$ is equal to $SL(n,R)$.
My question is as follows: Instead of $SL(n,R)$ I look ...

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**1**answer

85 views

### Is there something like an “f-regular” $K$-theory?

Let $R$ be a ring and $f\in R$. Is there something like an $f$-regular $K$-theory group of $R$ based on the category of $f$-regular $R$-modules, i.e. modules that do not have any $f$-torsion? If ...

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88 views

### Gersten complexes in Quillen's and Milnor's K-theories

Consider a good enough scheme $X$ (e.g. an algebraic variety over a field). Let $X_i$ be the set of points of dimension $i$ in $X$. Then we have the Gersten complex in Quillen's K-theory:
$$
...

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**1**answer

198 views

### Map between homotopy groups of O, related to J-homomorphism and K-theory of Z

Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi_{8s+1}^s = \pi_{8s+1} ...

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**1**answer

239 views

### The multiplication on $THH$ of finite fields

Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...

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**1**answer

472 views

### Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...

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139 views

### Do we have the following “devissage commutative diagram” in K-theory?

Let $X$ be a non-reduced Noetherian scheme. We define $K^0(X)$ to be the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ to be the Grothendieck group of the derived category ...

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**1**answer

151 views

### Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...

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433 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 ...

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344 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 $Z := \sum\limits_in_iZ_i$ ...

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**1**answer

228 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 ...

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201 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?

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### 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 ...

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726 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 ...

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99 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

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142 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 ...

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214 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 ...

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155 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 ...

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122 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 ...

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109 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.

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505 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 ...

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519 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 ...

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408 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 ...

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361 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 ...

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289 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 ...

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81 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 ...

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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 ...

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83 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, ...

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164 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 ...

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91 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
...

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609 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 ...

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### 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 ...

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370 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 ...

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### 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 ...

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357 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 ...

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### 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 ...

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155 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 ...

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201 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 ...