Questions tagged [algebraic-k-theory]
The algebraic-k-theory tag has no usage guidance.
51 questions
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What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
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10
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Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
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Universal cover of SL2(R) admits no central extensions?
Is it true that the universal cover of $\mathrm{SL}_2(\mathbb{R})$ has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective ...
- 43.2k
14
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2
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Motivic cohomology and cohomology of Milnor K-theory sheaf
Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$).
...
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When are representation rings special lambda-rings? (variations of an old question)
Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims.
On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...
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3
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What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?
More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
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When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
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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 ...
- 63.9k
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3
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Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,105
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For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ ...
- 6,614
14
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1
answer
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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 ...
- 18.7k
12
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0
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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 $I$), ...
- 5,901
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3
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The localisation long exact sequence in K-theory over an arbitrary base
If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory
... Kn+1(Dx) --> Kn(k) --...
- 8,866
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6
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Differences between reflexives and projectives modules
Let R be a normal noetherian domain.
What is the difference between a finitely generated reflexive module and a finitely generated projective module?
Can anybody recommend any references or make ...
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The saturation of Murray von Neumann relation
Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version:
Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $...
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1
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Z/48 and Moonshine Beyond the Monster
I am interested in pursuing an understanding of K-theory. Primarily, the
$K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the $\mathbb{Z}/...
- 515
4
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2
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Suslin's Stability Theorem for Chevalley Groups
I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...
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3
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K(F_1) = sphere spectrum?
I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?
- 10.8k
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Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?
Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
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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" ...
user1437
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What's about "quantum modular forms"?
Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling!
Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...
- 10.8k
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Why presheaves with transfer?
Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH ...
- 171
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3
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For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $...
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Entering to the K-theory realm
I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation
and interaction with the field of Algebraic Topology. I mainly had
concentrated on ...
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answer
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Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...
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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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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 ...
- 637
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Rationalised K-theory of number fields
Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...
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$K$-theory backwards
Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?
That is, ...
- 63.9k
11
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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 ...
- 63.9k
11
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Genus of smooth varieties with small Chow group
Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = CH(...
- 30.5k
11
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$K_0$ of a non-separated scheme
This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K_0(X)$ be the Grothendieck group of coherent ...
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Quasi-isomorphisms in exact categories
I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...
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Do symmetric monoidal groupoids model all connective spectra?
Thomason showed that symmetric monoidal categories model all connective spectra. But can it be done with groupoids? In order for this to be the case, the group completion prices process must be very ...
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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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Relative Chow groups
Most cohomologies have the notion of cohomology with support on a closed subspace, and also cohomology with compact support. In general, for any morphism $f\colon Y\to Z$ the inverse image fits into a ...
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Is the class of commutative generalized Euclidean rings stable under quotient and localization?
Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
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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 $F$. The ...
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Local property of split exact sequence
In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal?
Thanks!
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Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
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Atiyah class for non-locally free sheaf
Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$.
The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the ...
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When is $GL_m(R)$ generated by elementary and diagonal matrices?
Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary ...
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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 $...
- 5,901
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Can the trivial module be stably free for a monoid ring?
Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ ...
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Group completion of a monoid (braid groups)
Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...
- 140
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1
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Adams graded parts of rational K-theory of a number field.
Let $F$ be a number field and $r_{1}$ and $r_{2}$ the numbers of real and pairs of complex embeddings respectively of $F$. Then Borel computed that for $n\geq 2$
$$
K_{n}(F)_{\mathbb{Q}}\simeq
\...
- 41
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1
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When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
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1
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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 ...
user341790
2
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0
answers
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K-theory and completion
I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...
- 165
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0
answers
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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 ...
- 6,018