All Questions
Tagged with algebraic-k-theory ra.rings-and-algebras
20 questions
5
votes
2
answers
397
views
Ring with vanishing $K_0$
Suppose we have a ring $R$ such that the Grothendieck group $K_{0}(R)=0$.
Question 1: Does it follow that there exists two positive natural numbers $n\neq m$ such that
$R^{m}$ is isomorphic to $ R^{n}$...
- 855
1
vote
0
answers
123
views
Quiver representations and the standard matrix decompositions
Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form:
$$M = A D B$$
where $D$...
- 4,943
12
votes
1
answer
312
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 $\mathbb{Z}[G]$ ...
- 637
5
votes
1
answer
273
views
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) ...
- 1,307
5
votes
2
answers
470
views
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}(...
- 223
2
votes
1
answer
268
views
Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain
Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M_n(R)$-modules. Is it true that
$A\oplus M_n(R)\cong B\oplus M_n(R)\:\:\Rightarrow\:\:A\cong B$?
Here, the isomorphism is ...
- 348
3
votes
1
answer
236
views
For $T$ the $2\times 2$ triangular matrices over $R$, can we write $GL_2(T)=U(T)E_2(T)$?
Let $R$ be a commutative ring with identity, and let $T = T_2(R)$ be the ring of $2\times 2$ upper triangular matrices over $R$. Is it true that the following identity holds?
$$GL_2(T)=U(T)E_2(T)$$
...
- 348
7
votes
1
answer
212
views
What does $K_1(R)$ tell us about $GL_n(R)/E_n(R)$?
Let $D$ be a division ring, and $R=D[t_1,\ldots,t_n]$. If $GL_m(R)$ is the usual group of invertible matrices over $R$, then by $E_m(R)$ I mean the subgroup of $GL_m(R)$ generated by the elementary ...
- 348
7
votes
1
answer
518
views
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 ...
- 348
5
votes
1
answer
342
views
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 $mz=z=zm$ ...
- 38.6k
4
votes
1
answer
283
views
Geometric contractibility of noetherian rings
Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if
All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial.
There exist a non-trivial locally free sheaf of ...
- 493
6
votes
1
answer
302
views
Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?
Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...
- 255
8
votes
0
answers
307
views
How bad can $SK_1$ of a commutative ring be?
For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
- 3,186
7
votes
1
answer
1k
views
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 $...
- 356
12
votes
1
answer
458
views
Algebraic K-theory of a ring
I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is: What is the list of rings such that all their algebraic $K$-theory groups are known?
I ...
- 433
19
votes
2
answers
702
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" ...
user1437
6
votes
1
answer
550
views
A generalized K- theory via generalized idempotents
Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...
- 356
18
votes
4
answers
2k
views
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
7
votes
3
answers
1k
views
Group rings of infinite products of groups
Given a infinite family of groups $(G_i)$ for $i\in I$. Is there a ring theoretic construction, that produces $R[\prod_{i\in I} G_i]$ using only the rings $(R[G_i])_{i\in I}$ ?
For the case of a ...
- 11.1k
15
votes
2
answers
2k
views
Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?
I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...
- 2,814