All Questions
Tagged with algebraic-k-theory ac.commutative-algebra
46 questions
3
votes
2
answers
246
views
Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
4
votes
0
answers
112
views
The $K_1$-group of integer valued polynomials
Let $R=$ Int$(\mathbb{Z}) = \{f \in \mathbb{Q}[x]| f(\mathbb{Z}) \subset \mathbb{Z}\}$. I am interested to find $K_1(R)$. I list my trials below:
Let us construct a Milnor square $$\matrix{R&\...
- 141
0
votes
1
answer
186
views
Does going-down theorem hold for local homomorphism of finite flat dimension?
Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down theorem hold for $f$?
If yes, then by Theorem 15.1 in Matsumura’s ...
- 639
4
votes
1
answer
167
views
Does local homomorphism of finite flat dimension preserve Krull dimension?
Let $f:A\rightarrow B$ be a local homomorphism of Noetherian local rings, such that the $A$-module $B$ has finite flat dimension. Is it true that the Krull dimensions of $A$ and $B$ agree? If yes, ...
- 639
1
vote
0
answers
305
views
Presentation of Chevalley groups over Bezout domains
Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 ...
- 563
2
votes
1
answer
165
views
$K_0((k[x]/(x^2))[y])$
Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
- 167
4
votes
1
answer
194
views
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 $K_0(R)$ ...
- 167
5
votes
0
answers
107
views
Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
- 187
4
votes
0
answers
216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
3
votes
1
answer
132
views
How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?
$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...
- 639
1
vote
0
answers
151
views
$K_1(k[x]/(x^2))$ for a field $k$
$\DeclareMathOperator\GL{GL}$The definition of $K_1$ is stated in "The K-book" by Charles Weibel as a quotient of $\GL(R)$, where $\GL(R)$ is the union of the sequence $R^{ \times} = \GL_1(R)...
- 167
3
votes
1
answer
219
views
$K_1(\mathbb{Z}_4)$ and $K_1(\mathbb{Z_4}[t])$
I am an amateur in $K$-theory, I have just started reading from "The K-book" by Charles Weibel. I have only read the definition of $K_1$ which is stated as a quotient of $GL(R)$. The union ...
- 167
4
votes
1
answer
355
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 ...
- 2,017
3
votes
1
answer
185
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 ...
user139827
5
votes
1
answer
182
views
A question on $SK_1$ of rings
Let $B$ be a commutative ring with unity and $B/nil(B):=B_{red}$, where $nil(B)$ is the nilradical of $B$. Is $SK_1(B)=SK_1(B_{red}) ?$ In particular, is it true when $B$ is an affine algebra over an ...
4
votes
0
answers
135
views
$K$-group of category of bounded chain complexes of Projective modules with finite length homologies
For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
- 1,209
11
votes
1
answer
578
views
Are projective modules over a certain localised Laurent polynomial ring free?
Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[...
- 1,033
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
3
votes
0
answers
205
views
Extending vector bundles over a regular divisor in a regular affine scheme
This is more-or-less question (3) on page 170 of Quillen's "Projective Modules over Polynomial Rings" (link):
Let $A$ be a regular Noetherian ring and let $f \in A$ be an element of $A$ such that $...
- 2,017
3
votes
0
answers
122
views
Is the Milnor boundary map, a natural transformation?
Consider the Milnor $K_n$-functors for discrete valuiation fields. For any discrete valuation field $F$ we can associate an abelian group $K_n(F)$ and the construction is given thanks a universal ...
- 1,237
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
7
votes
2
answers
2k
views
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!
- 496
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
16
votes
3
answers
797
views
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 $...
- 563
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
20
votes
1
answer
1k
views
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 ...
- 7,893
9
votes
0
answers
426
views
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 ...
- 7,893
4
votes
0
answers
112
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 ...
- 161
6
votes
1
answer
1k
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 ...
- 21.3k
3
votes
1
answer
463
views
Endomorphism Ring of Indecomposable MCM Modules
Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
- 161
2
votes
0
answers
267
views
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
5
votes
0
answers
73
views
K-Theory and completion [duplicate]
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the $\...
- 165
9
votes
0
answers
260
views
Is the generation of rings by their units a question in K-theory?
Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...
- 13.3k
1
vote
0
answers
282
views
Computation of Quillen K-groups for mod R
The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the ...
2
votes
3
answers
832
views
Classify matrices up to similarity over arbitrary (commutative) ring.
One can define the K-theory space of a monoidal category $S$ in which every morphism is an isomorphism as the classifying space $B(S^{-1}S)$. Then we show that this definition coincides with the ...
- 1,558
8
votes
1
answer
472
views
Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
- 3,101
30
votes
2
answers
2k
views
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 ...
- 418
10
votes
1
answer
866
views
Current status of a conjecture of Bloch
In the seminal paper $K_2$ and algebraic cycles, Bloch make the following conjecture :
Suppose $A$ is a local Noetherian integral domain with quotient field $F$
$K_2(A)$ → $K_2(F)$ is injective
...
- 951
2
votes
1
answer
325
views
Gersten for homotopy invariant K-theory of non-singular varieties.
Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?
If $X$ is ...
- 1,347
4
votes
1
answer
579
views
Is there a clean definition of the residue map in Milnor K-theory?
If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
- 467
3
votes
0
answers
591
views
Algebraic description of double vector bundles.
It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
- 51
6
votes
1
answer
1k
views
Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings?
This question is motivated by some issue raised by David Speyer in this question.
Let $R$ be a ring. Let $K_0(R)$ and $G_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules ...
- 30.5k
6
votes
4
answers
959
views
Does every projective A/I-module come from A?
Let $A$ be a Noetherian commutative ring and $I$ an ideal in $A$. It is pretty much trivial to see that every free $A/I$-Module is obtained from a free $A$-module by tensoring over $A$ with $A/I$: ...
- 3,917
9
votes
1
answer
884
views
Isolated hypersurface singularities, Chow groups and D-branes
Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W,...
- 538
12
votes
2
answers
657
views
Maps between K-groups induced by rings homomorphism
Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free ...
- 30.5k
9
votes
6
answers
4k
views
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 ...
