All Questions
6,057 questions
1
vote
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260
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The intersection of Block Groups and R-trivial (finite) monoids
Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the ...
- 424
5
votes
1
answer
679
views
On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings $(A,\mathfrak{m})\...
- 3,101
16
votes
2
answers
3k
views
Quotients of number rings
Hi,
Here's a question that comes up every now and then. Of course, the quotient of a number ring (ring of integers of a number field) by an ideal $I$ is a finite (Artin) ring. If we take $I$ to be ...
- 385
0
votes
0
answers
166
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Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?
The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...
- 945
15
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6
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1k
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Conjugacy for $p$-adic matrices of finite order
$\DeclareMathOperator\GL{GL}$Say $p$ is an odd prime, and take two matrices $A,B\in \GL_n({\mathbb Z}_p)$ of finite order $m$. Is it true that they are conjugate in $\GL_n({\mathbb Z}_p)$ if and only ...
- 5,363
1
vote
0
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169
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Choosing generators of a submodule with divisibility properties
Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$.
Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the ...
- 1,391
16
votes
3
answers
2k
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Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?
Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...
- 33.8k
4
votes
1
answer
679
views
In what generality is the natural map $\operatorname{Hom}_R(L,M)\otimes S\to\operatorname{Hom}_{R\otimes S}(L\otimes S,M\otimes S)$ an isomorphism?
Let $k$ be a commutative ring, $R$ and $S$ commutative $k$-algebras. Let $L$ and $M$ be $R$-modules. Consider the natural map
$$\operatorname{Hom}_R(L,M)\otimes_k S \to \operatorname{Hom}_{R \...
- 7,338
5
votes
0
answers
238
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When does the normalization have regular special fiber?
Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume ...
- 4,487
5
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1
answer
898
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A little help with the unmixedness theorem?
I have two smooth subvarieties $Y$ and $Z$ of a smooth variety $X$. Their intersection $Y \cap Z$ has two irreducible components, both of the expected dimension and generically reduced. I want to ...
- 683
11
votes
2
answers
1k
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Is the support of an Artinian module finite?
Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite?
Thanks.
- 259
3
votes
1
answer
382
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Generalizing Krull's Principal Ideal Theorem to Modules
Let $R = \mathbf{C}[x_1, \ldots, x_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose
$
0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots
$
...
10
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3
answers
3k
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Counter-examples to Krull's intersection theorem
The more general form of Krull intersection theorem says:
Let $R$ be local and Noetherian and $I \subset R$ a proper ideal. If $M$ is finitely generated over $R$, and $N=\cap_1^{\infty} I^iM$, then ...
- 103
3
votes
1
answer
388
views
Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 7,338
4
votes
1
answer
398
views
A terminology question: formally finite ??
Is there a name for a local homomorphism $\varphi:A\longrightarrow B$ of local rings $A$ and $B$, whose completion $\hat{\varphi}:\hat{A}\longrightarrow\hat{B}$ is a finite homomorphism? (that is, $\...
- 3,101
9
votes
2
answers
781
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Additivity of projective dimensions, or, help me lower my blood pressure
Sorry for the shameless title. I'm rather stuck on a lemma in commutative algebra - namely, I have both a proof and a counterexample! I have tried rather strenuously and frustratingly to find the ...
- 13.1k
4
votes
0
answers
1k
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Grothendieck spectral sequence [duplicate]
Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as you ...
- 51
0
votes
1
answer
580
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Why is Ext^n(k,M) a vector space over k?
This might be a trivial question to experts but not to me whatsoever. Suppose that $(R,m,k)$ is a Noetherian local ring, $M$ is an $R$-finite module whose depth is $n$. One then defines the type of $M$...
- 149
9
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2
answers
1k
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Modules over Laurent series rings
Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...
- 66.8k
2
votes
4
answers
2k
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A proof for a statement about polynomial automorphism
I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...
- 149
-1
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1
answer
282
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Invertible matrices satisfying $[x,y,y]=x$ (take 2).
This is a simpler version of this question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\
0& 1 & 0\\\
0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with ...
user6976
2
votes
4
answers
2k
views
Closed-form for modified formal power series
This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.
We start with a formal power ...
- 195
2
votes
0
answers
152
views
Characterization of a "Jacobian pair" member
Consider the ring ${R}$ of Puiseux series in $Y$ with coefficients in the ring $\mathbb{C}((X^*))$ of Puiseux series in $X$ with coefficients in $\mathbb{C}$; $F \in R$ is said to be a member of a ...
- 96
8
votes
1
answer
725
views
Number of simplicial polytopes with a given f-vector
Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...
- 902
1
vote
2
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194
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Counting hyperplane cuts vs. projections. Combinatorial identity
I have checked the following combinatorial identity for several cases and it seems to work. I would like to know if this is known or if there is a counter-example. Note, i is a given constant.
$$(i+d)...
- 11
3
votes
1
answer
544
views
Injective modules and torsion functors
(This is a related question.)
Local cohomology is studied mostly over Noetherian rings. Parts of the machinery do in fact not rely on Noetherianness, but on some weaker properties, for example the ...
- 6,700
7
votes
2
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649
views
Characterization of locally free modules via exterior powers
Let $X$ be a scheme and $\mathcal{F}$ be quasi-coherent module on $X$. It is clear that if $\mathcal{F}$ is locally free of rank $n$, then $\det(\mathcal{F}) := \wedge^n \mathcal{F}$ is invertible, i....
- 63.1k
8
votes
2
answers
1k
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A name for "not quite saturated" graded modules
Let $M$ be a finitely generated graded module over a graded ring $R$. Let $\mathcal{F}$ be the corresponding coherent sheaf on $\operatorname{Proj} R$. There is a natural map of graded $R$-modules
$$\...
- 7,338
8
votes
1
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555
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Spectrum of an algebra object and Reconstruction of Schemes
In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.
In the introduction the ...
- 63.1k
7
votes
1
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735
views
Can we make Buchberger's algorithm faster for a given ideal if we are allowed to vary the monomial order?
Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ...
- 7,338
4
votes
2
answers
463
views
Hilbert-Kunz multiplicity of Cohen-Macaulay local domains
Is there an example of a Cohen-Macaulay local domain $R$ of characteristic $p>0$ for which the Hilbert-Kunz multiplicity $e_{HK}(R)$ is not equal to its Hilbert-Samuel multiplicity $e(R)$? If no ...
- 3,101
6
votes
1
answer
826
views
Rings with finitely generated nilradical
Let $\mathfrak{a}$ be a monomial ideal in a polynomial algebra over some commutative ring $R$. If $R$ is reduced, then the radical $\sqrt{\mathfrak{a}}$ of $\mathfrak{a}$ is again a monomial ideal, ...
- 6,700
7
votes
1
answer
266
views
Positive cone of a subgroup of $\mathbb{Z}^n$
This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a ...
- 16.9k
0
votes
1
answer
315
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Generalized Picard group (reflexive fractional ideals, principal ideals)
Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^...
- 1,391
17
votes
2
answers
1k
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Dimension 1 prime ideals in the intersection of two maximal ideals
This question/problem really comes from a fact in algebraic geometry, where it says that given an irreducible variety $V$ ($\dim V \geq 2$) then for any given pair of points $x,y\in V$ there is an ...
- 236
3
votes
1
answer
948
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Module of Kahler differentials of rings of integers of number fields
How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1_{B/A}$ can be generated by one element as a $B$-module?...
- 5,019
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2
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971
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Simple object in derived category or stable model category?
Exist any common definition of simple objects in derived categories, or even better, in stable model categories?
I was only able to find definition for abelian categories.
Thanks.
- 91
1
vote
3
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467
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$\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$
Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a_1,\dots,a_n$. Then there is an isomorphism $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ ...
- 671
1
vote
0
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263
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In a Noetherian local ring $(R,m)$ with a prime ideal $P\neq m$, $P^{(n)}=P^n:m^{\infty}$
I had asked this question on math.stackexchange.com, but I have not received a response. Hoping to get some help here.
I am trying to prove this result and I am stuck at one step.
Let $(R,m)$ be a ...
0
votes
0
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551
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sub ring of algebra over subfield
Let $k$ be a field and $k[a]$ an algebric extension.
If $A$ is a reduced commutative algebra over $k[a]$ and $B$ is a subring which is an algebra over $k$, then is the following true: if there exist ...
- 1
1
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0
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417
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Absolute Irreducibility in Characteristic 2
Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely ...
- 456
1
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1
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346
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Cohomology of the general linear group on punctured spectra of 2-dimensional power series rings
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Quot{Quot}\DeclareMathOperator\GL{GL}\DeclareMathOperator\char{char}$Let $(A,\mathfrak{m})=k[[x,y]]$ with $\char(k)=0$ and $K=\Quot(A)$. Set $X=\...
- 1,391
7
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1
answer
1k
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How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?
(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...
- 30.5k
7
votes
2
answers
1k
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The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
- 42.9k
1
vote
1
answer
470
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Status of Gao's Conjecture
In his 2001 paper titled "On The Deterministic Complexity of Factoring Polynomials", Shuhong Gao makes the following conjecture:
For any $a \in \mathbf{F}_q$ ($q$ is some prime power), we can write $...
- 11
7
votes
1
answer
2k
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An example of a rank one projective R-Module that is not invertible
Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...
- 345
6
votes
1
answer
616
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Projective modules over free groups
Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)
More generally, let $G$ be ...
- 1,243
3
votes
3
answers
461
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Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
4
votes
1
answer
246
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Paper by I. Swanson on symbolic powers
I am looking for a paper by Irena Swanson on a result on comparison of ordinary and symbolic powers of prime ideals in complete local rings.
The paper is referenced in problem 0.9 here
https://aimath....
- 43
45
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2
answers
3k
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Categorical definition of the ideal product within the category of rings
This is an extension of this question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I*J)$ out of $R \to R/I$ and $R \to R/J$ in ...
- 63.1k