All Questions
5,985 questions
17
votes
2
answers
1k
views
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 ...
CommunityBot
- 1
4
votes
0
answers
1k
views
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 ...
CommunityBot
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9
votes
2
answers
1k
views
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 ...
- 45.7k
0
votes
1
answer
580
views
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 $M$...
- 47.3k
2
votes
4
answers
2k
views
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. ...
- 38.6k
-1
votes
1
answer
282
views
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 ...
CommunityBot
- 1
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
1
vote
2
answers
194
views
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)...
- 85.6k
3
votes
1
answer
2k
views
Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 11.1k
7
votes
2
answers
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....
- 22.6k
5
votes
0
answers
204
views
A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
7
votes
1
answer
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 ...
- 151k
8
votes
1
answer
555
views
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 ...
- 13.6k
6
votes
1
answer
825
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, ...
- 47.5k
0
votes
1
answer
315
views
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^...
- 4,111
9
votes
2
answers
971
views
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.
- 39.3k
1
vote
3
answers
467
views
$\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$ ...
- 15.2k
1
vote
0
answers
263
views
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 ...
7
votes
2
answers
1k
views
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 ...
CommunityBot
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1
vote
0
answers
417
views
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
7
votes
1
answer
2k
views
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 ...
- 63.9k
6
votes
1
answer
616
views
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 ...
- 15.2k
3
votes
3
answers
461
views
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 ...
- 52.3k
52
votes
3
answers
5k
views
What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?
On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
CommunityBot
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2
votes
2
answers
402
views
Maximization of a matrix product by iterative methods
This might not be very difficult, but I think I may have gotten a little confused.
Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...
- 60.5k
7
votes
1
answer
800
views
Extensions of torsion modules
Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module.
Assume we have a nontrivial surjective map $f: M \...
- 30.3k
3
votes
1
answer
321
views
spurious torsion under compositions of linear maps
Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.
For $h = f, g, g \circ f$, let $c(h)$ be the ...
- 735
2
votes
1
answer
323
views
Presentation of finite modules with null annihilator
Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: $R^n\stackrel{\varphi}{\...
- 30.5k
1
vote
2
answers
639
views
Almost clean module
Please give me an example of an almost clean module $M$ over a ring $S$ so that if $x$ is a $M$ regular element then $M/xM$ is not almost clean.
CommunityBot
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24
votes
4
answers
4k
views
Is there a Galois correspondence for ring extensions?
Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...
CommunityBot
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1
vote
1
answer
2k
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The annihilator of the quotient module
Suppose that $(A,m)$ is a Noetherian local ring, $M$ is an $A$-finite module. Assume that $x_1, ..., x_n$ are elements in $m$. Is the following equality true:
$$
\mbox{ann}(M/(x_1, ..., x_n)M) = (x_1,...
- 30.5k
1
vote
2
answers
872
views
Rational power series
If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ ...
- 16.2k
8
votes
2
answers
1k
views
Algebra Counterexample Request: Linear Quotients
A result of Herzog, Hibi, and Zheng in "Monomial ideals whose powers have a linear resolution" states that:
Theorem: Let $I\subseteq\Bbbk[x_1,\ldots,x_n]$ be a monomial ideal generated in degree 2. ...
- 71
3
votes
0
answers
461
views
Krull dimension of non-integral extensions
Some hours ago, a question was posted, asking (citation by heart, not literally)
Let $R$ be a commutative ring (with unit). What can be said about the Krull dimension of an algebraic, non-integral ...
- 16.2k
1
vote
2
answers
355
views
Is there a relationship between the right global dimensions of R and R[1/v]?
A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
CommunityBot
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12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.5k
7
votes
2
answers
2k
views
Global dimension and localization
Is there any condition on a commutative ring $R$ so that the global dimension of $R$ coincides with the supremum of the global dimensions of the localizations $R_{\mathfrak{m}}$ at all maximal ideals $...
- 30.3k
12
votes
2
answers
1k
views
Cohen-Macaulay domain with non-Cohen-Macaulay normalization?
Is the normalization of a Cohen-Macaulay domain necessarily Cohen-Macaulay? I suspect that the answer is no, but I don't have a counterexample. I am most interested in "geometric" situations, so one ...
CommunityBot
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29
votes
2
answers
5k
views
Regular, Gorenstein and Cohen-Macaulay
All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
CommunityBot
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2
votes
2
answers
1k
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Irreducible component of a Cohen-Macaulay variety
Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?
- 20.5k
1
vote
1
answer
515
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Cohen Macaulay, free and finitely generated module
Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free ...
- 42.9k
7
votes
1
answer
2k
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structure theorem for modules
Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with ...
- 25.8k
8
votes
1
answer
3k
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Number of graphs with a given number of nodes, edges and triangles
Hi. Does anyone know if it is possible to enumerate the set of labeled/unlabeled graphs (loopless, undirected, only one edge between pairs of nodes) having a given number of nodes, edges and triangles?...
- 902
3
votes
0
answers
916
views
Unibranch rings
Let us call a Noetherian local ring $A$ unibranch if it is a domain and the normalization map is finite and induces a bijection on spectra.
My question is as follows: is this property preserved when ...
- 1,209
6
votes
3
answers
786
views
Trace of the identity map in a projective module
Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m_1,\ldots,m_n$ be a set of generators of $M$. The Dual ...
- 56.9k
4
votes
0
answers
811
views
$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
3
votes
0
answers
289
views
Terminal quasi-affine varieties?
Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically
closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular
functions on $U$. Write $Max(A)$ for the ...
- 42.9k
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
3
votes
1
answer
1k
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Does totally ramified extension really exist?
The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
- 37k
5
votes
2
answers
367
views
Invariant means on commutative magmas
It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem ...
- 6,034