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6,056 questions
5
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679
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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})\...
- 3,908
1
vote
2
answers
1k
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maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
- 3,101
5
votes
1
answer
504
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The ring of SL_2 invariants in sums of conjugation and tautological modules
Rings of Invariants
Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free ...
CommunityBot
- 1
16
votes
2
answers
3k
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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 ...
CommunityBot
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16
votes
3
answers
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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 ...
- 63.1k
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
$$\operatorname{Hom}_R(L,M)\otimes_k S \to \operatorname{Hom}_{R \...
CommunityBot
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5
votes
0
answers
238
views
When does the normalization have regular special fiber?
Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume ...
- 4,487
5
votes
1
answer
898
views
A little help with the unmixedness theorem?
I have two smooth subvarieties $Y$ and $Z$ of a smooth variety $X$. Their intersection $Y \cap Z$ has two irreducible components, both of the expected dimension and generically reduced. I want to ...
- 4,111
0
votes
1
answer
465
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What is lim⟶ I^n M?
Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits_{\begin{subarray}{c}
\longrightarrow \\
\...
CommunityBot
- 1
3
votes
1
answer
382
views
Generalizing Krull's Principal Ideal Theorem to Modules
Let $R = \mathbf{C}[x_1, \ldots, x_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose
$
0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots
$
...
- 30.5k
3
votes
1
answer
388
views
Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 546
2
votes
1
answer
505
views
graded noetherian module
Let M be a R graded module with $M= \oplus M_i$. If M is noetherian then $M_i=0 $ for i << 0. My question is this, isn't $M_i = 0$ for all i >> 0 as well? If $(M_{n_i})_{i} \neq 0, n_i > 0$ ...
CommunityBot
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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
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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 ...
CommunityBot
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9
votes
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 ...
- 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
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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. ...
- 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
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
2
answers
394
views
Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 38.6k
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
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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 ...
CommunityBot
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1
vote
0
answers
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
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1
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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 ...
- 63.9k
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 ...
- 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
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3
answers
5k
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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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7
votes
1
answer
801
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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
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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
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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
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639
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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
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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
- 1
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
0
votes
2
answers
172
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small extensions of the free semigroup of rank 1
Let N denote the free semigroup of rank 1. Say that a semigroup T is a small extension
of N if N embeds in T and |T - N| is finite. Is there some kind of classification
of small extensions of N? ...
- 38.6k
15
votes
5
answers
1k
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Monoids with infinite products
Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...
CommunityBot
- 1
1
vote
2
answers
872
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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
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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
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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
- 1
4
votes
2
answers
607
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Invertible elements in monoid rings of unital monoids without non-trivial invertible elements
This question is somewhat related to Tilmans notorious problem in this post. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \...
CommunityBot
- 1
7
votes
4
answers
1k
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Torsors for monoids
Torsors are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.
In general I'm interesting in the ...
- 53.3k