All Questions
5,985 questions
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3
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Maximal ideal in polynomial ring
Is it true that the intersection of a maximal ideal in $A[x]$ with $A$ is a maximal ideal in $A$?
Let's say A is Noetherian. I would be surprised if it isn't true but somehow I can't seem to show it. ...
CommunityBot
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2
votes
0
answers
5k
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A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
- 13.5k
5
votes
1
answer
631
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Showing an Ext^2 element is zero
If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
- 39.3k
3
votes
0
answers
766
views
Finite generatation of Ext
If $A$ is a Noetherian ring and $M$, $N$ are finitely generated modules over $A$, it is easy to see that $\mbox{Ext}_{A}(M,N)$ is finitely generated by taking a finitely generated projective ...
- 2,857
1
vote
1
answer
349
views
Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
- 65.4k
27
votes
4
answers
3k
views
Nilradicals without Zorn's lemma
It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$.
Every proof I found (e.g. in the classical "Commutative Algebra" by ...
- 44.4k
2
votes
3
answers
294
views
Necessary and sufficient criteria for non-trivial derivations to exist?
Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to ...
- 30.5k
4
votes
0
answers
790
views
Is it possible to use linear programming to solve this problem?
I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...
- 41
2
votes
0
answers
498
views
A question about the assassinator (={associated primes}) and the support of a module.
This question is motivated by a proof in Bruns' and Herzog's book on "Cohen Macaulay Rings".
Let $\(R,\mathfrak{m}\)$ be a Noetherian local ring, $M \neq 0$ a finitely generated $R$-module. Suppose ...
- 19.6k
5
votes
1
answer
499
views
software for computations on flag varieties in arbitrary characteristic
Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
- 5,846
20
votes
2
answers
15k
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Maximal Ideals in the ring k[x1,...,xn ]
Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field, are of the form $(x_1 - ...
CommunityBot
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0
votes
2
answers
563
views
Primary decomposition of zero-dimensional modules
(I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) ...
- 57.6k
9
votes
5
answers
3k
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Alternative proof of unique factorization for ideals in a Dedekind ring
I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
- 990
2
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0
answers
546
views
Ring objects in the category of cocommutative coalgebras (aka Hopf rings).
I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite ...
- 4,990
17
votes
1
answer
2k
views
Geometric interpretation of filtered rings and modules
Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...
- 15.6k
24
votes
3
answers
3k
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Origin of the term "localization" for the localization of a ring
I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source ...
- 14.5k
6
votes
3
answers
2k
views
A simple infinite dimensional optimization problem
I'd be grateful for a reference for the following result, which I believe to be true, and
should be well-known.
Let the continuous functions $f_0,f_1,\cdots,f_n: [0,1]\rightarrow [0,\infty)$ be ...
- 60.5k
15
votes
3
answers
2k
views
Why do modules with small support have high Exts?
Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...
- 30.5k
2
votes
3
answers
656
views
Connection: locally free - locally projective
Given a smooth projective variety $X$ over some algebraically closed field $k$
and a locally free sheaf $R$ of $O_X$-algebras, e.g. central simple algebras or orders.
If $M$ is a left $R$-module ...
- 39.3k
8
votes
1
answer
1k
views
Primes in a (commutative) Jacobson ring
Recall that a commutative ring is Jacobson if every prime ideal is the intersection of the maximal ideals that contain it.
In the exercises of a commutative algebra course I gave I asked the ...
- 4,790
7
votes
1
answer
730
views
Example sought of an atomic domain R such that R[t] is not atomic
Recall that an integral domain $R$ is atomic if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "...
- 19.6k
0
votes
2
answers
356
views
Can all induced maps be described categorically.?. (or at least as generally as possible)
Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol.
I am pretty confused about induced maps in different areas of algebraic
topology; I do know how these induced maps are ...
- 1,030
6
votes
2
answers
1k
views
Gaining intuition for how submodules behave
I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
CommunityBot
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5
votes
3
answers
980
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What is the coordinate ring of symmetric product of affine plane?
The symmetric product of a variety $M$ is the quotient of $M^n/S_n$ where $S_n$ is the symmetric group permuting components of n-fold product $M^n$. IF $M$ is an affine plane $C^k$ over complex ...
- 19.2k
9
votes
2
answers
1k
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Factorial Rings and The Axiom of Choice
It is shown in Lang's Algebra (and many other books I assume) that:
if A if a principal entire ring, then A is a factorial ring.
The proof uses Zorn's Lemma. Is this theorem equivalent to the axiom ...
- 57.6k
4
votes
2
answers
774
views
Converse of Principal Ideal Theorem
$(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a zero divisor. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?
- 5,846
6
votes
0
answers
577
views
Continuous homomorphisms between power series rings
Let $A$ be an arbitrary ring. In "Commutative Algebra" by Zariski and Samuel it is claimed that every continuous homomorphism $A[[Y_1,...,Y_m]] \to A[[X_1,...,X_n]]$ is a substiution homomorphism $Y_i ...
- 63.1k
17
votes
1
answer
2k
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Composing left and right derived functors
I would appreciate either an explanation or a reference for what is going on here.
Motivation:
Let $f : X \rightarrow Y$ be a morphism of algebraic varieties. The derived projection formula implies ...
- 39.3k
17
votes
0
answers
1k
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monomorphisms and epimorphisms of local rings
I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...
CommunityBot
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2
votes
2
answers
665
views
Z_p flatness and irreducible components.
I just used the following.
Lemma. Let $A$ be a $\mathbb{Z}_p$-flat ring, of finite type over $\mathbb{Z}_p$, and suppose that $A \otimes \mathbb{F}_p$ is a domain. Then $A$ is a domain.
Proof: ...
- 4,492
0
votes
3
answers
565
views
Homology of koszul complex is finitely generated?
$A$ a local ring and $a_{1}$, ..., $a_{n}$ elements in its maximal ideal, $M$ a finitely generated $A$-module. In this case apparently the homologies from the Koszul complex are finitely generated as $...
- 47.3k
1
vote
1
answer
312
views
Deformations of free modules
Where can I found a description of the deformation theory for modules?Is it possible to deform a free module in such way that each fibre of the deformation is still free?
- 301
2
votes
1
answer
412
views
derivative in the ring k[e]/e², chain rule
Let $k$ be a ring and $\overline{k} = k[\epsilon]/\epsilon^2$. For every $f \in k[t]$ there is a unique $f' \in k[t]$ such that $f(t+\epsilon)=f(t)+\epsilon f'(t)$ holds in $\overline{k}[t]$. It ...
- 22.6k
12
votes
2
answers
1k
views
Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?
Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has ...
- 7,237
9
votes
1
answer
1k
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Is formal smoothness a local property?
Is the following statement true?
Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally smooth ...
- 22.6k
1
vote
0
answers
1k
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Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
4
votes
4
answers
961
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Homological dimension of a graded ring which is like polynomial ring
Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$...
7
votes
2
answers
931
views
What is the German translation of "catenary ring"?
I am looking for the correct technical term in German for the notion of catenary ring in commutative algebra.
Does anyone know?
For those who don't know what a catenary ring is but would like to: ...
- 236k
7
votes
1
answer
2k
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Krull dimension of a completion
How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull ...
- 85.6k
11
votes
2
answers
869
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Why is the prime spectrum not useful in non-archimedean analytic geometry?
This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the notes of Conrad.
Reading Conrad's notes (and e.g. those of ...
- 57.6k
5
votes
3
answers
3k
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Generalized Chinese Remainder Theorem
Let $U,V$ be submodules of a $R$-module $M$. Then the diagonal induces an isomorphism
$M/(U \cap V) \to M/U \times_{M/(U+V)} M/V.$
This is a (useful!) generalization of the Chinese Remainder Theorem ...
- 12.3k
3
votes
1
answer
1k
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Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
- 161
3
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0
answers
325
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Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
- 30.5k
4
votes
1
answer
1k
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Non-existence ofintegral basis of integral closure in a finite extension of Frac(A), A Dedekind.
Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ ...
- 373
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votes
2
answers
610
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When is tensoring with a module representable by a scheme?
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...
- 27k
5
votes
3
answers
752
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
- 39.9k
14
votes
1
answer
1k
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Two questions about Cohen-Macaulay rings
The following questions seem basic, but I can't find them in the literature. I'm also unable to think of a counterexample.
Let $A$ be a local Cohen-Macaulay ring of dimension $d$.
Let $I$ be an ...
CommunityBot
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3
votes
1
answer
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Lifting results from smooth maps to essentially smooth maps.
Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is essentially finitely presented provided that $...
- 85.6k
9
votes
0
answers
281
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Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 22.1k
15
votes
6
answers
1k
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An example of a series that is not differentially algebraic?
Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
CommunityBot
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