All Questions
6,053 questions
4
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elementary classification of artinian rings
this may be too elementary for mathoverflow, but I'll give it a try.
rings are commutative here. it is well-known that every $0$-dimensional noetherian ring is artinian. the standard proof uses a ...
CommunityBot
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5
votes
2
answers
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Why is the fibered coproduct of affine schemes not affine?
I am confused about the following issue:
Let $X=SpecS$, $U_1=SpecR_1$, $U_2=SpecR_2$. and suppose we have maps $S \rightarrow R_1$, $S \rightarrow R_2$. Let $U_3=Spec (R_1 \otimes_S R_2)$. We have ...
- 39.3k
1
vote
0
answers
2k
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Formal power series ring & completion
I encountered the following passage in Matsumura's Commutative Ring Theory :
A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(...
- 2,857
1
vote
2
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340
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Infinite collection of elements of a number field with very similar annihilating polynomials
Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number
of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and
let $B$ be the subset of $R$ ...
CommunityBot
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13
votes
1
answer
990
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Is -1 a sum of 2 squares in a certain field K?
Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a ...
- 65.4k
2
votes
1
answer
430
views
0 dimensional Dedekind domain?
It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
- 12.6k
16
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2
answers
2k
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Extra principal Cartier divisors on non-Noetherian rings? (answered: no!)
On the way to defining Cartier divisors on a scheme $X$, one sheafifies a presheaf base-presheaf of rings $\mathcal{K}'(U)=Frac(\mathcal{O}(U))$ on open affines $U$ to get a sheaf $\mathcal{K}$ of "...
CommunityBot
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9
votes
2
answers
2k
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Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 14.5k
2
votes
2
answers
349
views
Subrings of rational functions invariant under change of sign
Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$ over the real numbers.
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which ...
- 12.3k
5
votes
1
answer
443
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Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set?
"Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree....
- 538
1
vote
2
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Hilbert Syzygy Theorem - Induction step
Does someone know in which books, lecture notes or ... I can find the induction step of the proof of Hilbert Syzygy Theorem? I'd only found the proof for R[x] (e.g. Weibel) and I haven't really an ...
CommunityBot
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6
votes
1
answer
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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 ...
CommunityBot
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17
votes
1
answer
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Do these conditions on a semigroup define a group?
As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...
CommunityBot
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10
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5
answers
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$K_{0}(R) =\mathbb{Z}$ but some f.g. projective not stably free?
This question is motived by this recent question.
$K_{0}(R)=\mathbb{Z}$ is often used as a euphemism for saying that every finitely generated projective module is stably free; however, there are some ...
CommunityBot
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4
votes
6
answers
665
views
Number of A Subset of Monomials
I need to count the number of monomials of degree $n$ in $k$ variables, $x_1,\ldots ,x_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. ...
CommunityBot
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9
votes
3
answers
4k
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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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5
votes
0
answers
350
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Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
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
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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
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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
2
votes
0
answers
498
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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
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563
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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
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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
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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
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1
answer
2k
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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
views
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
15
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3
answers
2k
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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
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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