All Questions
6,056 questions
5
votes
1
answer
655
views
an example of a semigroup with solvable word problem but unsolvable power problem
We say that a semigroup $S$ has solvable power problem if there is an algorithm that takes as input an element $s \in S$ and decides whether or not there exist $m,n \in \mathbb{N}$ with $m \neq n$ and ...
- 549
7
votes
1
answer
801
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 \...
- 1,391
3
votes
1
answer
546
views
Center of the category of $R$-algebras
Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See this recent question ...
- 63.1k
7
votes
1
answer
747
views
Non-normal domain with algebraically closed fraction field
I am looking for an integral domain $A$ with the following properties:
$A$ is not integrally closed
$A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
There is an ...
- 2,275
4
votes
0
answers
1k
views
An example of a noetherian ring in which the integral closure of a finite extension of its field of fractions is not noetherian
Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.
- 113
3
votes
2
answers
667
views
Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 7,037
15
votes
4
answers
6k
views
how to determine whether an ideal is prime or not by an algorithm
Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume ...
- 1,528
3
votes
1
answer
320
views
Decision problem about the existence of solution for an integer matrix equation
Given $A,B,C \ $ integer matrices of dimensions $l \times m$, $l \times n$ and $l \times m$, we want to decide (algorithmically) about the existence of $X$ (unimodular) and $Y \ $ integer matrices ...
- 61
2
votes
1
answer
504
views
Zero-dimensional algebras of infinite vector space dimension
Consider an algebra $A$ over a field and suppose that $A$ is zero-dimensional as a ring. It is well-known that if, in addition, $A$ is finitely generated, it has a finite vector space dimension. ...
2
votes
2
answers
406
views
Extending a polynomial function from an open subset
I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, ...
- 7,037
8
votes
1
answer
289
views
Top degree local cohomology under action by a non-zerodivisor
Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does ...
- 19.6k
7
votes
2
answers
2k
views
An effective way to tell if the saturation of a homogeneous ideal is the irrelevant ideal
Let $\Bbbk$ be an algebraically closed field, let $R$ denote the graded ring $\Bbbk[x_0, \dotsc, x_N]$, and let $f_1, \dotsc, f_n \in R_m$ be nonconstant homogeneous polynomials. Then the common ...
- 7,338
2
votes
1
answer
456
views
Generic liftings of a regular sequence on the initial ideal
Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is ...
- 103
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 1,871
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.
- 287
4
votes
1
answer
2k
views
Unsolved problems concerning Artinian Rings and Artinian Modules
I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to ...
- 2,855
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 $...
- 15.2k
8
votes
0
answers
494
views
"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,595
2
votes
1
answer
186
views
Behaviour of Primes under Regular Coefficient Extensions
Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
- 125
3
votes
3
answers
549
views
Castelnuovo-Mumford Regularity of Ideals of Maximal Minors
I have an $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $$M = ( x I - A z -B w \mid y I - C z - D w).$$ Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let $...
- 431
3
votes
1
answer
901
views
Behaviour of Hilbert functions
Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...
- 31
4
votes
1
answer
284
views
When is Out$(SL_n(R))$ a torsion group ?
This question is a follow up question to this question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...
- 11.1k
2
votes
1
answer
578
views
An example of a noetherian N-1 ring that is not N-2 and/or a Nagata ring
Hello is there anyone that would know where I can find an example of a noetherian N-1 ring that is not a Nagata ring. (See the Wikipedia article "Nagata ring" for the definitions of N-1 ring and ...
- 113
4
votes
3
answers
2k
views
Chevalley's valuation extension theorem and the axiom of choice
Hello,
Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend ...
- 43
8
votes
1
answer
721
views
Is this a characterization of Dedekind domain?
Let $R$ be an integral domain. Suppose that for any two nonzero ideals $I$ and $J$, we have $I \oplus J$ is isomorphic to $R \oplus IJ$ as $R$-modules. Does this implies $R$ is a Dedekind domain?
- 1,373
0
votes
0
answers
544
views
isomorphism between vector spaces and modules - Commutative Algebra
Hi, Let $M_i$ be A modules. Then we know that $Ass (\oplus M_i) = \bigcup Ass(M_i) $. We consider here isomorphisms between modules.
Now consider a stanley ...
- 287
4
votes
2
answers
820
views
What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$?
What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$? Anything of the form $\pm 1 + 2 x p(x)$ for $p(x) \in \mathbb{Z}_4[x]$ works, and is in fact its own inverse. It's easy to see that any unit must ...
5
votes
3
answers
412
views
CM for primary ideal
Let $R$ be a regular local ring, $I$ a prime ideal and $J$ an $I$-primary ideal in $R$. Is it true that if $R/I$ is CM then also $R/J$ is CM?
This question is in some way the inverse of this one.
- 205
2
votes
2
answers
282
views
ring of idempotents of the integral extension of a ring
For any commutative ring $A$, the set of idempotents of $A$ will be denoted as $E(A)$. This set has a (canonical) ring structure. With addition defined by:
$$e+'f=e(1−f)+f(1−e)$$
where $+$ and $−$ are ...
- 2,275
5
votes
1
answer
504
views
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 ...
- 13k
4
votes
2
answers
1k
views
Diagrams consisting of triangles and squares
S. Lang gives a statement on page x of his 'Algebra':
Most of our diagrams are composed of triangles and squares as above, and to verify that a diagram consisting of triangles and squares is ...
- 2,934
12
votes
0
answers
2k
views
Finding ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer
As the title says, I'm trying to find ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer. I suspect $(2^k, 2^{k-1} x, 2^{k-2} x^2, ..., x^k)$ is generated by no fewer than $n=k+1$ ...
9
votes
1
answer
1k
views
Fixing a mistake in "An introduction to invariants and moduli"
On page 13 of the book "An introduction to invariants and moduli" of Mukai
http://catdir.loc.gov/catdir/samples/cam033/2002023422.pdf there is a mistake, in the end of the proof of Proposition 1.9. ...
- 14.3k
5
votes
0
answers
769
views
Looking for a reference for a generalization of the Weierstrass preparation theorem
I am looking for a reference for the following generalization of the Weierstrass preparation theorem for formal power series. Suppose that $A$ is a noetherian complete local ring with residue field $k$...
- 27k
2
votes
2
answers
983
views
Torsion in tensor products over noncommutative rings
I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things ...
- 1,391
3
votes
2
answers
344
views
Pseudo-idempotent matrix generating a free module
Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
- 33.8k
4
votes
2
answers
468
views
Maximal separable extensions of residue fields
Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose ...
- 43
1
vote
1
answer
307
views
A problem on Moebius transformations
We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (...
- 73
3
votes
2
answers
804
views
A problem for finite dimensional commutative algebra
Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and
(1) $A$ is finite dimensional as vector space
(2) for any ...
- 73
10
votes
3
answers
2k
views
A question about an application of Molien's formula to find the generators and relations of an invariant ring
In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai
proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. ...
- 14.3k
13
votes
1
answer
3k
views
When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
5
votes
0
answers
420
views
Ring of invariants of a finite subgroup of $GL_2(\mathbb{C})$
In the paper:
Kac, Victor; Watanabe, Keiichi, Finite linear groups whose ring of invariants is a complete intersection. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221–223
it is said in Remark 2 ...
- 221
28
votes
2
answers
3k
views
Maximal Ideals in Formal Laurent Series Rings?
Setup: Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x_1,\ldots,x_n]]$ denote the commutative ring of formal power series over $k$ in $x_1,\ldots,x_n$. We know that there is ...
- 283
0
votes
0
answers
254
views
What is Castelnuovo-Mumford regularity of this algebra?
Let $M=\mathbb{C}[f_1,f_2,\ldots,f_r]$ is finitely generated algebra, $f_i \in S:=\mathbb{C}[x_1,x_2,\ldots,x_n],$ $\deg(x_i)=1, 1<\deg(f_i)<99.$ Suppose that minimal free resolution of $...
- 301
3
votes
1
answer
293
views
Freeness of modules along ring homomorphisms
This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-...
- 30.5k
2
votes
1
answer
996
views
Count the number of homogeneous polynomials
Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some ...
- 248
7
votes
1
answer
757
views
Characterizing intersection of zero sets of elementary symmetric polynomials on R^n
Stated simply, the question is:
Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb i_{j}...
- 83
4
votes
5
answers
2k
views
What properties define open loci in families?
This question is somehow related to the question What properties define open loci in excellent schemes?.
Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over ...
- 16.1k
9
votes
3
answers
2k
views
If a polynomial f is irreducible then (f) is radical, without unique factorization?
Is there a short way to prove that for each irreducible polynomial $f$ in $k[x_1,...,x_n]$ the principal ideal $(f)$ is radical without using unique factorization of polynomials? A short proof of this ...
- 14.3k