All Questions
5,985 questions
2
votes
1
answer
521
views
Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
- 3,968
6
votes
1
answer
970
views
Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 156k
9
votes
2
answers
1k
views
Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 47.1k
8
votes
1
answer
3k
views
When a tensor product of two local rings is a local ring?
This is a follow-up to Is tensor product of local algebras local?.
Let $A, B$ and $C$ be local rings (commutative and noetherian). Suppose that we have local ring maps $C \to A$ and $C \to B$.
What ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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
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 ...
- 11.6k
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
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 ...
CommunityBot
- 1
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, ...
- 481
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. ...
- 31.2k
2
votes
1
answer
577
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
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 ...
- 7,328
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
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 ...
CommunityBot
- 1
8
votes
0
answers
493
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 ...
- 1,645
33
votes
2
answers
7k
views
Noetherian rings of infinite Krull dimension?
Since Noetherian rings satisfy the ascending chain condition, every such ring must contain infinitely many chains of prime ideals s.t. the heights of these chains are unbounded.
The only example I ...
- 161
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?
- 6,262
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
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
4
votes
2
answers
817
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 ...
- 63.1k
3
votes
1
answer
494
views
universal finite differential module of affinoid algebra
Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field.
The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| \...
CommunityBot
- 1
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
4
votes
1
answer
643
views
An application of Zorn's lemma.
Suppose that $R$ is a commutative ring, an $R$-module $M$ is said to be finitely embedded if $M$ has a finitely generated essential socle. Now Let $M$ be finitely embedded and not
artinian, let $S$ ...
- 21
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 ...
- 85.6k
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 $n=k+1$ ...
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 ...
- 3,702
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
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=\...
- 6,919
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 ...
- 19.6k
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 ...
- 6,262
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 (...
- 66.3k
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 $\mathbb C^n$. ...
- 11.6k
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 ...
CommunityBot
- 1
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 $x \in \mathbb{C}$ ...
- 17.1k
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
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 ...
- 11.6k
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}...
- 544
5
votes
5
answers
4k
views
Unique factorisation and the fact that $\mathbb A^2-0$ is not an affine variety?
While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question:
...
CommunityBot
- 1
4
votes
0
answers
367
views
criteria for reduced fibres
I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic ...
- 1,347
0
votes
1
answer
379
views
Is a tensor product of two dvrs semilocal?
Under what conditions is the tensor product of two dvrs semilocal?
The same question about being reduced.
Tensor product is taken over another dvr or over a field to make things simpler.
For ...
- 343
14
votes
2
answers
2k
views
Explicit ring of differential operators for polynomial algebras over the integers?
Does anyone know of a reference or have any idea for an explicit description of the ring of differential operators for polynomial algebras over the integers? I'm hoping there is something analogous to ...
- 3,702
12
votes
2
answers
1k
views
Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 44.7k
12
votes
4
answers
940
views
Factorizing polynomials in $\mathbf{Z}[[x]]$
Let $f(x)\in\mathbf{Z}[x]$ be a non-constant, irreducible polynomial, and let $\alpha \in\mathbf{C}$ be a root of $f(x)$. Denote by $\varphi_\alpha:\mathbf{Z}[x]\rightarrow\mathbf{C}$ the ring ...
- 20.8k
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
- 20.5k
9
votes
2
answers
1k
views
Projective resolution of modules over rings which are regular in codimension n
All rings are Noetherian and commutative, modules are finitely generated.
It is a theorem of Serre that over a regular ring $R$, every module has a finite projective resolution.
More generally, if $...
- 30.5k
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 423
8
votes
2
answers
425
views
Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 20.5k