Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,316
questions
-2
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2
answers
731
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Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
1
vote
1
answer
332
views
Is this algorithm for primary decomposition correct?
I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right.
Since Singular (the ...
2
votes
1
answer
100
views
Higher degree of Hilbert's irreducibility theorem
A basic form of Hilbert's irreducibility theorem can be formulated as follows:
Let $f(t,x)\in\mathbb{Q}[t,x]\setminus\mathbb{Q}[t]$ be an irreducible polynomial. There exist infinitely many linear ...
4
votes
0
answers
182
views
Prime ideal generated by two quadratic polynomials
Let $q_1$ and $q_2$ be two irreducible quadratic homogeneous polynomials in $\mathbb{C}[x_0, \ldots, x_n]$.
Consider the ideal $\langle q_1, q_2 \rangle$.
When this ideal is prime?
I am ...
9
votes
1
answer
429
views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
3
votes
0
answers
233
views
Is there a converse of Abhyankar-Moh-Suzuki theorem?
The following question is the same as this question; I ask it here, since I have not got any comments there (I really apologize if I should have waited some more time before asking it here; it is just ...
3
votes
0
answers
132
views
Hilbert's irreducibility theorem for prime ideals
A typical formulation of Hilbert's irreducibility theorem is like this (see [1]):
Let $k=\mathbb{Q}$ and $f\in k[x_1,\ldots,x_n,y_1,\ldots,y_m]$ be an irreducible polynomial. There exists a Zariski ...
7
votes
1
answer
237
views
Is being a Frobenius algebra a rare condition for local algebras?
Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the ...
3
votes
0
answers
171
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
4
votes
1
answer
250
views
Embedding a finite morphism into a finite morphism of smooth varieties
Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (...
28
votes
4
answers
4k
views
What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
2
votes
0
answers
393
views
Intersection of all positive powers of prime ideal in an integral domain with all ideals of finite height
Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ? If that is not true in general, then ...
5
votes
1
answer
404
views
Does Fermat's last theorem hold in the Grothendieck ring of the ordinals?
Inspired by this question and its answer, I am curious whether or not Fermat's last theorem holds in the Grothendieck ring of the ordinals under Hessenberg (commutative) operations.
The excellent ...
11
votes
1
answer
797
views
Is $\mathbb{R}$ a $\mathbb{C}$-module without AC?
Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial (scalar multiplication is not identically zero) $\mathbb{C}$-module.
Now my questions are?
0.Is it consistent with $ZF$ that $\mathbb{R}$ ...
6
votes
1
answer
274
views
Binomial coefficients in discrete valuation rings
Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer.
For any integer $d,n\ge 0$, define:
$${\pi^d \choose n} := \...
1
vote
1
answer
242
views
Coefficients of the monomials appearing in a Schubert polynomial
It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
4
votes
1
answer
345
views
Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$
Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$).
(1) Is there a ...
0
votes
0
answers
105
views
Kelly's theorem for quadratic polynomials
Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \...
8
votes
1
answer
316
views
Which ideals have standard Hilbert series?
Let $m$ and $d$ be two positive integers.
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...
3
votes
1
answer
971
views
How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle
I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{...
6
votes
1
answer
632
views
Jordan form on an invariant vector subspace
Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\...
3
votes
0
answers
278
views
Witt vectors with $p$-torsion
Let $R$ be a commutative ring, $W(R)$ is a ring of Witt vectors over $R$. Can you give an example of a ring $R$ such that $W(R)$ has $p$-torsion?
3
votes
0
answers
173
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
6
votes
3
answers
791
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Algebraic characterization of commutative rings of Krull dimension 1,2, or 3
A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of zero-...
1
vote
2
answers
189
views
A question arising in the distribution theory of L. Schwartz
Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ...
1
vote
0
answers
130
views
What can be said about $\{\deg(f),\deg(g),\deg(h)\}$, such that $k[f,g,h]=k[t]$?
Let $f=f(t),g=g(t),h=h(t) \in k[t]$, $k$ is a field of characteristic zero.
Denote $\deg(f)=a$, $\deg(g)=b$, $\deg(h)=c$. Assume that $a \geq 2, b \geq 2, c \geq 2$.
Assume that $k[f,g] \neq k[t]$, $...
2
votes
1
answer
286
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
5
votes
1
answer
1k
views
local ring all whose non-maximal ideals are finitely generated
Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ?
It is easy to see ...
4
votes
0
answers
94
views
Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases
Suppose that $k$ is a field and I have two ring homomorphisms
$$
\phi, \psi :k[x_1,...,x_m] \to k[y_1,...,y_n].
$$
How can I use Gröbner bases (or other computational tools) to compute the subring of ...
3
votes
1
answer
168
views
Hermitian forms over $K\times K$
Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$.
When $V$ is a free module, ...
1
vote
1
answer
439
views
Valuation ring whose maximal ideal and every ideal of finite height are principal
Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
7
votes
0
answers
262
views
Hilbert series for invariant ring
I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with).
To put it in some broader context, let $G$ ...
8
votes
1
answer
1k
views
Adjoints of scalar extension and scalar coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts):
$h^*$: Scalar extension by ...
9
votes
2
answers
345
views
Restricted extension of prime ideals of the ring of polynomials over $\mathbb{Q}$
Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ be a ring of multivariate polynomials over $\mathbb{Q}$. Let $V=\mathop{\rm span}\{x_1,x_2,\ldots,x_n\}$ be the linear subspace spanned by the indeterminates. ...
4
votes
0
answers
109
views
On properties of affinoid algebras preserved under reduction
Let $R$ be a complete valuation ring of rank one, with maximal ideal $m$ and residue field $k$. Consider a $K$-affinoid algebra $\mathcal{A}$ and its reduction given by $\widetilde{\mathcal{A}}:=\...
3
votes
0
answers
63
views
Structure of $k[X,Y,X^a/Y^b]$, name for such rings
Rings of the form $k[X,Y,\frac{X^a}{Y^b}]$, where $k$ is a finite field and $a$ and $b$ are relatively prime natural numbers, are showing up as residue rings in a problem I'm studying, and I'm ...
5
votes
3
answers
407
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.
7
votes
1
answer
531
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Domain in which every ideal is generated by at most three elements
Does anyone know of a domain in which
every ideal is generated by at most three elements
at least one ideal is generated by no less than three elements?
What if three is replaced by n: a positive ...
13
votes
3
answers
552
views
UFD containing element with finite quotient
Since this question could not be answered at math.stackexchange, I would like to try my luck here now:
Does anyone know an example of a unique factorization domain $R$ that is
(i) not a Dedekind ...
5
votes
1
answer
277
views
When is a zero dimensional local ring a chain ring?
A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ...
4
votes
1
answer
129
views
The existence of $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}1)v=(1\otimes_{\mathbb{K}}a)v$
If $A$ is a finite dimensional commutative, associative, unital algebra over a field $\mathbb{K}$ then does there exist a non-zero vector $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}...
1
vote
1
answer
154
views
A property similar to arithmetical property
By an arithmetical ring is understood a commutative ring $R$ with
identity for which the ideals form a distributive lattice, i.e., for which
$(I+J)\cap K=(I\cap K)+(J\cap K)$ for all ideals $I, J$ ...
1
vote
0
answers
65
views
Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
5
votes
1
answer
328
views
When is a linear combination of the elementary symmetric polynomials reducible?
Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...
2
votes
1
answer
439
views
Polynomials with no multiple root
Let $a,d$ be polynomials of $\mathbb Z[X]$ with $\deg a>\deg d\ge0$ and $P$ be a polynomial of $\mathbb Z[X]$. Consider an infinite sequence of integers $(\lambda_n)_n$. Can one assert there exists ...
10
votes
1
answer
353
views
Is the product of Jacobson rings a Jacobson ring?
I asked this question on Mathematics Stackexchange, but got no answer.
Is the product
$$
\prod_{i\in I}A_i
$$
of a family $(A_i)_{i\in I}$ of Jacobson rings
a Jacobson ring?
(Here "ring" means "...
1
vote
1
answer
386
views
Tensor of Hom spaces over an algebra
Given a field $k$ of null caracteristic, an associative/commutative/unital $k$-algebra $A$ and two free $k$-modules $M,N$, denote by $Hom_k(M,A)$ the space of $k$-linear maps from $M$ to $A$.
I want ...
2
votes
1
answer
414
views
Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?
Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or ...
4
votes
0
answers
737
views
When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...