Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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-2 votes
2 answers
731 views

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 views

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 views

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 ...

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