# 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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### Valuation with values in a semiring?

The notion of "valuation" on a ring $R$ is peculiar in that as typically presented, it is really two notions, neither of which subsumes the other.
A valuation can be a homomorphism $v: (R,\times) \to ...

**7**

votes

**1**answer

135 views

### The Image of a Derivation is Contained in the Jacobson Radical

Let $A$ be a finite-dimensional unital commutative associative algebra over a field $K$ of characteristic $0$. Is it true that for any derivation $D$ of $A$ we have $D(A) \subseteq J(A)$ where $J(A)$ ...

**3**

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95 views

### Cohen structure theorem with explicit equations

By Cohen structure theorem, a complete regular equicharacteristic Noetherian local ring is isomorphic to a power series. In particular, this should hold for finite extensions of power series $k[[t]][\...

**10**

votes

**1**answer

226 views

### Iteration of a morphism and flatness

Let $A$ be a Noetherian local ring, $f:A \rightarrow A$ be a local ring morphism. Assume some power of $f$ is a flat morphism, must $f$ be flat as well?
Motivation: Kunz's theorem shows the result is ...

**4**

votes

**1**answer

219 views

### For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?

Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?

**1**

vote

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99 views

### Intersection condition for polynomial ring and maximal ideals

In ring theory, there is interest in a condition known as the intersection condition. There is a brief comment in McConnell-Robson along these lines: Consider the ring $R = k[x,y]$ where $k$ is a ...

**2**

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52 views

### Linearity of a canonical morphism related to scalar extension and coextension

Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...

**12**

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316 views

### If a polynomial ring is a finite flat module over some subring, is that subring itself a polynomial ring?

A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring ...

**16**

votes

**1**answer

597 views

### Lang's Jacobian identity: slicker, elementary proof?

In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...

**3**

votes

**1**answer

84 views

### What is K+M structure?

In the following paper (Example 2.1), it has been mentioned to K+M to provide an example of a pseudo valuation domain which is not a valuation ring, and its reference is Gilmer's book, but I have no ...

**0**

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83 views

### How to write the involution in the new coordinates?

Let $f= xy^3+y^4-x^2+xy$. Using the following codes in Maple,
f := xy^3+y^4-x^2+xy;
v := Weierstrassform(f, x, y, x0, y0);
I obtain the following result:
\begin{align}
& f_0 = {{ x_0}}^{3}+{{...

**16**

votes

**1**answer

531 views

### Counting real zeros of a polynomial

I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...

**8**

votes

**1**answer

188 views

### The different gradings of a graded ring, and their schemes

Let $(A,g)$ be a graded commutative ring, where $A$ denotes the commutative ring, and $g$ its grading. What can be said about the set $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...

**1**

vote

**1**answer

107 views

### Inverse limit of $p^n$-torsion abelian groups

Let $p$ be a prime and let $\{A_n\}_{n > 0}$ be an inverse limit of abelian groups such that $A_n$ is $p^n$-torsion with $A_n/p^{n - 1} \cong A_{n - 1}$ (these isomorphisms are part of the data). ...

**16**

votes

**1**answer

431 views

### Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?

I asked this question on Mathematics Stackexchange (link), but got no answer.
Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall ...

**2**

votes

**1**answer

111 views

### Ideal of the union of two zero loci

Let $X$ be a smooth (complex) projective variety and $\mathcal E$ a globally generated vector bundle on $X$ of rank $< dim(X)$. I would like to know, please, if there is a nice description (exact ...

**3**

votes

**2**answers

341 views

### Finite maps and jacobian condition

Let $k$ be an algebraically closed field and take $f_{1}, ..., f_{n} \in k[X_{1},..., X_{n}]$ with the jacobian condition: $\det J_{f} = 1$. Let $A:= k[X_{1},...,X_{n}]/(f_{1},...,f_{n})$ and ...

**6**

votes

**1**answer

316 views

### The soccer splitting problem in arbitrary commutative ring

There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...

**7**

votes

**0**answers

153 views

### Reference request: A commutative variant of the Exterior Algebra

Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...

**7**

votes

**3**answers

268 views

### Infinite Galois descent for finitely generated commutative algebras over a field

Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$.
Write $G={\rm Gal}(k/k_0)$.
Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit.
Then ...

**0**

votes

**1**answer

140 views

### How to classify a plane complex curve?

Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates)
\begin{align}
& {p_1}^2 {p_2}^2 c_1 {t_1}^2 ...

**1**

vote

**1**answer

80 views

### Finding a characteristic for which the zero-locus of an ideal is not empty

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...

**7**

votes

**1**answer

214 views

### Minimal resolution of local cohomology module

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...

**4**

votes

**0**answers

152 views

### Can nonflat deformations of singularities always produce Cohen-Macaulay rings?

To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...

**1**

vote

**2**answers

270 views

### A relation between an ideal and its radical

Let $I$ be an ideal of a commutative ring $R$ with $1$ such that $\sqrt{I}=I_1\cdots I_n$ where $I_i's$ are pairwise comaximal ideal of $R$. Are there ideals $J_1,...,J_n$ of $R$ such that $V(I_i)=V(...

**2**

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147 views

### Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...

**7**

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194 views

### Embedding a given affine variety as a divisor

Let $A$ be a finite type algebra over $\mathbb{C}$. Does there exist a finite type $\mathbb{C}$-algebra $B$ and a nonzero divisor $b \in B$ such that $B/b \cong A$ and $B[1/b]$ is Cohen-Macaulay (or, ...

**11**

votes

**1**answer

894 views

### How to visualize the Frobenius endomorphism?

As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked some people in real life and they said they didn't know and that I could go and ask on MO and possibly get ...

**2**

votes

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25 views

### Does the $G$-norm coincide with the ordinary norm for “quasi-$G$-Galois” extensions

Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...

**3**

votes

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69 views

### Normal set of points in the plane

When defining the normality of a scheme in the book The Geometry of Syzygies, Eisenbud says that there are just a few facts that are known for their bounds.
Given $X \subset \mathbb{P}^r$, we say ...

**0**

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194 views

### Are the integers a vector space or algebra over “some” field or over “some” ring?

Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...

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134 views

### Derivations of special rings

Consider $p$-adic field $\mathbb{Q}_p$ and let $A$ be any finite dimensional $\mathbb{Q}_p$ algebra without zero divisors (and maybe without $1$). Now we can look on $A$ as on a ring $A_{\mathbb{Z}}$ (...

**3**

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143 views

### Ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, ...

**5**

votes

**1**answer

174 views

### about morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$

It is well known that there is a correspondence between homomorphism of rings $A\to B$ and morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$.
Question: (1) In analogy, is there ...

**4**

votes

**1**answer

136 views

### Betti numbers of a Cohen-Macaulay Module in small projective dimension

I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...

**7**

votes

**2**answers

469 views

### A geometric proof of Krull's Principal ideal theorem

Krull's height theorem states that in a Noetherian, local ring $(A,\mathfrak m)$, for any $f \in \mathfrak m$, the minimal prime ideal containing $(f)$ is at most height $1$.
This is a very geometric ...

**3**

votes

**1**answer

125 views

### Solutions to a system of homogeneous equations (inequalities)

Let $f_1,\ldots,f_r \in \mathbb{R}[x_1,\ldots,x_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$.
For which values of $r,n,d$ there exists a real ...

**7**

votes

**1**answer

256 views

### Intersection of free/affine submodules, comparison with vector spaces

If $W_1,W_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d_1, d_2 \leq d$, respectively, then $d_1 + d_2 > d$ suffices to guarantee $W_1 \cap W_2 \neq \{0\}$. There are ...

**1**

vote

**1**answer

93 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

**2**answers

361 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 ...

**2**

votes

**1**answer

79 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 ...

**3**

votes

**0**answers

59 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 ...

**7**

votes

**2**answers

289 views

### Generalized Smith Theorem for the torsion of cokernels

Let $R$ be a (commutative) domain and let $Q$ be its fraction field.
Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.
Let $I_k=(\det \...

**4**

votes

**0**answers

209 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 ...

**1**

vote

**0**answers

98 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

182 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 ...

**9**

votes

**1**answer

328 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 ...

**4**

votes

**1**answer

155 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 (...

**6**

votes

**0**answers

285 views

### Curious anti-commutative ring

Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
By $\Lambda$ I mean the free anti-commutative algebra, $x_i x_j = - x_j x_i$,
either ...

**5**

votes

**1**answer

277 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 ...