# Questions tagged [ac.commutative-algebra]

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

**4**

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110 views

### Ultrapower of a field is purely transcendental

Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$?
According to Chapter VII, Exercise 3.6 from Barnes, Mack "...

**6**

votes

**2**answers

267 views

### Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...

**11**

votes

**3**answers

780 views

### Homologically nice commutative rings

Let $R$ be a commutative regular local ring. Is it true that for every $\mathfrak p \in \mathrm{Spec}(R)$ there is a finitely generated $R$-module $M$ such that $\mathrm{projdim}(M)=\mathrm{ht}(\...

**1**

vote

**1**answer

101 views

### Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...

**2**

votes

**1**answer

80 views

### Are integral extensions of a catenary ring still catenary?

A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\...

**3**

votes

**1**answer

124 views

### Complete local rings, automorphisms and approximation

Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective.
Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...

**2**

votes

**0**answers

71 views

### Does a surjection between polynomial rings lose one Krull dimension per generator quotiented?

Let $k$ be a Noetherian commutative ring with unity (I am happy adding hypotheses, as I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a ...

**2**

votes

**1**answer

105 views

### Are the ring of power series and the ring of germs of holomorphic functions catenary?

A (commutative unitary) finite dimensional Noetherian ring $R$ is said to be catenary if for any prime ideal $\mathfrak{p}$ of $R$, one has $\dim R =ht(\mathfrak{p})+\dim(R/\mathfrak{p})$.
I am ...

**0**

votes

**0**answers

150 views

### Reconstructing almost known polynomial from a system of polynomials with common roots

We have $n$ algebraically independent degree $2$ homogeneous system of polynomials with $\mathbb Z$ coefficients in $n$ variables with exactly $t$ primitive (gcd of coefficients is $1$) integer roots ...

**3**

votes

**0**answers

136 views

### What is wrong with this argument that $ \mathbb{A}^{2}_{k} $ is not cancellative in positive characteristic?

I have a question about a result of Abyankar, Heinzer, and Eakin, and a similar result in Russell. One of the results in the first paper is that if $ Y $ is a variety such that $ \mathbb{A}^{1}_{k} \...

**3**

votes

**0**answers

44 views

### Ideals generalizing maximal ideals and ideals generated by regular sequences

Let $R$ be a local commutative Noetherian ring with maximal ideal $m$.
My questions concern
ideals $I \subseteq m$ of $R$ such that for any non-zero number $n \in \mathbb{N}$
the $R/I$-module $I^n/I^...

**9**

votes

**2**answers

445 views

### Relations between homogeneous polynomials

Edit: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation:
Let $\mathbb{P}$ be a projective space, and $V$ a general linear subspace ...

**6**

votes

**0**answers

90 views

### Standard reference/name for “initial ideals $\Leftrightarrow$ associated graded rings”

Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...

**6**

votes

**2**answers

221 views

### Maximize $L^p$ norm over sphere

For $p \in \mathbb{R}$, consider the function
$$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda_1^2 +...

**1**

vote

**0**answers

125 views

### Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...

**7**

votes

**1**answer

172 views

### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has this two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...

**42**

votes

**8**answers

8k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...

**1**

vote

**1**answer

387 views

### Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...

**2**

votes

**4**answers

1k views

### Closed-form for modified formal power series

This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.
We start with a formal power ...

**4**

votes

**0**answers

87 views

### Invariants of linear endomorphisms of tensor products

Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero.
Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...

**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 ...

**0**

votes

**1**answer

97 views

### Symmetric polynomials in two sets of variables

Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...

**2**

votes

**1**answer

191 views

### Étale fibration for $K[[X_1,…,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...

**-1**

votes

**0**answers

42 views

### Does $\mathrm{grade}(J,M) = \mathrm{depth} M$?

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay ring, $J$ an ideal of $R$ such that $\dim R/J >0$ and $M$ a finitely generated $R$-module.
Is $\mathrm{grade}(J,M) =\mathrm{depth} M$ true?
Here $\...

**1**

vote

**1**answer

92 views

### For a holonomic $D_X$-module $M$, can $gr M$ have embedded primes?

Let $M$ be a holonomic $D_X$ module. This means that the minimal primes in $\sqrt{Ann(gr M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But what about the embedded primes?
...

**2**

votes

**0**answers

114 views

### Noether’s “set theoretic foundations” of algebra. Reference

In [C Mclarty] we read
[Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...

**1**

vote

**1**answer

132 views

### Bound on number of proper ideals of norm equal to n

I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand:
Let $d$ be a positive non-square interger and set let $K = \...

**1**

vote

**0**answers

156 views

### Primes of the power series rings

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...

**3**

votes

**0**answers

102 views

### What is the $Ass(Ext^p_R(M,R))$?

Let $R$ be a Noetherian commutative local ring, $M$ a finitely generated $R$-module with $p=pd M<\infty$ (projective dimension of $M$). What is the relation between $Ass(Ext^p_R(M,R))$ and $Ass(M)$?...

**4**

votes

**0**answers

61 views

### A presentation for a subalgebra

Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$.
Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...

**3**

votes

**1**answer

140 views

### On a relation between the Hessian and the catalecticant matrix of a binary quartic form

I am currently working on a paper that requires using the theory of invariants of binary quartic forms. Playing around, I have found an interesting identity that gives the Hessian from the minors of ...

**3**

votes

**0**answers

43 views

### Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique
Reductions for Mészáros's Subdivision Algebra (published version in
SIGMA 2018, and slightly updated preprint
version with ...

**1**

vote

**0**answers

116 views

### Algebraic generalization of Pascal's identity

Let $R$, $S$ be rings with identity. A map $f: R \times R \to S$ is said to be an a $R_S$-Pascal map if, for all $r_1, r_2 \in R$, the following relations are satisfied :
$$\begin{align*}
f(r_1-1_G, &...

**5**

votes

**1**answer

143 views

### Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$.
Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...

**3**

votes

**0**answers

103 views

### Injective resolution of the ring of entire functions

Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...

**1**

vote

**0**answers

106 views

### Infinite-dimensional representation theory of $K[x]$

Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...

**21**

votes

**4**answers

2k views

### Does Smith normal form imply PID?

Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...

**2**

votes

**0**answers

50 views

### Algorithm to find the minimal number of multiplications

Start with the $\mathbb{Q}$-vector subspace $V_0$ of the polynomial ring $Q[x_1,\ldots,x_n]$ spanned by $\{1,x_1,\ldots,x_n\}$. In each step, we can choose an element of the form $v_iv_i'$ for $v_i,...

**2**

votes

**1**answer

101 views

### If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a finite-...

**0**

votes

**0**answers

80 views

### Reference request: J-P Serre, “Groupes finis d'automorphismes d'anneaux locaux réguliers”

Does anyone have, or know a link to, a copy of the paper named in the title? It is published in Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11. I do not have ready access to Serre's ...

**3**

votes

**0**answers

91 views

### Flatness through parametrization

Let $A$ be a $\mathbb{C}$-algebra. Let $\phi:\mathbb{C}[X_1,...,X_n] \otimes_{\mathbb{C}} A \to \mathbb{C}[t] \otimes_{\mathbb{C}} A$ be a ring homomorphism, sending $X_i$ to say $f_i \in \mathbb{C}[t]...

**7**

votes

**0**answers

93 views

### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...

**3**

votes

**1**answer

150 views

### Graded Grothendieck Group and Hilbert Polynomial

I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group.
Let $A$ be a noetherian graded $K$-...

**1**

vote

**1**answer

297 views

### primary regular sequences

Let $R$ be commutative regular local ring. Is it true, that for every $\mathfrak p \in \mathrm{Spec}(R)$, there is a $\mathfrak p$-primary $R$-regular sequence? (I.e. an $R$-regular sequence $\bf x$ ...

**2**

votes

**2**answers

1k views

### Projective modules over semi-local rings

Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.

**4**

votes

**1**answer

242 views

### Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$.
Let $L \subset \hat{K}$ be a separable finite ...

**23**

votes

**2**answers

1k views

### Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do ...

**1**

vote

**0**answers

89 views

### Fine tuning the growth rate of the degrees of polynomials

Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$.
Suppose that
$$\...

**1**

vote

**0**answers

109 views

### Factorizations of etale morphisms

Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated.
By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...

**0**

votes

**0**answers

53 views

### Normality of certain subrings of polynomial rings in characteristic p

Let $k$ be an algebraically closed field of characteristic p. Let
$Z\subset k[x_1,\cdots,x_n]$ be a graded $k$-subalgebra of a
polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in
$k[...