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Questions tagged [ac.commutative-algebra]

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

4
votes
0answers
107 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 "...
1
vote
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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 ...
3
votes
0answers
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
0answers
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^...
0
votes
0answers
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 ...
9
votes
2answers
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 ...
1
vote
0answers
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
1answer
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^...
4
votes
0answers
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 ...
6
votes
0answers
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 ...
0
votes
1answer
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,...
-1
votes
0answers
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
1answer
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
0answers
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 ...
2
votes
1answer
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 ...
6
votes
2answers
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
1answer
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
0answers
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 \...
4
votes
0answers
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
0answers
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 $Ass(M)$?...
3
votes
1answer
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
0answers
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
0answers
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, &...
3
votes
0answers
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
0answers
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 ...
2
votes
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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 $K$-...
6
votes
2answers
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 $...
4
votes
1answer
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 ...
1
vote
0answers
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
0answers
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 ...
23
votes
2answers
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 ...
0
votes
0answers
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[...
1
vote
0answers
53 views

Computing Gröbner basis elements of some constant degree

I'm wondering if there is any way or any special set of ideals such that there is an efficient way to compute elements of degree at most $d$ in a Gröbner basis for that ideal. If you have any paper ...
5
votes
1answer
301 views

Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?

Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime. I have two questions: Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
1
vote
2answers
178 views

Computing Groebner basis for a complicated systems of polynomials

I am trying to solve complicated systems of polynomial equations. The first step is to determine maximal sets of independent variables for the solution manifold (ideal) or the number of isolated ...
8
votes
0answers
171 views

Is $[JK:(x)][JK:(y,z)]\subseteq JK$ in $k[x,y,z]$?

Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $J,K\subseteq m$ be proper homogeneous ideals in $S$. Is this true that we always have: $$[JK:(x)][JK:(y,z)]\subseteq JK \ ?$$ Some ...
2
votes
0answers
92 views

Flatness of modules over dual numbers

Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...
1
vote
0answers
93 views

Separable extensions & topology vs inseparable extensions and algebra

In the note Properties of fibers and applications, Osserman writes above Definition 1.5: Intuitively, the point is that phenomena relating to topology can only change under separable extensions, ...
27
votes
2answers
1k views

A sum involving roots of unity

Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that \begin{align*} \sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}. \end{align*} Since $\...
7
votes
0answers
202 views

Perfectly balanced sets of complex numbers

Suppose that $X$ is an inclusion-minimal finite set of non-zero complex numbers such that $\sum\limits_{x\in X}x^n=0\ $ for infinitely many integers $n$. Can the cardinality of $X$ be a ...
-3
votes
1answer
185 views

A common name for a functorial construction of Commutative Algebra?

I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name. Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...