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(...
