Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,310
questions
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Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
0
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0
answers
16
views
Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
7
votes
1
answer
497
views
Original proof of Hilbert irreducibility theorem
Does there exist a modern exposition of Hilbert's original (1892) proof of the Hilbert irreducibility theorem? Of course, I can (and will) read Hilbert's original article, but I would feel more ...
0
votes
1
answer
215
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Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia
I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
1
vote
1
answer
134
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Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
1
vote
1
answer
169
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Shrinking the base field of an affine variety
This is a question on algebraic geometry/commutative algebra.
Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
2
votes
0
answers
101
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How to find a single-variable polynomial in a zero-dimensional ideal?
Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal?
If we ...
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0
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33
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If $G$ is a connected bipartite graph, then the edge ideal $I(G)$ is normally torsion free
I am studying the paper "On the Ideal Theory of Graphs" by Simis, Vasconcelos and Villarreal, Journal of Algebra 167, No. 2, 389-416 (1994), MR1283294, Zbl 0816.13003. I got stuck at theorem ...
1
vote
0
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42
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
2
votes
0
answers
152
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A direct proof that every projectivity between parallel lines is affine
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
18
votes
5
answers
7k
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Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
21
votes
1
answer
568
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Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors
For all primes up to $p=89$ there exists a product $Q=\prod_{j=1}^d(x-a_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a_j$ in $\mathbb F_p[x]$ such that $Q'$ has all its roots in $\mathbb ...
4
votes
0
answers
214
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When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?
Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$.
It is known that for two elements the following result holds:
$\langle u,v \rangle$ is a maximal ...
2
votes
1
answer
161
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Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
3
votes
3
answers
495
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Cohen-Macaulay property for reducible schemes
I have the following question about certain schemes being Cohen-Macaulay.
Let $X$ be the union of all coordinate $k$-planes in
${\mathbb A}^N$. Is it CM?
Let $R$ be a collection of $k$-element ...
-1
votes
0
answers
77
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Zero of gradient of a sum of squares of polynomial
Let $f_1,\ldots,f_n\in \mathbb{R}[x_1,\ldots,x_n]$ have non zero constant jacobian determinant.
Then is it true that $f_1^2+\cdots+f_n^2$ has any critical point?
Let me explain a motivation.
If ...
1
vote
0
answers
113
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A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
6
votes
0
answers
564
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Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
2
votes
0
answers
61
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On the closed subring of valuation rings
Let $(K,v)$ be a (not necessarily discrete) valued field of characteristic $p$ with valuation ring $\mathcal{O}$ and residue field $k$. We endow $K$ and $\mathcal{O}$ with the valuation topology. ...
-2
votes
0
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78
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System of polynomial equations and its Jacobian determinant
double post
https://math.stackexchange.com/questions/4875170/system-of-polynomial-equations-and-its-jacobian-determinant
Does this propostion hold?
Proposition
Let $\mathbb{C}[x_1,...x_n]$ be a ...
11
votes
1
answer
2k
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geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings
Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...
3
votes
1
answer
183
views
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...
6
votes
0
answers
489
views
Finite extensions of $\mathbb Q_p$
Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$?
Analogously in equicharacteristic, if $k=\overline {\mathbb F_p}$...
1
vote
0
answers
92
views
Recovering a ring from its localization and completion with respect to a fixed element
Suppose I have a commutative ring $k$ and an element $x \in k$. Then I can form the localization $k[x^{-1}]$ of $k$ at the multiplicative subset $\{ 1, x, x^2, ... \}$, and I can form the completion $\...
12
votes
3
answers
4k
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Existence of prime ideals and Axiom of Choice.
One of the must obvious equivalences of Axiom of Choice is the converse of Krull Theorem.
Bernhard Banaschewski in the Article titled by A New Proof that “Krull implies Zorn” showed a very simple ...
23
votes
7
answers
3k
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When can we prove constructively that a ring with unity has a maximal ideal?
Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
1
vote
1
answer
458
views
Family of curve singularities whose generic members are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
1
vote
0
answers
81
views
Factorial surfaces and smoothness
It is well-known that normal curves are smooth. Moreover, a UFD of Krull dimension one is regular. Is there any higher-dimensional analog?
For example, given a normal projective surface $S$ over $\...
2
votes
0
answers
51
views
When does an algebraically independent set "satisfy Noether normalization"?
Let $k$ be a field, $A$ a finitely generated $k$-algebra.
By Noether normalization, we know that there exists a finite morphism of $k$-algebras $\varphi : k[x_1, \ldots, x_d] \hookrightarrow A$, with $...
5
votes
0
answers
91
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
0
votes
0
answers
165
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
3
votes
0
answers
59
views
Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$
I have a (fairly large) system of polynomial equations, of the form
$$
c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots
$$
(In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
3
votes
2
answers
331
views
$R$-Module objects in cartesian closed categories
I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...
5
votes
2
answers
1k
views
Surjectivity of the natural map of injective module to its localization
Lemma 3.3 page 214 in Hartshorne's Algebraic Geometry book states: "If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its localization $...
2
votes
0
answers
142
views
Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology
Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
2
votes
1
answer
121
views
Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow A$ must ...
2
votes
1
answer
285
views
When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?
For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...
2
votes
1
answer
392
views
Finite generation of certain graded sequences of ideals
Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
4
votes
1
answer
233
views
Height of a conductor ideal
We say an extension of Noetherian rings $R\subset S$ is elementary subintegral if $S=R[b]$ for some $b\in S$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...
3
votes
0
answers
121
views
Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
13
votes
4
answers
2k
views
Does smoothness descend along flat morphisms?
Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$?
If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after ...
2
votes
2
answers
1k
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when tensor complex resolves S/I+J?
Assume that $I\subset k[x_1,\ldots,x_n]$ and $J\subset k[y_1,\ldots,y_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F_\cdot$ and $G_\cdot$, are ...
0
votes
0
answers
63
views
Hensel lifting of roots of a biquadratic polynomial
Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
12
votes
2
answers
749
views
Description of p-adics tensor the reals
What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to?
where $\mathbb{Z}_p$ are the p-adic integers.
I am specially interested in the case $p=2$.
Do know that $\mathbb{Z}_p\otimes_{\...
2
votes
1
answer
136
views
The presentations of finite complete local rings
Suppose that $R$ is a commutative ring such that there is a surjection $ \pi:\mathbf{Z}_p[[T_1,\cdots,T_n]]\to R$ of rings where $\mathbf{Z}_p[[T_1,\cdots,T_n]]$ is the ring of formal power series ...
0
votes
1
answer
266
views
Derivations and ideals
Let $R$ be a regular local ring and $I$ and ideal of $R$. If $D$ is a derivation of $R$, let
$$\lambda_D:I/I^2\to R/I$$
be the composition of the restriction of $D$ to $I$ and the quotient map $R\to R/...
21
votes
6
answers
3k
views
A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
5
votes
1
answer
1k
views
$\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$
Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\...
6
votes
2
answers
308
views
Are finite projective modules over $R[t]$ free when $R$ is DVR?
Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$?
As far as I understand, this should be a ...
6
votes
1
answer
331
views
Integral group rings on which stably free modules are free
Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...