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
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16
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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 $R$ ...
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
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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 ...
7
votes
1
answer
497
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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 ...
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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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 ...
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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 $\mathbb{P}^2$...
4
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0
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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 ...
-1
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0
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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
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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 ...
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?
2
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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
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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 ...
3
votes
1
answer
183
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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 ...
1
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0
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92
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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 $\...
1
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0
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81
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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
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0
answers
51
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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 $...
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165
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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
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0
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59
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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 ...
6
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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{...
5
votes
0
answers
91
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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 $y=f_m(x)$. ($x,y,a,b$ ...
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 ...
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
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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, ...
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 ...
2
votes
1
answer
121
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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
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 ...
1
vote
0
answers
84
views
Regarding the common zeros of the system of equations
Consider the following two systems of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
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\...
1
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0
answers
48
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Can a surjective morphism between complete intersection rings be given by adding terms to a regular sequence?
Given a surjective morphism
$$\frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I}\twoheadrightarrow \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J}$$
where $I,J$ are genereated by regular sequences.
Question
Can ...
0
votes
1
answer
215
views
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 ...
6
votes
3
answers
819
views
A polynomial identity for $\displaystyle \sum_{k=0} ^m (-1)^ka^{m-k}b^k$
I asked this question on MSE here.
I was solving this problem:
Show that if $\gcd(a, b) = 1$ and $p$ is an odd prime, then $\displaystyle \gcd\left(a+b, \frac{a^p+b^p}{a+b}\right)=1$ or $p$.
This ...
4
votes
0
answers
270
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What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?
In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside:
One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
1
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0
answers
93
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Algorithms to decompose a graded module over $R[x]$, where $R$ is a PID
There is a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors. A few equivalent definitions are given below. The question is what computer algorithms ...
10
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1
answer
321
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How is Taylor-Wiles patching "horizontal Iwasawa theory"?
I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...
1
vote
1
answer
419
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
3
votes
0
answers
98
views
Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
0
votes
1
answer
99
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$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
3
votes
0
answers
134
views
Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
1
vote
1
answer
247
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Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
0
votes
0
answers
55
views
Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
1
vote
0
answers
44
views
generating set of polynomial ring
I am considering the polynomials $P=P[x_1,x_2,\ldots,x_n]$ with coefficients in a ring $R$. Consider a subset $S=\{p_1,p_2,\ldots,p_k\}$ of $P$. There is a map $f\colon P[x_1,x_2,\ldots,x_k] \to P$ ...
0
votes
1
answer
130
views
On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables
Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
0
votes
0
answers
82
views
Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
2
votes
1
answer
110
views
Projective dimension and subrings
$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...
3
votes
0
answers
160
views
Amalgamation of commutative subrings
Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...
1
vote
1
answer
83
views
Perturbing pole of Laurent polynomial/series in a single summand
I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
1
vote
0
answers
315
views
Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
8
votes
1
answer
584
views
A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...