Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
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The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?
In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
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Factoring a polynomial into linear factors by ring extension
The following sounds so natural, I'm surprised I have never asked it before:
Question 1. Let $R$ be a commutative ring. Let $P \in R\left[X\right]$ be a polynomial. Can we find a commutative ring $S$ ...
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An omission in K. Conrad's notes on the conductor ideal
I am referring to the very useful K. Conrad's notes on the conductor ideal of an order in a Dedekind domain: https://kconrad.math.uconn.edu/blurbs/gradnumthy/conductor.pdf
$\DeclareMathOperator\Cl{Cl}$...
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Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)
Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
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Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
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Does going down property imply a corresponding map is open without "finiteness"?
Does the following proposition hold?
Proposition
Let f:A$\rightarrow$B be a ring homomorphism
If f has going down property then the corresponding map
$f^*$:Spec B$\rightarrow$Spec A is open map.
I ...
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Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
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When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
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Torsion of modules
Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
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Ideal quotient and regular sequences
Let $(R,m)$ be a Noetherian local ring. $(x_1,...,x_d)\subset (y_1,...,y_d)$ are two $R$-regular sequences. If we write $x_i=\sum_{j=1}^{d}a_{ij}y_j$, then we have $(x_1,...,x_d):(y_1,...,y_d)=(x_1,......
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Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
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Explicit construction for Cohen’s $p$-ring with imperfect residual field
Apologize if this is a below-research-level question. Asked in stack exchange but no response yet.
Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of ...
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Ideal membership and change of fields
Let $R=k[x_1,...,x_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f_1,...,f_m)$ a homogeneous ideal.
With Macaulay2, one can compute the Groebner basis of $I$ when $...
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For a finite-type $\mathbb{Z}$-algebra $A$, is the intersection of all ideals $I$ such that $A/I$ is finite and local necessarily zero?
Background: A referee has suggested a shorter proof of one of my results, but I'm having trouble justifying one of their assertions. The setting is that $A$ is a commutative ring, and the referee's ...
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Is $C^r(M)$ non-isomorphic to $C^s(N)$ for $r\neq s$ and nontrivial manifolds $M,N$?
This is an obvious continuation of an MO question. Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a ...
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Examples of integral ring extensions that $\operatorname{ht}P \lt \operatorname{ht}P\cap A$
$\DeclareMathOperator\ht{ht}$All rings are commutative Noetherian with identity.
Exercise 9.8 of Matsumura's book Commutative ring theory: Let $A$ be a ring and $A\subset B$ an integral extension. If $...
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For a finite locally free $A\to B$ when does the different equal the Noether different?
(Cross-posted from MSE.)
All rings commutative with $1$. Let $A\to B$ be an $A$-algebra which is finite projective, meaning $B$ is finitely generated projective as an $A$-module, so there is the trace ...
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Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
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On linear schemes
Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...
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Local cohomology and residues of rational functions at 0 and $\infty$
Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where
$s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector
space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...
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Examples of compressed Gorenstein ring
Let $(R,\mathfrak{m},k)$ be a Gorenstein local Artinian ring of socle degree $s$ and embedding dimension $e>1$. We set
$$
\varepsilon_i=\min\left\{ \binom{e-1+s-i}{e-1}, \binom{e-1+i}{e-1}\right\} \...
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R-module hom a direct summand of Z-module hom?
$\DeclareMathOperator\Hom{Hom}$Fix a commutative ring $R$. For $R$-modules $M$ and $N$, there is an inclusion of abelian groups $\Hom_R(M,N) \to \Hom_{\mathbb{Z}}(M,N).$ Are there conditions on $R$ ...
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Example showing that $\mathbb{P}^1$ does not preserve monics
Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(...
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Symmetric powers, localisation and Frobenius
I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
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Constructive proof of univariate McCoy theorem without Dedekind-Mertens?
McCoy's theorem (one of them) says that for any commutative ring $A$, $f\in A[x]$ is a zero-divisor iff it's annihilated by a scalar in $A$.
There's a widespread proof by contradiction. There's also a ...
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Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
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Brauer group of rational numbers
Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...
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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 ...
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Characters of algebra of Schwartz functions
Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.
Question: Does there exist some character (non-zero multiplicative ...
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Kernels of actions on truncated polynomial algebra
Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
- 5,039
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Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
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Latest "A Term of Commutative Algebra" by Altman and Kleiman? [closed]
Where can I find the latest revision of A term of Commutative Algebra by Allen B. ALTMAN and Steven L. KLEIMAN? Is my 2013 version ok?
It is hard to locate the latest one; many old revisions and ...
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Undecidability of irreducibility of infinite families of integer polynomials?
A recent question, Is irreducibility of polynomials $\in\mathbb{Z}[X]$ over $\mathbb{Q}$ an undecidable problem? was quickly answered in the negative. I am wondering if there is a simple example of a ...
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Top local cohomology - recommendations
I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand ...
user340793
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Absolute integral closure of Noetherian local domain
Let $(R,\mathfrak{m})$ be a Noetherian local domain, let $K(R)$ be the total fraction field of $R$ and let $R^{+}$ be the absolute integral closure of $R$. Consider the $R^{+}$-module $K(R)\otimes_{R} ...
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What is the right level of generality for $(R/a) \times (R/b) \cong (R/\gcd(a,b)) \times (R/\operatorname{lcm}(a,b))$?
Let $R$ be a principal ideal domain. Let $a$ and $b$ be two elements of $R$. Let $g$ be a greatest common divisor of $a$ and $b$, and let $\ell$ be a least common multiple of $a$ and $b$. (Of course, ...
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Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?
Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{...
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Maximal ideals of the ring $\mathbb C \{T\}$
Consider the Banach $\mathbb C$-algebra
$$
\mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace
$$
With the norm given by $\| \sum a_i T^i\| = \sum |...
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Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?
Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?
...
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How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?
$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...
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Is every additive, left exact functor isomorphic to a hom functor?
Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{...
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On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
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Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
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Do power sums determine the variables?
In my analysis research, I came across the following problem. Given $n$ positive real numbers $x_1,\dots,x_n$, consider the $n$-many power sums
$$ p_3 = x_1^3 + x_2^3 + \dots + x_n^3 , $$
$$ p_5 = ...
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101
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Theta functions in acyclic cluster algebras
Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
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Number of solutions of overdetermined quadratic polynomial equations
Given $m$ linearly independent quadratic polynomials over the complex field in $n$ variables with $m>n$ and such that the number of zeros, say $N$, is finite, is there a known or conjectured strict ...
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What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?
$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...
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