# Questions tagged [ac.commutative-algebra]

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

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### derived completion and flat base change

Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings.
We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete.
For a ...

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### Dimension inequality for primary groups

Let $p$ be a prime number and $G$ an abelian group.
The group $G$ is said to be $\textbf{primary}$ if every element of $G$ has order power of $p$.
For every natural number $n$, we define
$$\ker(p^n)=\{...

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### Solvability of derivation Lie algebras of local finite-dimensional commutative algebras

Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...

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### Integral closure in the total ring of fractions of reduced ring. Is it finitely generated?

Let $R = \bigoplus_{i = 0}^\infty R_i$ be a reduced finitely generated graded $\mathbb{C}$-algebra, $R_0 = \mathbb{C}$. Let $\overline{R}$ be the integral closure of $R$ in its total ring of fractions....

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### Presentation of Chevalley groups over Bezout domains

Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 ...

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### The eventual number of generators of modules of which $M$ is a subquotient

Let $R$ be a (commutative) ring and let $M$ be an $R$-module. Say that $M$ is subfinitely generated if $M$ is a submodule of a finitely-generated module. Write
$$\mathcal F(M) = \{ M \rightarrowtail N ...

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### Do graded-commutative rings satisfy the strong rank condition?

Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$.
It is ...

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### Characterisation of even characteristic quadratic system

$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that
$f_1(\bar{x}) = x_1 + x_5^2 + q_1$,
$f_2(\bar{x}) = x_2 + x_1^2 + q_2$,
$f_3(\bar{x}) = ...

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### A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?

One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...

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### Example of a ruled, CM, $ \mathbb{Q} $-factorial, normal, Mori dream space whose Cox ring is integral but not CM,

This question is related to one I asked here in Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM. In ...

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### Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM

Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...

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### For an element in the integral closure of an ideal $I$ - which power is in $I$?

Consider an ideal $I$ in a ring $R$. If $f \in R$ belongs to the integral closure of $I$, then there is $k_0 \geq 0$ such that $f^k \in I^{k-k_0}$ for all $k \geq k_0$. Are there any known upper ...

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### Lengths and additive invariants which preserve positivity

The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...

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### Commutant of irrep of $S_n$ (over local field)

Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...

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### How can I define and construct a module of integers of Grassmann variables?

This is a question that I asked in mathematics stack exchange :
I am interested about constructing a certain space where the coordinates are "integers" of grassmann variables. I know that ...

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### A question on classification of quadratic polynomials in even characteristic

$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...

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### Generalizations of Chevalley–Shephard–Todd's Theorem?

Major Edit
I will reformulate my question signicantly, given Anton Geraschenko's comment. The old version of the question is bellow.
For simplicity, my base field is $\mathbb{C}$. If $G<\...

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### Let $A, B$ be matrices with elements in $\mathbb{Z}_n$, does $\ker A = \ker B$ imply that they are row equivalent?

Let $A, B$ be matrices with elements in $\mathbb{Z}_n$. If $A x = 0$ and $B x = 0$ have the same set of solutions, where the vectors also have elements in $\mathbb{Z}_n$, does this mean that there is ...

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### Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...

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### Multiply an integer polynomial with another integer polynomial to get a "big" coefficient

I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...

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### Existence of a symmetric matrix satisfying certain irreducible conditions

Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...

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### Lazard module structure of rings with formal elliptic curve

Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...

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### Invariants of primary groups

In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...

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### Locally compact rings with reciprocals

A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...

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### An example of radical ideal which is irreducible but not prime

$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal.
In case $R$ is Noetherian, the radical of $I$ being ...

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### Quotient of a polynomial ring with a prime ideal is Cohen$-$Macaulay

[Bruns-Herzog, Exercise 2.1.17] Let $k$ be a field and $R = k[x_1, . . . , x_n]$. Suppose $\mathfrak{p} \subset R$ is a prime ideal, $ht\mathfrak{p} \in \{0, 1, n − 1, n\}$. Show that $R/\mathfrak{p}$ ...

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### Multiplication map by a ring element on an object vs. all its suspensions in singularity category

Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...

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### Is the integral closure of a henselian local domain of dimension $1$ again local?

Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$.
If $R$ is henselian, then is $\overline R$ also a local ...

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### Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced

I have a question about correctness of following statement claimed here in $\boxed{2} \ $:
Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...

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### Error in Altman Kleiman A Term of Commutative Algebra? [migrated]

Proposition 1.6 says that if $R$ is a ring and $a \in R$, then the map $\pi:R[X]\rightarrow R$ given by $\pi(X) = a$ has kernel generated by $(X-a)$ and $R[X]/(X-a)$ is isomorphic to $R$. But if $a$ ...

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### Existence of cyclic subspace decompositions for pairs of commuting matrices

Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...

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### Vanishing of self-hom in Spanier–Whitehead stabilization category

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...

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### Cohen–Macaulayness of $k[[x^2, x^3, xy, y]]$ over $k[[x^2, y]]$

Let $k$ be a field and $R = k[[x^2, y]]$ and $S = k[[x^2, x^3, xy, y]]$. Since $R \subset S$, is $S$ Cohen–Macaulay as $R$-module?
To check this, what I have observed is that in $S$, the maximal ...

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### Unitary representation of a group of automorphism on an abelian algebra

Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...

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### DG algebra structure on minimal free resolution of modules over regular local ring

Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...

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### Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring

Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...

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### If every ascending chain of ideals leading up to an ideal stabilises, is the ideal finitely generated?

I'm a fourth-year undergraduate currently studying a master's degree. In the last couple of weeks, I have been wondering about the interaction of the Noetherian condition with the prime ideals of a ...

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### When is a module a filtered colimit of finitely presented submodules?

For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both:
a filtered colimit of finitely generated $R$-submodules (by considering all finite subsets of $M$ and ...

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### Does the category of integral domains admit a symmetric monoidal structure?

Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...

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### On a generalized homotopy transfer theorem

In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...

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### Intersection of commutative factorial domains: completely integrally closed and Krull domain

Let $A=\bigcap_{t\in T}D_t$ be an integral domain such that $D_t$ is a commutative factorial domain for every $t\in T$. It is quite natural to see that $A$ is a completely integrally closed domain. ...

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### Understanding the relations without the knowledge of Plucker relations [duplicate]

Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so ...

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### Length of $\text{Tor}$ modules in complete intersection rings with characteristic $p$

Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$, it follows that $l(F(M))\geq p^nl(M)$, where $n=...

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### Estimation for dimension of support and associated primes of a module of depth zero

Let $(R,\mathfrak{m})$ be a local Noetherian ring and $M$ a finitely generated $R$-module of depth zero, ie $\operatorname{depth}(M):=\text{depth}_{\mathfrak{m}}(M)=0$.
Can we make some interesting ...

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### Degree three, codimension one subvarieties lying on a quadratic hypersurface

Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that
$V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...

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### Is every character of the algebra of continuous functions on a locally compact space some evaluation?

Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$.
Question. Given an arbitrary character (i.e. a non-zero ...

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### (non)reduced stabilizer scheme

A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...

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### Correspondence between fundamental group and geometric properties of $X$

At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...

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### Ext for commutative Gorenstein algebras

Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...

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### Serre subcategories of the category of chain complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...