# 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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### 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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### 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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### 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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### 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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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 $\... • 301 6 votes 0 answers 168 views ### Ext for commutative Gorenstein algebras Let$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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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 ...