# Questions tagged [ac.commutative-algebra]

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

3,821
questions

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### Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...

**2**

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153 views

### What generalizes symmetric polynomials to other finite groups?

Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...

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73 views

### Automorphisms of rational functions of two variables

Let $k$ be a field. In 1941, Jung showed that all polynomial $k$-algebra automorphisms of the rational (polynomial) functions in two variables, denoted by $k(x,y)$ can be written as compositions of ...

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64 views

### Eventually non vanishing tors

Let $A$ be a commutative $k$-algebra, for $k$ a field of characteristic $0$. Let $Perf_{A}$ denote the dg category of cohomologically graded $A$-modules and let $M\in Perf_{A}$ be a classical perfect ...

**1**

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**1**answer

66 views

### Jacobson radical of a derived $I$-complete ring

Let $A$ be a commutative ring and $I \subseteq A$ a finitely generated ideal (I am not assuming that $A$ is Noetherian).
Assume that $A$ is derived $I$-complete, meaning, let's say, that $\mathrm{...

**3**

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100 views

### Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...

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109 views

### When is a maximal ideal isolated?

Let $R$ be a commutative ring with 1, and let $0\not=a$ be an element of $R$ such that there exists an ideal $I$ of $R$ with the property that $a\not\in I$ and if $J$ is an ideal with $a\not\in J$ ...

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141 views

### $\mathcal{C}$-filtering of modules inherited by submodules

I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology.
DEFINITION: Let $\mathcal{C}$ be a ...

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145 views

### Induced morphism of completions of local rings

Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced ...

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79 views

### Nice forms for the Weierstrass equation

The motivation for this question comes from the study of elliptic curves but the question itself is about choosing convenient algebraic transformations.
Fix a base ring (commutative, with a ...

**3**

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**1**answer

121 views

### A local ring with a unique minimal ideal

Let $R$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $I$ and $m$ of $R$ such that for each ideal $K$ of $R$ with $0\not=...

**6**

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235 views

### Flat maps and Zariski tangent spaces

Let $f:A \to B$ be a finite flat local homomorphism of noetherian local rings.
Are there some nice conditions on $A$ and $B$ which guarantee that the dimension
of the Zariski tangent space of $...

**4**

votes

**1**answer

125 views

### Condition for a monomial to belong to a particular ideal

Consider the polynomial ring $R[x_1,x_2,\ldots,x_n]$, where $R$ be an algebraically closed field (preferably $\mathbb{C}$) and the ideal $J=\langle m_1, m_2,\ldots,m_n\rangle$ generated by monomials . ...

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65 views

### Surjective maps between power series rings

Suppose that $A$ is a complete neotherian local ring and that we're given a surjective homomorphism
$f: A[[x_{1}, \ldots, x_{n}]] \rightarrow A[[t_{1}, \ldots, t_{m}]]$.
Can we always find a ...

**1**

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58 views

### Infinitely generated modules with weirdly jumping ranks

Assume we have a commutative Noetherian ring $R$ with a unit and a connected spectrum and a module $M$ over it. The following is known:
$\mathrm{Spec}(R)$ has a finite stratification (in the sense of ...