# 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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### On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)

I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
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### Chevalley restriction theorem: group vs lie algebra version

Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as (1) $k[G]^G \xrightarrow{\sim} k[T]^W$ and ...
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### The cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term?

I recently studied the structure of the AR quiver of Dynkin type $\mathbb{A}_n/I$, $\mathbb{D}_n/I$ with $I$ any admissable ideal, and found that the cokernel of an irreducible monomorphism is always ...
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### When annihilator of ideal and ideal is co maximal

Let $R$ be a commutative ring with identity. It is not always true that ideal $J$ and annihilator of ideal $ann(J)$ are co maximal (ex: integral domain) Is there a sufficient (necessary) condition( ...
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### Concerning $\mathbb{C}[a,b,c,d]$, with special generators $a,b,c,d$

The following is a question I have asked in MSE; hopefully, it is ok to ask it here. Let $a,b,c,d \in \mathbb{C}[x,y]$. Assume that: (1) Each two of $\{a,b,c,d\}$ are algebraically independent. In ...
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### Some basic definitions on the ring $\mathbb{Z}_n[x]$ [closed]

Let $n\in\mathbb{N}^*$ be a composite number. So $\mathbb{Z}_n[x]$ is not an integral domain. Recently, we need to research the Chinese remainder theorem (CRT) on $\mathbb{Z}_n[x]$, we need to find ...
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### Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?

Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism? As I ...
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### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
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### Infinite sum in power series ring

Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
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### When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...
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### A characterization for a commutative ring with a special intersection property for prime ideals

Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
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### Formally étale at all primes does not imply formally étale?

All rings are assumed to be commutative and unital, with all homomorphisms unital as well. On last week's homework, there was a mistake in one of the questions: (2.5) Let $R\to S$ be a ...
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### Uniqueness of $\delta$-structure on a $p$-torsion ring

I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...
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### Automorphisms of the completion of a strict henselian local ring $R$ which come from automorphisms of $R$

Let $A\rightarrow R$ be a local homomorphism of Noetherian strict henselian local rings with completions $\hat{A},\hat{R}$. Let $u\in R^\times, x\in R$ be such that there is a unique $\hat{A}$-linear ...
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### Non-zero divisors on an $I$-completely flat module

Let $A$ be a commutative ring (not necessarily Noetherian), $I=(f_1, f_2, \dots, f_n) \subseteq A$ a finitely generated ideal that is generated by a regular sequence. Let $M$ be an $A$-module, and ...
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### Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty.$ How can we construct an ...
### A $q$-analogue of a characterization of polynomials by binomial coefficients
Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely ...