# 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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### Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$. Let $f: k[x,y] \to R_{-1}$ be ...
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### Is completion of isolated singularity isolated?

Let $K$ be an algebraically closed field and let $A=K[x_1,\dots,x_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x_1,\dots,x_n)$ and ...
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### Ideal in ring of power series

Let $K$ be a field of characteristic $p$ and $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring over $K$ such that $n, p \geq 3$. Consider the ideal $I$ defined by \begin{...
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### Reduced Noetherian ring is the intersection of its localizations at primes associated to a nonzero-divisor

I shall quote proposition 11.3 of Eisenbud: Commutative algebra If $R$ is a reduced noetherian ring then an element $x\in K(R)$ belongs to $R$ if and only if the image of $x$ in $K(R)_P$ belongs to ...
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### Formally smooth algebra of a field

Let $k$ be a field of characteristic zero and $R$ a local $k$-algebra. By Stacks \tag 00TX, if we assume that $R$ is of finite type, then the $R$ is smooth over $k$ if and only if $\Omega_{R/k}$ is ...
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### A question about the span of a sequence of polynomials satisfying a linear recurrence

Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...
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### Open Morphism of Schemes

Let $f: X \to S$ a finite morphism between affine schemes $X=Spec(A), S= Spec(R)$. Denote by $\phi:R \to A$ the corresponding ring map. I'm looking for pure ring theoretical/algebraic tools/...
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### Is every universally catenary ring a going-between ring?

This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions. A ring $R$ is ...
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### Structure theorem for etale algebras over a more general ring than a field

I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition). In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite ...
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### Finite-dimensional algebras isomorphic as commutative unital rings

For which fields $k$ do there exist two finite-dimensional $k$-algebras of different dimension which are isomorphic as commutative unital rings? Some thoughts: for a finite field this can not happen, ...
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### Nice Form of Vector Field

Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...
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### Global to local principle for f.g. $\mathbb{Z}[x]$ modules

In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
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### Additive group of local rings

Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
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### depth and extension of sections

Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two. Suppose that we have a function on $U$, can we extend it to $X$?
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### Computing the codimension of the variety defined by a system of quadratic forms

Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...
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### Koszul -regular sequences (reference request)

Let $R$ be a ring and let $f:P\rightarrow P'$ be a surjective morphism of smooth $R$-algebras. Let $J$ be the kernel of this map. If $R$ is Noetherian, one can show that $J$ is locally generated by a ...
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### Locality of a tensor product of two fields

Assume that $K$ and $L$ are two field extensions of a field $k$. Is it known when a tensor product $K \otimes_k L$ is local? (A unital commutative ring is said to be local if it contains a unique ...
Let $L, K$ be fields, $L$ an algebraic extension of $K$, $R$ a valuation ring of $K$, and $\bar{R}$ the integral closure of $R$ in $L$. Is it true that for any maximal ideal $\mathfrak{m}$ of $\bar{R}$...