# 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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346 views

### Why are there elementary equations that are not solvable in closed form?

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.
$\log\colon x\mapsto\log(x)$; $x\...

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263 views

### Finite dimensional commutative algebras containing infinitely many nilpotents whose $d$-way products are nonzero

I'm interested in the following strange question: for some $d > 1$, what is the minimum dimension of a commutative $\mathbb{C}$-algebra containing infinitely many elements that square to zero, but ...

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90 views

### Catenarity and epimorphisms of rings

Let $R$ be a commutative ring. The following are well-known:
If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary.
If $R$ is catenary and $S\subseteq R$ is ...

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192 views

### What is the difference between total integral closure and integral closure?

I was advised here to make this a new question:
What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...

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132 views

### Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...

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41 views

### Non-separated Nygaard filtration

Let $S$ be a quasiregular semiperfectoid ring, then on its prism we may define the Nygaard filtration (Definition 12.1 in this preprint). What is an explicit example where it is not separated?

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126 views

### Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...

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175 views

### Generalised CRT - How to compute the cokernel?

Let $R$ be a commutative ring of dimension one with minimal prime ideals $P_1,\ldots,P_n$. We have the canonical injective map
$$\phi_n: R/(P_1 \cap \ldots \cap P_n) \to \prod_{i=1}^n R/P_i.$$
My ...

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74 views

### Converging sequence of polynomials

Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$ of degree $2$. Does there exist two polyomials $\alpha,\beta\in\mathbb F_q[T]$ (not both zeroes) such that the sequence $(\beta T^{q^{2n}}-\...

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229 views

### 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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87 views

### Intersection of an ideal and a subring

Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...

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247 views

### Example of a ring where every module of finite projective dimension is free?

I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says ...

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118 views

### Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules
For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...

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104 views

### 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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68 views

### Proper ideals are invertible

I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...

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276 views

### Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...

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109 views

### Etale map has image whose complement is the vanishing locus of a finitely generated ideal

While working through a proof of this paper, at the end of page 46, the author seems to claim along the lines that the following is true:
Let $A\rightarrow B$ be an etale map of rings. Then the ...

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34 views

### Graded commutative PBW bases

A Poincaré–Birkhoff–Witt (PBW) basis is a particularly nice basis of a quadratic algebra that can be used to prove that it is Koszul (see Priddy's 1970 paper "Koszul resolutions", Trans. Amer. Math. ...

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110 views

### Special submodules over almost Dedekind domains

An integral domain $R$ is an almost Dedekind
domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind
domain, where $R_m$ is the localization of $R$ at $m$.
Question: Let $M$ ...

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83 views

### Finite dimensionality of fibers of etale ring map

While working through a proof of this paper, at the middle of page 46, the author introduces a dimension notion which seems to claim that the following is true:
Let $A\rightarrow B$ be an etale map ...

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107 views

### Map between localizations induces map on underlying modules for Zariski covering

While working through a proof of this paper,1 at the middle of page 45, the author's claim of a short exact sequence seems to amount to the following problem:
Let $A$ be a commutative ring and let $...

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75 views

### Etale algebra whose local rank is constantly zero is the zero algebra

While working through a proof of this paper, at the middle of page 46, the author seems to claim the following is true:
Let $A\rightarrow B$ be an etale map of rings. Suppose that for every prime $...

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102 views

### Noetherian affine schemes for which localization computes the values of the structure sheaf

Is there a non-tautological characterization of the class of commutative unital Noetherian rings satisfying the following property: for any open subset $U$ of the topological space of $\mathrm{Spec}\,...

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74 views

### Preimage of a constructible set in spectrum of a subring

While working through a proof of this paper, at the beginning of page 42, the author seems to claim the following is true:
Let $R\subset S$ be rings, where $R$ is a finite type algebra over $\...

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130 views

### Structure of Complete Local Rings

Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$.
...

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79 views

### Structures like vector spaces but closed under heterogeneous products

The category of (pseudo-)Euclidean vector spaces (vector spaces with a nondegenerate but not necessarily positive-definite quadratic form) is not closed under products because $R^n$ over $R$ and $Z_2^...

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82 views

### Is a Maximal Cohen-Macaulay sheaf on a Cohen-Macaulay scheme locally free?

Let $F$ be a Maximal Cohen-Macaulay module on a Cohen-Macaulay scheme $X$.
By definition $\mathrm{depth}_{\mathcal{O}_{X,x}}(F_x)=\mathrm{dim}(\mathcal{O}_{X,x})$ and $\mathrm{depth}(\mathcal{O}_{X,x})...

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180 views

### Power series ring and monomials

Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables.
For a given positive number $\epsilon > 0$ we call a monomial $X_{...

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96 views

### 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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458 views

### Involutions in $\mathbb{F}_p[[x]]$

Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ with a zero constant term a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?
Here involution in $A[[x]]$ means $f\in ...

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423 views

### Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$.
Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...

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280 views

### Algebras such that the tensor product with any Noetherian algebra is Noetherian

Let $R$ be a Noetherian commutative unital ring. It is generally speaking not true that the tensor product of two Noetherian $R$-algebras is Noetherian (e.g. take $R$ to be a field, and consider the ...

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458 views

### “Noetherian” and “finitely generated” for polynomial algebras

Let $k$ be a field. Does there exist a positive integer $n$ such that there is $k$-subalgebra of $k[x_1, \dots, x_n]$ which is Noetherian but not finitely generated?

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102 views

### Geometric regularity for infinitely generated field extensions

Let $k$ be a field. Suppose that for a finite type $k$-algebra $A$, we define two following properties:
$A\otimes_k k'$ is a regular ring for all finitely generated field extensions $k\subset k'$.
$...

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168 views

### 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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115 views

### 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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99 views

### 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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255 views

### 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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265 views

### 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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81 views

### 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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303 views

### 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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203 views

### 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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78 views

### 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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115 views

### 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 ...

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115 views

### Two commuting matrices over a commutative ring

I would like to know if there are results about the dimension of the Algebra generated by two commuting Matrices over a ring (as there are in the case of a Field).
The good news is that "my" ring is ...

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51 views

### Localization at a maximal ideal of the integral closure of a valuation ring

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 $\bar{R}$...