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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113 views

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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1answer
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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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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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 ...
4
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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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1answer
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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79 views

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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1answer
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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1answer
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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1answer
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, ...
2
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1answer
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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1answer
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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107 views

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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1answer
83 views

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 $\bar{R}$...