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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Splitting a nilpotent into square-zeros by ring extension

Let $R$ be a commutative ring. It is well-known that if $b \in R$ and $c \in R$ are two nilpotent elements with $b^k = 0$ and $c^\ell = 0$ (where $k$ and $\ell$ are positive integers), then $b+c$ is ...
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7 votes
1 answer
176 views

Algebraic proof that the monoid ring of a torsion-free monoid is reduced

In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result: Claim: if $M$ is a torsion-free commutative ...
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-4 votes
0 answers
63 views

Can a non-distributive algebraic system be represented as matrices?

For instance, can the following 4-dimensional "number system" (which I would call "anti-split numbers") be represented as matrices? It is commutative and associative but not ...
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Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?

Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$). I will denote the ...
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10 votes
1 answer
211 views

Finite coverings by closed subschemes

Let $X$ be a scheme. Assume we have two closed subschemes $Y_1$, $Y_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'_1$, $Y'_2$ with the same underlying sets, such that the ...
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0 votes
0 answers
34 views

Examples of almost complete intersection ideals

I am stutying some articles about perfect ideals and there are interesting results when $I$ is an ideal over a noetherian local ring that satisfies the following conditions : $\operatorname{grade}(I,R)...
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1 vote
1 answer
106 views

Cohen-Macaulay quotient ring and symbolic power

Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{a} \subset R$ be an ideal. Let $$ \mathfrak{b} = \bigcap \{R \cap \mathfrak{a} \cdot R_\mathfrak{p} \text{ } \colon \mathfrak{p} \in \...
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3 votes
4 answers
443 views

$R$ is a UFD iff $R_{\frak{m}}$ is a UFD?

Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
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1 vote
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When is the product of two elements in algebraic closures of rational functions a constant function?

I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields. My question is as follows: Let E and F be ...
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1 answer
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Injective resolutions of the module of Kähler differentials

Let $k$ be a field, $A=k[x_1,\dots,x_n]/I$ an affine algebra and $\Omega_{A|k}$ the $A$-module of Kähler differentials. By abstract nonsense there exists an injective resolution $\mathcal I$ of $\...
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2 votes
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Root systems and subroot systems

Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...
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-1 votes
1 answer
43 views

Convertor matrices

Let us consider $\mathbb{C}^n$, the set of all n-tuples of complex numbers. We know that $(\mathbb{C}^n,+,\cdot)$ is a unital commutative complex algebra. Suppose that $\diamond$ is an associative ...
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6 votes
2 answers
261 views

Algebraic numbers which prescribed degree which does not belong to some fields

In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
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2 votes
0 answers
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Decidability of the solvability of quadratic systems

Let a finite collection of (complex) unknowns $\{x_1,\ldots,x_n\}$ be given, as well as an affine system $AX=B$ in the quadratic variables $X:=[x_i x_j : i\leq j]$, with entries in a computable ...
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0 votes
0 answers
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Exponent of the scalar part of the finite part of the logarithm of an object, or hypermodulus

I will call it "hypermodulus". In simple words, hypermodulus is the exponent of the scalar part of the finite part of the logarithm of the object: $H(A)=\exp (\operatorname{scal} \...
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2 votes
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A note on perfect ideals

I am reading an article that cites this note: E.S. Golod, A note on perfect ideals, in: A.I. Kostrikin (Ed.), Algebra Collection, Nauka, Moscow, 1980, pp. 37–39. E. S. Golod, A note on perfect ideals, ...
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A question about mapping cone and resolutions

I am studying this papper https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full By Daniel ...
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0 votes
0 answers
47 views

Commutative unitary ring A such that the quotient A/fA is finite for each prime element f of A [migrated]

We say that A is a residual ring if A is a commutative unitary ring such that the quotient A/fA is finite for each prime element f of A. For example : Z, Z[i], the set of polynomials with ...
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12 votes
1 answer
480 views

Does every map from a noetherian ring to a valuation ring factor through a DVR?

Let R be a noetherian ring and V a valuation ring with maximal ideal $\mathfrak{m}_V$. Does every morphism of rings $\varphi: R \rightarrow V$ factor through a discrete valuation ring? One may ...
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-2 votes
0 answers
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Anti-dual numbers and what are their properties?

I have asked this question before in Math.SE. It got upvotes but no answer so far. In this post user William Ryman asked what would happen if we try to build "complex numbers" with shapes ...
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0 votes
1 answer
55 views

Changing base field for sum of polynomials

Let $L/\mathbb{Q}$ be a finite extension and $f_{1},\dotsc,f_{n}\in L[x_{1},\dotsc,x_{k}]$ be degree $d$ homogeneous polynomials. Is there a way to find homogenous degree $d’$ polynomials $g_{1},\...
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3 votes
0 answers
190 views

Symmetric powers, localisation and Frobenius

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
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2 votes
0 answers
53 views

$\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$-modules extended from $\mathbb Z_n$

Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\...
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  • 504
0 votes
1 answer
157 views

Factor $\sum_{n=1}^{N} x^n$ [closed]

I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation $$\sum_{n=1}^{N} x^n$$ Although the Galois group for anything beyond a ...
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  • 485
-1 votes
1 answer
160 views

Why we can analytically define $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers?

Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this? ...
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1 vote
0 answers
57 views

Buchsbaum-Eisenbud-Horrocks conjecture for finitely generated modules

Someone know a version for conjecture of Buchsbaum-Eisenbud-Horrocks whithout the assumptions that the $R$-module $M$ has length finite?
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-1 votes
0 answers
29 views

Is it possible to solve this equation for x, [migrated]

$\frac{z}{y}$= $(1-\frac{xa}{m_1})$ $(1-\frac{xb}{m_2})$ $(1-x)^{n-2}$ Can we solve this equation for $x$ such that, in the new equation we have, $x$ is the dependent variable, and $n$ is the ...
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1 vote
0 answers
76 views

A question about minimal system of generators and regular sequences

Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $I$ an ideal. Suppose that: $\mu(I)=\operatorname{grade}(I,R)+1$ and $\operatorname{pd}_R(R/I)=\operatorname{grade}(I,R)$. (Some people says $I$ is ...
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1 vote
0 answers
61 views

Some properties for height 1 prime ideals in the local ring

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $R=\mathbb{K}[x_0,x_1,\dotsc,x_n]/I$ be the coordinate ring of an affine variety/projective variety. Also, assume that $I$ ...
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5 votes
1 answer
182 views

Localization of a ring and the Hom functor

Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
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  • 201
-1 votes
0 answers
149 views

Lemniscate numbers and others - what would be the properties?

Recently I noticed a question about systems of alternative "complex" numbers, defined by shapes other than circle, hyperbola and straight lines. I even made and answer, telling that this is ...
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1 vote
1 answer
89 views

On the trivialization of the sheaf of kahler differentials on the G-invariant topology

Let $X$ be a connected, smooth affine algebraic variety over an algebraically closed field $K$ of characteristic zero. Assume we have a finite group $G$ acting on $X$ by morphisms of $K$-schemes. ...
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4 votes
1 answer
549 views

Brauer group of rational numbers

Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
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1 vote
0 answers
71 views

Characterise set of polynomials which are zero over an ideal

This is not a specific question, but rather a question about possible techniques approaching a problem. Although this question came from research, it might not fit this forum; in which case I will ...
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2 votes
0 answers
73 views

Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?

Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...
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3 votes
0 answers
155 views

Does a torsion-free coherent sheaf embed into a locally free sheaf?

Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
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4 votes
0 answers
60 views

The index of an order defined by a binary form

In his well-known paper, Nakagawa generalized a construction due to Birch and Merriman to arbitrary binary forms and orders. In particular, his construction gives a canonical algebraic order $\mathcal{...
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9 votes
0 answers
195 views

Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?

Originally asked and bountied at MSE without success: Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
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1 vote
0 answers
80 views

Cohomological dimension and height of ideals

Let $I$ be an ideal in a Noetherian ring $R$. We define the cohomological dimension of $I$ to be $\operatorname{cd}(I)=\operatorname{sup}\{i\in \mathbb N:\operatorname{H}_I^i(R)\neq0\}$ and it is ...
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4 votes
1 answer
147 views

Faithful module cancellation with maximal ideal

Let $k$ be a field of characteristic $0$ and $R = k[[x_1, \dotsc, x_n]]$. Suppose that $M$ is a faithful, finitely generated $R$-module and $\mathfrak{a} < R$ is an ideal such that $\mathfrak{a} M =...
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3 votes
0 answers
207 views

Algebras which admit tensor calculus and (pseudo-)Riemannian geometry

It's an often observed fact that the basic notions of analysis on manifolds and (pseudo-)Riemannian geometry, such as tensors, connections and curvature, can be defined in purely algebraic terms. The ...
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10 votes
1 answer
234 views

Rational even polynomials maximally tangent to the unit circle

This question is motivated by College Mathematics Journal problem 1196, proposed by Ferenc Beleznay and Daniel Hwang. My solution to this problem (pre-publication version here) uses Chebyshev ...
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3 votes
1 answer
169 views

Is this a true weakening of the quasi-coherence property?

Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition (#) For all containments $V \subseteq ...
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  • 1,660
1 vote
0 answers
141 views

Meaning of "cut out (scheme-theoretically)"

Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
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  • 341
1 vote
0 answers
195 views

Source for conjectures in commutative algebra

Do you know some books/survey papers/ websites on conjectures or open problems in commutative ring theory? I want to see if there are very famous open problems or conjectures in commutative ring ...
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1 vote
0 answers
100 views

Strict henselianization of complete intersections

As far as I understand (and tbh for my purposes), one of the main points of strict henselisation of a local ring is that it computes the stalk at a point of a scheme in the étale topology. In the ...
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6 votes
1 answer
188 views

On the Artin-Rees Lemma for non-commutative rings

Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
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2 votes
0 answers
104 views

Theorem on formal functions when the initial data is a proper map of formal schemes

Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$. Set $S_0=\{x\}$ be a closed point of $S$ and $...
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  • 1,016
2 votes
0 answers
71 views

Jacobian ideal as primary idea;

Let $R = \mathbb{C}\{x_1, \dots,x_n\}$ be the ring of germs of analytic functions and let $f \in R$ be a homogeneus polynomial of degree $p$ such that $\sqrt{Jac(f)}=\langle x_1, \dots, x_{n-1}\rangle ...
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1 vote
0 answers
118 views

Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently. So far, I only found MAGMA with its ...
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