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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About Homotopy Transfer Lemma

If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ ...
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133 views

Question about basis of $\text{Der}_{k}(k[X])$

Let $k[X] = k[x_1,\ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Assume that $(D_1,\ldots, D_n)$ is a $k[X]$-basis of $\text{Der}_k(k[X])$. Suppose that the vector space $\...
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1answer
77 views

On a sum of squares representation

We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$ $$r^2=|4pq|$$ holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
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39 views

On interpolation of endomorphisms

Let $R$ be a commutative ring and let $M$ be a $R$-module. Fix a non zero $m\in M$. We are given a family of endomorphisms of $M$ $(f_i)_{i \in I}$ and two functions $\alpha : I\times I \to I$, and $\...
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Is there a Swan-style description of topological K-homology?

A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\...
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Associativity equation for topological rings and logarithms

Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,...
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95 views

Flatness over local Artin rings

Let $A$ be a local Artin ring and $M$ an $A$-module with a finite flat resolution. Is $M$ a flat $A$-module? Can the assumption on $A$ be changed (relaxed?) while keeping the assumption on $M$ the ...
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66 views

Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots

Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
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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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95 views

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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114 views

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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113 views

Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
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1answer
260 views

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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169 views

Open covers by ind-affine ind-schemes

Many apologies if this is totally standard! I couldn't find it in the literature. Background definitions: A presheaf $X: \textbf{Aff}^\text{op} \to \textbf{Set}$ is an ind-scheme if it is a filtered ...
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85 views

$K$-group of category of bounded chain complexes of Projective modules with finite length homologies

For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
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72 views

Reference request: Cohen-Macaulay representations

Question: Are there good references of Cohen-Macaulay (CM) representation theory with concrete calculations of examples. In particular, I look for ones which contatin the classification of CM modules ...
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1answer
116 views

Is a ideal of direct limit of rings is itself a direct limit of ideal?

By Ńeron-Popescu desinaglarization theorem, If R is any regular semi-local ring containing $\mathbb{Q}$. Then R is a direct limit of regular semi-local rings $R_{i}$, where each $R_{i}$ is essentially ...
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Set-theoretic generation by circuit polynomials

Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
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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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1answer
132 views

Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
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66 views

Rational functions with trivial Weil symbols at every point

Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
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1answer
135 views

Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective

Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective? I'm looking for conditions ...
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1answer
55 views

Question about Jacobian subalgebra

Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$. Is it true that $\mathbb{C}[x, ...
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0answers
100 views

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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1answer
79 views

An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
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1answer
39 views

Continuations of derivations of Jacobian subring

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...
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0answers
75 views

F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex

Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex? Thank you
2
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1answer
281 views

In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal?

I have asked the following question at MSE and got one answer. Any further ideas are welcome: Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$. Let $\lambda, \...
7
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2answers
493 views

Determining the kernel of the localization map when defining the localization by generators and relations à la Serre

All rings considered will be commutative and unitary. Let $A$ be a ring, $S \subseteq A$ a multiplicatively closed subset. The localization $\lambda_S : A \longrightarrow A[S^{-1}]$ can be ...
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0answers
113 views

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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60 views

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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0answers
214 views

Polynomial objects in any concrete category

EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
2
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2answers
147 views

Commuting nilpotent matrices and conjugation isomorphisms

Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices. Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
3
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0answers
144 views

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 ...
3
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1answer
117 views

Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map is surjective?

Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map $x\mapsto x^2$ is surjective? This is certainly true for fields. For DVR's, you can ...
3
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1answer
197 views

Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite

I was told that if $A$ is the subring of $\mathbb{C}[x_1,\ldots, x_n]$ generated by the polynomials $p_1(x_1,\ldots, x_n),\ldots, p_1(x_1,\ldots, x_n)$, then the preimage $p^{-1}(c)$ via the map $p = (...
5
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1answer
152 views

Closure of the product of subfunctors

Background: Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every ...
4
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1answer
134 views

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 ...
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1answer
149 views

For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$?

Suppose we have $a^2 \equiv b^2 \bmod 4R$ where $R$ is an integral domain. Under what conditions on $R$ can we conclude that $a \equiv b \bmod 2R$? This would hold if $2 \in R$ is a prime or the ...
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0answers
82 views

Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...
3
votes
1answer
214 views

Prime ideals of formal power series ring that are above the same prime ideal

Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$, then $I[[X]]$, the set of power ...
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3answers
199 views

non-associative but commutative algebra [closed]

Is it possible(or may be easier) to give an example of non associative algebra but commutative? At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...
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0answers
105 views

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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0answers
126 views

A series that is algebraic?

This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic ...
1
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1answer
48 views

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 ...
9
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1answer
806 views

A series that is rational?

Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no ...
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0answers
66 views

Fast double exponentiation in finite fields

Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
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0answers
134 views

Is it possible to compute the basis of this module?

Let $A$ be a polynomial algebra in $n$ variables over field $\mathbb{F}$ of characteristic zero which is algebraically closed. Assume that $a_1,\ldots, a_n, b_1,\ldots, b_n\in A$ are such that $a_1b_1+...
2
votes
0answers
99 views

Computing whether a set of polynomials cuts out a projective variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....
4
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0answers
154 views

Variant of the Ringel-Kotzig graceful labeling conjecture

I'm looking for some tips on the following problem I faced with. Imagine we have a tree $T$, i.e. acyclic connected graph. We can label its edges using $\{0,1\}$. Note that such labelings are just ...

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