# Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

3,791
questions

**-2**

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101 views

### Annihilator of an element and Jacobson radical

Let $R$ be a commutative ring with 1. Is there any characterization for an element $a$ of $R$ such that $\operatorname{ann}(1-a)\subseteq J(R)$ and $a\in J(R)$, where $\operatorname{ann}(x):=\{r\in R\...

**-3**

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138 views

### Glue DVR to itself, get a separated non-affine scheme

Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...

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194 views

### Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?

Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...

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vote

**1**answer

72 views

### Monomials in products in power series ring on several variables

Let $A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $m + n$ variables and ${\frak m}$ be the unique maximal ideal of $A$.
For arbitrary two elements $\alpha ...

**4**

votes

**2**answers

532 views

### A DVR algebra with weird automorphisms

Denote by $k$ an algebraically closed field. Can one produce a DVR $A$ over $k$ such that
the fraction field of $A$ has an automomorphism not preserving $A$
no non-trivial field extension of $k$ maps,...

**0**

votes

**1**answer

133 views

### Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...

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93 views

### Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
Remark: ...

**-1**

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**0**answers

49 views

### Infinitely generated PID algebra with infinitely many prime ideals

Given a field, is there a functorial construction of a PID algebra over it that has infinitely many prime ideals and is not finitely generated? This excludes the ring of univariate polynomials and the ...

**4**

votes

**2**answers

354 views

### When does glueing affine schemes produce affine/separated schemes?

Let $X$ be an affine scheme with an open affine subscheme $U\subset X$. Given an automorphism of $U$, we can glue $X$ with itself along $U$ to get a new scheme. Is there a description in terms of ...

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202 views

### Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...

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90 views

### Are Noetherian algebras over a field catenary?

A commutative unital ring that can be given the structure of a finitely generated algebra over a field is catenary. Is a Noetherian integral domain that can be given the structure of a (possibly ...

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32 views

### Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$

Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...

**7**

votes

**1**answer

294 views

### A ring of generalized power series

Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like
$$ \sum_{n=0}^{\infty} a_n x^{b_n} $$
where $a_n\in \...

**2**

votes

**1**answer

255 views

### Simple object of $k[X,Y]/(Y^2)$

Let $k$ be a field. Let $A=k[X,Y]/(Y^2)$ be the quotient of polynomial ring $k[X,Y]$. Let $\mathcal{C}$ be the category of finite-dimensional $A$-modules $M$ with the action of $X$ nilpotent, and of ...

**1**

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**1**answer

122 views

### A property for primitive idempotents

Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...

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94 views

### Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...

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149 views

### Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...

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**1**answer

131 views

### Special idempotents in a commutative ring

Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...

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50 views

### On the milnor number of analytic germ map

If $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic germ with an isolated singularity, then the Milnor number of $f$, denoted by $\mu(f)$ can be defined as $\dim_{\mathbb{C}} \mathcal{O}_n/\text{...

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**1**answer

195 views

### Definition of dualizing complex

Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way.
This question refers ...

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62 views

### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...

**3**

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**1**answer

164 views

### How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $p + 1$, but I'm not quite sure.

**3**

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**1**answer

239 views

### The relationship between a finite field and a quotient ring in $\mathbb{F}_p[x]$

Let $ f$ be an irreducible polynomial of degree $q$
over $\mathbb{F}_p$.
Let ${\bf F}=\frac{\mathbb{F}_p[x]}{f}$ be the finite field which contain $p^q$ elements.
Assume $k>1$ is an integer and ...

**0**

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**1**answer

61 views

### Change chain of prime ideals so that $a \in P_1$

A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof.
Lemma: Let $n\in\mathbb{N}_{>0}$, and let $P_0\subsetneq P_1\subsetneq\cdots\...

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111 views

### Locus of trivialization of an extension of a vector bundle

Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...

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131 views

### Extensions of rings

Let $K$ be a commutative but not necessarily unital ring. Let $C$ be a unital commutative ring.
An extension is the data of a unital commutative ring $R$, a homomorphism $\phi:K\rightarrow R$, a ...

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166 views

### A non-commutative analog of a result concerning a Jacobian pair

Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...

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53 views

### Non-minimal Krull associated primes of a PF-ring

A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if
for every element $x\in P$ ,...

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90 views

### Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...

**10**

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**1**answer

225 views

### Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$.
Indeed let $P(X)=\...

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70 views

### Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...

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103 views

### Converse to Tannaka duality for rings

Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...

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114 views

### Algorithm telling when an affine curve is planar

I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question.
Let $\mathfrak{p}$ be a prime ideal of $\mathbb{C}[x, y, z]$...

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230 views

### Concerning certain Keller maps of $k[x,y]$

Let $k$ be a field of characteristic zero.
Let $(x,y) \mapsto (p,q) \in k[x,y]$ be a Keller map, namely, a $k$-algebra endomorphism of $k[x,y]$ with $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}...

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**2**answers

361 views

### On a morphism from the Brauer group to the Picard group

Suppose that $k$ is a commutative ring and that $A$ is an Azumaya
$k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...

**3**

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**3**answers

604 views

### Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...

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50 views

### Homogeneous basis on a polynomial subalgebra

Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...

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69 views

### Polynomial equations parametrized by binary forms

Consider the equation
$$\displaystyle Ax^p + By^q = Cz^r, A,B,C \in \mathbb{Z}, \gcd(x,y,z) = 1, p,q,r \geq 2.$$
When $p^{-1} + q^{-1} + r^{-1} > 1$, the above equation is called spherical and ...

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81 views

### A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper:
Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...

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votes

**1**answer

107 views

### Module such that every finitely generated submodule is semisimple [closed]

Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely ...

**2**

votes

**1**answer

170 views

### Every radical ideal in the ring of algebraic integers a finite intersection of prime ideals

Is every radical ideal in the ring of algebraic integers (i.e. the integral closure of $\mathbb{Z}$ considered as a subring of $\mathbb{C}$ via the unique homomorphism of unital rings $\mathbb{Z}\...

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46 views

### Homomorphism or derivation conserving irreducibility

Let $R$ be a integral domain and $\phi$ be an automorphism of $R$. For a given element $x \in R$, we consider a sequence $(\phi^n(x))_{n=0}^{\infty}$.
I wonder if there is any related theory to ...

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64 views

### Inverse limit and graded functor commute?

I am trying to understand a proof where there are graded algebras and inverse limit involved.
In one of the steps it seems to commute this two elements. Is there any reference where this is stated.
$...

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150 views

### Simple description of a Grothendieck topology on the opposite of f.p. complex algebras

Let ${\cal A}$ be the category of finitely presented $\mathbb{C}$-algebras.
Let $J$ be the largest subcanonical Grothendieck topology on ${{\cal A}^{op}}$ such that the local algebras in $\cal A$ are ...

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73 views

### Differential criterion for regular sequences

Let $R$ be a subalgebra of the polynomial ring $\mathbb{C}[X_1,\ldots,X_n]$.
Suppose $\theta=(\theta_1,\ldots,\theta_p)$ is a sequence of of elements in $R$. If I want to test for their algebraic ...

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vote

**2**answers

225 views

### Regularity of certain schemes

In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (https://arxiv.org/abs/1207....

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259 views

### On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...

**0**

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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 ...

**2**

votes

**0**answers

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\...

**7**

votes

**1**answer

262 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 ...