# Questions tagged [ac.commutative-algebra]

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

**37**

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### The roots of unity in a tensor product of commutative rings

For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...

**2**

votes

**1**answer

117 views

### $(x + y + z)…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity.
The ...

**4**

votes

**0**answers

96 views

### Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem:
Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...

**0**

votes

**0**answers

94 views

### gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...

**9**

votes

**1**answer

397 views

### Do commutative rings without unity have the IBN property?

Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.
Does $R$ still have the Invariant Basis Number (IBN) property?
Recall that a ring is said to have the IBN ...

**5**

votes

**0**answers

167 views

### Fraction fields of strict henselizations of DVRs

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...

**5**

votes

**1**answer

75 views

### Reflection-invariant monomial ideals and Alexander duality

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...

**2**

votes

**1**answer

67 views

### Steinberg components of local deformation rings

Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...

**4**

votes

**0**answers

67 views

### On ideals in Noetherian rings, isomorphic to the trace of some finitely generated module

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, let $tr_R(M):=Im(\tau_M)$, where $\tau_M:M\otimes Hom(M,R)\to R$ is the map defined as $\tau_M(m\otimes f)=f(m)$.
Let $I$ be a ...

**7**

votes

**1**answer

185 views

### How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...

**1**

vote

**0**answers

55 views

### Persistence of saturation closure

Let $R$ be a ring. A closure operation $cl$ on a set of ideals $\mathcal{I}$ of
$R$ is a set map $cl: \mathcal{I}\longrightarrow \mathcal{I}( I\mapsto I^{cl})$ satisfying the following conditions:
(i)...

**5**

votes

**0**answers

87 views

### Cohen-Macaulay local ring and its quotient by minimal prime ideal

Assume $R$ is a Noetherian local ring which is Cohen-Macaulay, must there exist an ideal $I$ of $R$ such its radical is a minimal prime and the quotient ring $R/I$ is still Cohen-Macaulay? If the ...

**2**

votes

**0**answers

127 views

### Prime ideals being finitely-generated implies coherence?

Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$
$(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...

**3**

votes

**0**answers

98 views

### Can one characterize power series that have polynomial inverses?

Going through my Commutative Algebra notes I found out that I don't know the answer to this question.
Let $A$ be a commutative ring with unit and let $f(T):=\sum_{k=0}^\infty a_k T^k \in A[[T]]$ be a ...

**0**

votes

**1**answer

117 views

### Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...

**0**

votes

**0**answers

101 views

### Krull dimensions and regular sequences

I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting:
Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...

**2**

votes

**0**answers

136 views

### Solving solutions to systems of polynomial equations over $\mathbb Z$

Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...

**5**

votes

**0**answers

189 views

### Ideals of orders in number fields

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...

**2**

votes

**1**answer

103 views

### Transitivity of an invariant of finitely generated field extensions

For a finitely generated extension of fields $K/k$, let us define "$S_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with ...

**7**

votes

**1**answer

261 views

### An infinite dimensional local domain whose chains of primes are finite

Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite?
Of course, such a ring must be neither noetherian nor catenary.
(This question arose while ...

**1**

vote

**2**answers

159 views

### Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]

Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$.
Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...

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vote

**0**answers

79 views

### A special family of prime ideals

I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...

**1**

vote

**1**answer

46 views

### Decomposing semihereditary rings

Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...

**1**

vote

**1**answer

75 views

### Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?

**3**

votes

**1**answer

227 views

### Are quotient varieties local complete intersections?

Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is ...

**0**

votes

**1**answer

214 views

### When does a subspace of the affine space form a regular sequence in a ring of regular functions?

Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$.
Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...

**4**

votes

**1**answer

117 views

### On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...

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vote

**0**answers

44 views

### What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?

Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...

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31 views

### Numerator-cancellable Modules

I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following :
Let $R$ be a ...

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vote

**0**answers

118 views

### On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...

**14**

votes

**2**answers

341 views

### Ostrowski's Theorem for topological rings?

Ostrowski's theorem classifies all absolute values on a number field $K$.
Questions:
More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?
In ...

**3**

votes

**1**answer

231 views

### Local ring of infinite dimension

Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional?
Longer version: Let $...

**4**

votes

**1**answer

212 views

### Resultants for compactly represented product form polynomials?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...

**2**

votes

**0**answers

117 views

### derived symmetric powers of an ideal

Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$.
We have a canonical map $Lsym^k(m)\to m^k$ from the ...

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vote

**0**answers

71 views

### On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module.
...

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**0**answers

110 views

### Flatness of Endomorphism sheaf

Let $X$ be a nodal curve and $T$ be any scheme. Let $F$ be a flat family of torsion-free coherent sheaves on $X$ parametrized by $T$. Is there an example where $\mathcal{E}nd ~F$ is not flat over $T$? ...

**7**

votes

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192 views

### Counting exceptional divisors

Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to ...

**5**

votes

**0**answers

129 views

### Descending chain of subalgebras of $k[x,y]$

Let $k$ be a field of characteristic zero.
Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$,
such that ...

**3**

votes

**0**answers

82 views

### Infinite-dimensional wild commutative algebras with non-trivial units

Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....

**1**

vote

**1**answer

90 views

### An ideal and its J-radical

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $...

**2**

votes

**0**answers

203 views

### Eudoxus real numbers

I recently remembered the eudoxus construction of the real numbers.
Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction?
Clearification: The ...

**1**

vote

**1**answer

140 views

### Regular local artinian k-algebra with residue field k is k

I am reading an article. There is a step in which I suspect that they use a "result" that "Let $A$ be a local artinian $k$-algebra with residue field $k$. If $A$ is regular then $A$ is nothing but $k$....

**2**

votes

**1**answer

112 views

### Homomorphisms of ring extending nicely ideal intersections

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$.
Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$.
Is there some "natural" assumption on $\varphi$ to ...

**8**

votes

**1**answer

139 views

### Reference for Kakutani result on power sum bases of symmetric functions

Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...

**12**

votes

**1**answer

354 views

### Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...

**3**

votes

**1**answer

167 views

### difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...

**0**

votes

**1**answer

124 views

### Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?

If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...

**2**

votes

**1**answer

184 views

### radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set
$$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...

**2**

votes

**1**answer

149 views

### Heights of contracted ideals

Let $R$ be a non-noetherian domain. $S$ be a multiplicatively closed subset of $R$. Let $S^{-1}R$ be a localisation of $R$, where all element of $S$ is invertible. Suppose we have an ideal $I$ of $R$. ...

**7**

votes

**1**answer

380 views

### Vector space objects in schemes - confusion

Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$.
$\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...