**0**

votes

**0**answers

63 views

### Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...

**2**

votes

**0**answers

26 views

### Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on ...

**0**

votes

**0**answers

101 views

### Non-finitely generated, non-projective flat module, over a polynomial ring [migrated]

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective.
Therefore, the only hope to find a flat ...

**3**

votes

**0**answers

128 views

### Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...

**0**

votes

**0**answers

89 views

### Degrees of generators of radical ideals

Let $I \subseteq \mathbb{C}[x_1,\ldots,x_n]$ be an ideal generated by polynomials $f_1,\ldots,f_r$ of degree at most $d$. Is it possible to generate the radical $\sqrt{I}$ of this ideal with ...

**0**

votes

**1**answer

90 views

### section of reduced structure map

Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ...

**-3**

votes

**0**answers

57 views

### The Zariski closure of subset connected [on hold]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ (where $Spec(R)$ is the set of all prime ideals of $R$) such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is ...

**-3**

votes

**0**answers

44 views

### non local indecomposable commutative ring [on hold]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $spec(R)$ such that $\frac{R}{\bigcap_{i\in I}{P_i}}$ is an indecomposable ring, is any relation among $P_i$'s? Thanks a ...

**0**

votes

**0**answers

73 views

### I want an example of a monomial curve [closed]

Example of a monomial curve and its defining ideal.
I am looking for precise definition of a monomial curve and its defining ideal. If possible please tell the minimal generator of this ideal

**4**

votes

**1**answer

178 views

### Milnor descent for ring spectra

Suppose given a homotopy cartesian square of (commutative) ring spectra (or (c)dgas)
$\begin{matrix}A & \to & A_1 \\
\downarrow & & \downarrow \\
A_2 & \to &A'\end{matrix}.$
...

**0**

votes

**1**answer

203 views

### Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...

**0**

votes

**1**answer

58 views

### Some references for f-ring

A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there
is a $b\in I$ such that $ab = ...

**0**

votes

**0**answers

99 views

### Question about Castelnuovo-Mumford regularity

let $R$ be a Noetherian ring an $I$ an ideal of $R$.
If $n,m\in N$ and $reg(G(I))=n$, then what can we say about $reg(G(I^m))$?
Here $G(I)$ is the associated graded ring.

**2**

votes

**1**answer

94 views

### Classification of commutative ring ideal closure operators?

First, some setup:
So: given a commutative ring $R$, let $Ideals(R)$ be set of ideals of $R$ and let $IdealClosure(R)$ be the set of closure operations $cl: \mathcal{P}(R) \rightarrow Ideals(R)$. In ...

**0**

votes

**0**answers

81 views

### When is a homomorphic image of a polynomial ring self injective [closed]

Following Utumi (but adjusting the definitions for commutative rings):
A commutative ring with a unit element is self injective if it is injective as a module over itself. A commutative ring $R$ with ...

**0**

votes

**1**answer

74 views

### Irreducible components of a cone

Suppose $B=A\oplus S^1\oplus S^2\dots$ is a graded ring, $B$ is generated by $S^1$, $C=\textrm{Spec}B$ is called a cone over $X=\textrm{Spec}A$. We have natural projection $\pi\colon C\to X$. ...

**0**

votes

**1**answer

100 views

### Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutativr ring with 4 or more maximal ideals?

**4**

votes

**2**answers

106 views

### Lie Algebras over DVRs and basechange to the completion

Let $R$ be a discrete valuation ring containing an algebraically closed field $K$ of characteristic zero and let $L$ be a Lie algebra over $R$ whose underlying $R$-module is finitely generated and ...

**1**

vote

**0**answers

31 views

### some sort of 'saturation' of module quotients

Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. ...

**0**

votes

**0**answers

399 views

### Commutative algebra books representing the edge of research

Recently I have come across the books Combinatorial Commutative Algebra by Miller and Sturmfels along with Combinatorics and Commutative Algebra by Stanley. I will soon own a copy of each. I also ...

**5**

votes

**1**answer

528 views

### How to compute this $\mathrm{Ext}^1$?

Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, ...

**-2**

votes

**0**answers

55 views

### powers of largest graded ideal of R contained in I

Let $I$ be an ideal of a graded ring $R$. By $I^{*}$ we mean an
ideal of $R$ generated by all homogeneous elements contained in $I$.
Clearly $I^{*}$ is the largest graded ideal of $R$ contained in ...

**1**

vote

**0**answers

145 views

### Separability and smoothness

Let $A \subseteq B$ be commutative noetherian rings.
I have found the following claim: "Separability implies smoothness" with the following explanation:
"The natural thing is to prove that a separable ...

**3**

votes

**1**answer

145 views

### Example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$

Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that ...

**1**

vote

**0**answers

132 views

### How do I check if a sequence of R-modules is exact?

Let R be a ring. For example, take $R=k[x_1,\ldots,x_n]$ or, if possible, $R = \Bbb{Z}[x_1,\ldots,x_n]$.
Consider a sequence of free R-modules
$$R^a \stackrel{f}\to R^b \stackrel{g}\to R^c$$
where ...

**0**

votes

**0**answers

48 views

### Writing a module as a direct sum

Let $q_1, q_2, q_3 \in \mathbb{Z}[x,y]$ such that $q_1, q_2$ are algebraically independent and let $S$ be an algebra generated by $q_1, q_2, q_3$ over $F_p$. If writing $S$ as a module over $F_p ...

**0**

votes

**0**answers

96 views

### Intersections of ideals and nilpotence

Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...

**7**

votes

**2**answers

93 views

### Differential operators between modules, $\mathcal{D}_A(M, M)$ necessarily a filtered, almost commutative ring?

See here. Does it follow immediately that $\mathcal{D}_A(M, M)$ as defined in the link is a filtered, almost commutative ring? How can I visualize this geometrically?

**-3**

votes

**0**answers

98 views

### Non-flat $R \subseteq S$, which is integral, separable, $R$ is a noetherian (not integrally closed) integral domain

On ramification theory in noetherian rings,
of Auslander and Buchsbaum say:
"Chapter 4 is devoted to showing that under various conditions if $S$ is unramified over $R$, then $S$ is $R$-projective. ...

**21**

votes

**2**answers

409 views

### Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...

**2**

votes

**2**answers

170 views

### Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...

**1**

vote

**0**answers

75 views

### A question about a specific inverse proposition of Combinatorial Nullstellensatz

From the Hilbert's Nullstellensatz, we have the following consequence which is usually called Combinatorial Nullstellensatz:
Let $F$ be an arbitrary field, and let $f = f(x_1,x_2,\cdots,x_n)$ be a ...

**3**

votes

**2**answers

165 views

### Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...

**1**

vote

**1**answer

82 views

### integral closure of m-primary ideals

I need help with this excercise
Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that ...

**2**

votes

**2**answers

113 views

### A question about “large” indecomposable injectives over commutative rings

Here is my question:
Does there exist an infinite commutative ring $R$ with identity with an indecomposable injective (unitary) $R$-module $M$ of larger cardinality than $R$ with the additional ...

**0**

votes

**0**answers

102 views

### book for help on problems with noetherian rings

Can you please introduce to me a book which would help me to prove the two following problems?
In a noetherian ring, every integrally closed ideal is unmixed.
Let $R$ be a noetherian ring, $P$ a ...

**0**

votes

**0**answers

72 views

### The universal property of ring of big Witt vectors

Let $X_1,X_2,\cdots$ be infinite many indeterminates. Define
\begin{equation*}
W_n = \sum_{d\mid n} d X_d^{n/d}.
\end{equation*}
A big Witt vector over a commutative ring $R$ is a sequence ...

**5**

votes

**1**answer

250 views

### Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...

**0**

votes

**0**answers

61 views

### Can it occur that $q^{ce}$ is a prime ideal (of $S$), while $q^{ce}\neq q $?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$) and for an ideal ...

**23**

votes

**2**answers

650 views

### How slowly can a power of an ideal grow?

For a polynomial ideal $I\subset \mathbb{C}[x_1,x_2]$, let $D(I)$ be the smallest degree of any polynomial in $I$.
How slowly can $D(I^n)$ grow as a function of $n$? For example, if $D(I^n)\leq ...

**0**

votes

**0**answers

88 views

### If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?

Let $R$ be any ring with unit over some field $k$ and let $M_1$ and $M_2$ be cyclic left $R$-modules with $dim_k(Ext^1_R(M_2,M_1))\geq 1$.
Assume $M_1\oplus M_2$ is a cyclic left $R$-module. Given ...

**6**

votes

**1**answer

235 views

### Definitions of the module $R/(x_0^\infty,x_1^\infty,\ldots,x_{n-1}^\infty)$

There are several constructions of the Prüfer group $\mathbb{Z}/p^\infty$; here are two that are relevant for this question.
It can be constructed via the short exact sequence
$$
0 \to \mathbb{Z} ...

**0**

votes

**1**answer

76 views

### How to find ideals of finite length in a power series ring with special properties?

Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.
Assume we are given two ideals $I,J$ of finite length in $A$ such that:
$xJ\subseteq I\subseteq J$
Is it possible to find ideals of finite ...

**1**

vote

**0**answers

144 views

### Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...

**1**

vote

**1**answer

127 views

### local cohomology and radical of ideal

Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, ...

**0**

votes

**1**answer

71 views

### In what conditions every ideal is an extension ideal? Is every prime ideal extension of prime ideal?

Let $R$ and $S$ be commutative rings (with $1$), and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...

**1**

vote

**1**answer

129 views

### M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$

Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of BRODMANN-SHARP's Local Cohomology ...

**4**

votes

**0**answers

71 views

### Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms

Let $K$ be an algebraically closed field of characteristic $p>0$
and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by
(homogeneous) polynomials of degree $d$. Assume that $I$ is ...

**0**

votes

**0**answers

55 views

### An additive question on polynomials

Consider $S\cup T=\{0,1\}^n$ where $S\cap T=\emptyset$.
Consider real multilinear (only monomials of form $x_ix_jx_k$) polynomials $P,Q$ such that:
$$Q(S)=0\quad Q(T)\neq0\quad P(S)\neq0\quad ...

**6**

votes

**0**answers

239 views

### Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...