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

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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
0answers
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 ...
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0answers
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
0answers
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 ...
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0answers
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 ...
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1answer
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 ...
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0answers
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
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0answers
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 ...
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0answers
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
1answer
178 views

Milnor descent for ring spectra

Suppose given a homotopy cartesian square of (commutative) ring spectra (or (c)dgas) $\begin{matrix}A & \to & A_1 \\ \downarrow & & \downarrow \\ A_2 & \to &A'\end{matrix}.$ ...
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1answer
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 ...
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1answer
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 = ...
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0answers
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
1answer
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 ...
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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 ...
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1answer
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 $\pi\colon C\to X$. ...
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1answer
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
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2answers
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 ...
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0answers
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. ...
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0answers
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
1answer
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 $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, ...
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0answers
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 ...
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0answers
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
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1answer
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 ...
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0answers
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 ...
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0answers
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 ...
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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
2answers
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?
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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. ...
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2answers
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
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2answers
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 ...
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0answers
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
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2answers
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
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1answer
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 ...
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2answers
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 ...
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0answers
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 ...
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0answers
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
1answer
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 ...
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0answers
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
2answers
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 ...
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0answers
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 ...
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1answer
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} ...
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1answer
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 ...
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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 ...
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1answer
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, ...
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1answer
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
1answer
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
0answers
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 ...
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0answers
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
0answers
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 ...