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

learn more… | top users | synonyms (1)

0
votes
2answers
198 views

Does there exist an Affinization or Projectivization process for Varieties?

Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...
1
vote
1answer
276 views

Automorphisms of rings fixing all prime ideals

Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...
1
vote
0answers
68 views

Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark: Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where $...
0
votes
0answers
223 views

On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$ be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$ $A$ consists of such formal sum elements as $\sum c_{e_1,.....
2
votes
2answers
301 views

Irreducible algebraic sets via irreducible polynomials

There are many results about irreducible polynomials over finite fields: we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
4
votes
1answer
296 views

an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$. I mean there is polynomial reduction $F$ such that for every boolean ...
0
votes
1answer
111 views

Reduction of ideal in noetherian local ring

Let $R$ be a noetherian local ring and $I$ an ideal with $\operatorname{ht}I=\mu(I)$. Prove that $I$ is basic. (Recall that an ideal $I$ is basic when it has no proper reduction.)
0
votes
0answers
169 views

Exact sequence of vector bundles

Consider the short exact sequences below; \begin{equation} 0\longrightarrow H^0(\mathbb{P}^4,\mathcal{O}_{\mathbb{P}^4}(d-1)^{\oplus 4})\longrightarrow H^0(\mathbb{P}^4,\Omega_{\mathbb{P}^4}(d+1))\...
2
votes
1answer
149 views

The center of a(n endomorphism) ring is a PID

Let $A$ be a torsion-free Abelian group, and let $\mbox{End}(A)$ be its endomorphism ring. Denote by $R\doteq\mathbf{Z}(\mbox{End}(A))$ the center of $\mbox{End}(A)$. What would be necessary and/or ...
2
votes
2answers
113 views

Computer algebra system that test zero divisors in a quotient algebra

I have an algebra $A$ over a Noetherian ring and an ideal $I=(x,y)$, where $x,y \in A$. I need to examine whether a polynomial $h \in A$ is a zero divisor in $A/I$ or not. Is there a computer algebra ...
1
vote
0answers
212 views

How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?

Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much. For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...
0
votes
0answers
101 views

Noetherian almost Dedekind domain

A Dedekind domain is an integral domain in which every nonzero proper ideal factors into a product of prime ideals, and an integral domain $R$ is called almost Dedekind whenever $R_m$ is Dedekind ...
4
votes
1answer
306 views

How to compute the tangent space of a quotient by a finite group

Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
1
vote
0answers
152 views

Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain. The question is all in the title: is there ...
1
vote
0answers
107 views

A strong form of Bezout theorem

Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
1
vote
1answer
132 views

Canonical module of a Buchsbaum ring

Is the canonical module of a Buchsbaum ring a Buchsbaum module?
4
votes
1answer
96 views

The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
3
votes
1answer
122 views

Number of free summands of finite local extensions

Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is: etale on the punctured spectrum not flat / etale at the origin and such that the residue fields $R/m = S/...
0
votes
0answers
52 views

Decomposition of PID modules

This is (probably) the culmination of a series of questions I posted recently that have lead me to this (probably) final question. As usual, I aplogize for my illiteracy in algebra. Recall that ...
6
votes
0answers
254 views

Geometric interpretation of minimal number of generators of a module

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \...
5
votes
0answers
165 views

A question on symmetric functions

Let $0 \leq m \leq n$ be integers. The group $S_n$ of permutations acts on the ring $\mathbb{Z}[X_1,\dots,X_n]$ by permuting the coordinates, with fixed subring $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
2
votes
0answers
127 views

When is the torsion submodule a direct factor?

Let $\mathbb{F}$ be a field (of characteristic 0, if needed) and $\mathbf{V}$ an $\mathbb{F}$-vector space. Let $T\in\mbox{End}_\mathbb{F}(\mathbf{V})$ be an endomorphism, and let (following Bourbaki) ...
1
vote
0answers
29 views

Is an weakly finite R-module Serre subcategory of the category of R-modules?

A definition for weakly finite $R$-modules is as follow: Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties: (1) If $M \in S$...
6
votes
0answers
79 views

Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra. Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
1
vote
0answers
48 views

Modification of nonfree locus

Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...
1
vote
0answers
89 views

Transitivity for algebraic extensions of integral domains?

I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...
2
votes
0answers
66 views

Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?

Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}...
0
votes
1answer
176 views

Rings with a property similar to integral domains

For an integral domain $R$, the intersection of two non-zero ideals is also non-zero, because the product of any two non-zero elements is non-zero. Is the converse true, i.e. if $R$ has the ...
1
vote
0answers
70 views

Cyclic decomposition of an infinitely generated module

My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance. Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a ...
6
votes
0answers
92 views

Relative variants of the Jacobson radical

Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as $$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...
1
vote
1answer
247 views

When are Segre- and Veronese embeddings Gorenstein?

Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m_1}\times\cdots\times\mathbb P^{m_n}$, is there a characterization in terms of $m$ and $n$, or the $m_i$, for the ...
0
votes
1answer
123 views

When Hom(M,E) is injective? [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module and $E$ be an injective $R$-module. When $Hom(M,E)$ is injective? Thanks.
3
votes
0answers
101 views

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
0
votes
0answers
147 views

Chow classes of non-reduced sub-schemes

I am trying to understand the geometric picture of primary ideals, in particular if and how one can define a notion of multiplicity for the underlying geometric 'sets'. My understanding of the subject ...
3
votes
1answer
148 views

is the normalization of a smooth curve in a tamely ramified finite separable extension of the function field also smooth?

Let $X$ be a smooth proper curve over a field $k$, with function field $K$. Let $L$ be a finite separable tamely ramified extension of $K$, and let $Y$ be the normalization of $X$ in $L$. Is $Y\...
1
vote
1answer
96 views

Does local cohomology commute with taking the degree-zero component?

Let $S = \oplus_{d \geq 0} S_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S_{(f)}$ the subring of degree-$0$ ...
3
votes
1answer
73 views

Prime ideal ramified in extension if and only if certain polynomial divides another one?

Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that ...
3
votes
1answer
131 views

Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$ be positive integers $\geq 2$. I want to prove that there exists a primitive polynomial $$F(x) = x^{mn}-\sum\limits_{j=0}^{...
7
votes
1answer
267 views

What does the notation $[b_1,b_2]$ in M. Hochster's “Prime Ideal Structure in Commutative Rings” mean?

I'm reading the article M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43--60. Freely available here on the journal's website. But, I can not find the ...
9
votes
1answer
290 views

Intersection between integral closures is algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Does it follow that $...
7
votes
1answer
178 views

How fine an invariant of a representation is its quotient singularity?

This is a refinement of a question asked on MSE. Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In ...
1
vote
0answers
83 views

Dimension of a module (which is not necessarily finite)

Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
0
votes
1answer
96 views

$0 :_M I^n$ is finitely generated for all $i\ge 1$?

I see the remark that: "Let $R$ be a Noetherian commutative ring, $M$ an $R$-module and $I$ an ideal of $R.$ Assume that $0 :_M I$ is finitely generated. Then $0 :_M I^n$ is finitely generated for all ...
0
votes
1answer
143 views

Hilbert function and numerical polynomial

Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
3
votes
0answers
75 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
5
votes
2answers
358 views

A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$

Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote $w(f)=f(x_{w(1)},x_{w(2)},\ldots,...
3
votes
1answer
225 views

etale localization reference request

I'm looking for a reference for the following statement: Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes ...
0
votes
1answer
131 views

The injectivity of Noetherian ring

Let $R$ be a ring with 1, $M$ be a left $R$-module. Then $M$ is fp-injective if every $R$-homomorphism from a finitely presented left ideal to $M$ extends to a homomorphism of $R$ to $M$ i.e. if $\...
6
votes
1answer
138 views

What is the cokernel of $O_S \to F_\infty/O_\infty$?

Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...
3
votes
2answers
113 views

The injective hull of cyclic modules and self injective ring

It is well-known that if $R$ is a Noetherian ring, and the injective hull of every finitely generated $R$-module is projective, then $R$ is a self-injective. My question is that could one replace "...