# Tagged Questions

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

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### 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 ${\mathbb CP}^1$ ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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.)
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### Exact sequence of vector bundles

Consider the short exact sequences below; 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))\...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{C}[GL_3]$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
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### Canonical module of a Buchsbaum ring

Is the canonical module of a Buchsbaum ring a Buchsbaum module?
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### 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-...
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### 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 $\mathbb{Z}[\sigma_1,\dots,\sigma_n]$, ...
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### 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) ...
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### 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 $M \in S$...
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### 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 $\mathrm{Pic}(A)$...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $... 1answer 178 views ### How fine an invariant of a representation is its quotient singularity? This is a refinement of a question asked on MSE. Let$G$be a finite group and let$V$be a finite-dimensional faithful complex representation of$G$. Consider$V$as an affine complex variety. In ... 0answers 83 views ### Dimension of a module (which is not necessarily finite) Let$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 ... 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 ... 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 ... 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 ... 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,...
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 ...