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

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4
votes
1answer
127 views

computing the nonnegative part of a $\mathbb{Z}$-graded ring

Let $R = \bigoplus_{n \in \mathbb{Z}} R_n$ be a $\mathbb{Z}$-graded commutative ring with nonnegative part $R^+ = \bigoplus_{n \geq 0} R_n$ and nonpositive part $R^- = \bigoplus_{n \geq 0} R_{-n}$. By ...
0
votes
0answers
134 views

On degrees of polynomials with matching zeros in a subset

Let $S\subsetneq \Bbb R^n$ such that $|S|<\infty$ and for all partitions $S_1$ and $S_2=S\backslash S_1$ of $S$, there exits a multilinear polynomial $h$ such that $h(s)=1-h(s'),\mbox{ }\forall ...
-1
votes
0answers
52 views

On multilinear linear combinations

If $K$ is algebraically closed field, then consider $m$ multilinear polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ ...
26
votes
4answers
1k views

Why Cohen-Macaulay rings have become important in commutative algebra?

I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects. I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm ...
0
votes
0answers
52 views

Criterion for global dimension of subring

All rings are assumed to be associative and unital. If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
0
votes
1answer
209 views

Projective Modules/Algebras: decomposition of linear functions, and the rank formula

Let $A$ be a ring, $B$ a finite projective $A$-algebra, and $C$ a finite projective $B$-algebra. We can show that $C$ is also finite and projective when regarded as an $A$-algebra (by, for instance, ...
0
votes
1answer
125 views

When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map $$Mor_{Sch}(S,X)\to ...
-2
votes
0answers
51 views

Is it true that standard R-algebra remains standard after going modulo a homogeneous ideal [on hold]

Is it true that any standard R-algebra remains standard after going modulo a homogeneous ideal?
1
vote
1answer
221 views

When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
4
votes
1answer
188 views

Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...
0
votes
0answers
60 views

Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...
0
votes
0answers
227 views

Can we find a Groebner Basis?

I would like to ask the following. Given only the leading terms of an ideal $I$, namely the set $LT(I)$, is it possible to find a Groebner Basis of $I$? If not always, then when is it possible? We ...
2
votes
1answer
284 views

geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring. What is the geometric interpretation of 1- Gorenstein rings 2- Complete intersections 3- Regular rings? and how can I realize differences by geometric ...
1
vote
1answer
232 views

Is every polynomial ring over any field regular?

Is every polynomial ring over any field regular? For a field that is algebraically closed, it is true as any maximal ideal of $k[x_1,...,x_n]$ corresponds to a point $(t_1,...,t_n)$ in ...
0
votes
2answers
158 views

Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?
1
vote
0answers
69 views

Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...
0
votes
1answer
150 views

A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...
1
vote
0answers
113 views

Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...
4
votes
6answers
2k views

an easy example of valuation ring which is not noetherian? [duplicate]

Is there an easy example of valuation ring which is not noetherian?
-2
votes
0answers
33 views

Associated primes of generic deformation [closed]

Let $I \subset S$ be a monomial ideal and $I_{\epsilon}$ it's generic deformation. Is it true that Ass $S/I$ = Ass $S/I_{\epsilon}$ ?
6
votes
3answers
475 views

Is this formally étale morphism of schemes an isomorphism?

In the last days I came to consider the following question which I'd be happy to see answered by the affirmative: if $f:X\to S$ is a morphism of schemes which is formally étale, quasicompact, ...
0
votes
1answer
138 views

Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...
7
votes
1answer
451 views

Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking ...
1
vote
0answers
42 views

Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...
16
votes
4answers
4k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...
4
votes
0answers
140 views

How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...
9
votes
3answers
1k views

State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...
2
votes
1answer
101 views

Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is ...
1
vote
0answers
38 views

Bass' stable range for Bezout rings

As discussed in this MO topic, every principal ideal domain has stable rank at most 2. The proof in the accepted answer uses the fact that PID is a unique factorization domain, but there can be no ...
3
votes
0answers
67 views

Is there some algorithm for calculating the least number of generators of an ideal in a polynomial ring?

My question comes from the context of algebraic geometry. By Krull's principal ideal theorem, the number of generators of the defining ideal of a variety gives an upper bound of its codimension. ...
2
votes
1answer
172 views

for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
0
votes
0answers
54 views

Isomorphy of finite $R$-algebras under special conditions

Let $R=k[x_1, \ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Let $R \subseteq S$ be a finite ring extension i.e. $S$ is finitely generated as $R$-module (free if that helps) ...
1
vote
3answers
209 views

Under the condition specified below, is $\mathcal{O}_X(X-V(I))=R$, where $X=\mathrm{Spec}R$?

Is it true that given a noetherian normal domain $R$ and an ideal $I$ of height $\geq 2$ we have $\mathcal{O}_X(X-V(I))=R$, where $X=\mathrm{Spec}R$?
3
votes
1answer
159 views

Canonical presentation of pro-modules over pro-rings

Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i ...
25
votes
0answers
1k views

A short proof for $\dim(R[T])=\dim(R)+1$

For a commutative ring $R$ we clearly have $\dim(R[T]) \geq \dim(R)+1$. If $R$ is noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
0
votes
0answers
100 views

Checking smoothness of complex schemes by reduction to characteristic prime

Let $X=V(F_1,\ldots,F_n)$ be a closed subscheme of the complex projective space $\mathbb{P}^N_{\mathbb{C}}$ defined by homogeneous polynomials $F_i\in\mathbb{Z}[x_0,\ldots,x_N]$ with integer ...
0
votes
1answer
136 views

Theorem 2.5 in “Castelnuovo-Mumford regularity of products of ideals” by Conca & Herzog

Theorem 2.5 in Conca and Herzog, Castelnuovo-Mumford regularity of products of ideals http://arxiv.org/abs/math/0210065, says that if $R$ is a polynomial ring over a field $k$, $I$ a homogeneous ideal ...
1
vote
0answers
108 views

Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed ...
4
votes
3answers
780 views

An example where GCD depends on the domain

First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$. I want to find an example of an GCD-domain $R$, a subdomain $S ...
0
votes
0answers
134 views

Is this essentially of finite type algebra actually of finite type?

Let $R$ be a discrete valuation ring with a uniformizer $\pi$ and $(A, \mathfrak{m}_A)$ a local $R$-algebra that is essentially of finite type (i.e., is a localization of a finite type $R$-algebra), ...
1
vote
1answer
86 views

Given a locally nilpotent derivation over a field of characteristic 0 and a local slice, how is the ring homomorphism below defined?

Let $K$ be a field of characteristic 0 and $A$ a $K$-domain. Let $D:A\longrightarrow A$ be a locally nilpotent K-derivation, that is, $D(k)=0$ for all $k\in K$, $D(ab)=(Da)b+a(Db)$ for all $a,b\in A$, ...
3
votes
1answer
99 views

“Unramified” extension of DVRs and permanence of excellence

Recall that a discrete valuation ring $R$ is excellent if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = ...
3
votes
1answer
203 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
0
votes
1answer
209 views

When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
3
votes
1answer
140 views

When two determinantal ideals together generate a power of the maximal ideal?

(A somewhat technical question, but maybe it is well known.) Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal ...
0
votes
0answers
39 views

Subextensions of separable field extensions [migrated]

Let $F/L/K$ be a tower of (possibly nonalgebraic) field extensions. If $F/K$ is separable, does it follow that $F/L$ is also separable? I recall that a field extension $a/b$ is separable if for any ...
14
votes
2answers
498 views

Nice Algebraic Statements Independent from ZF + V=L (constructibility)

Background and Motivation I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering ...
1
vote
1answer
169 views

Self-similarity for simple algebraic structures [closed]

I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...
6
votes
2answers
262 views

Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function $P_i = px^i + qy^i + rz^i$ where $x,y,z$ are coordinates. I have a few related ...
3
votes
1answer
175 views

Flatness of Normalization of regular schemes

I have a followup to the following question: Flatness of normalization. Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in ...