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

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11 views

$I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals), an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$. [See Comm. Rings by Kaplansky, ...
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0answers
14 views

An ideal that contained in finitely many maximal ideals but all of its elements contained in infinitely many maximal ideals [migrated]

Is it possible that an ideal I in an integral domain D is contained in only finitely many maximal ideals but each element of I is contained in infinitely many maximal ideals? I am quit sure that it is ...
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1answer
240 views

Who defined and who coined “module”?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...
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1answer
101 views

Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$. Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring ...
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1answer
122 views

When is a local subring of a number field a valuation ring?

Do we have some good examples of local subrings of number fields which are not valuation rings? Do we have an easy criterion for determining whether a local subring of a number field is a valuation ...
3
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0answers
88 views

Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height. Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
3
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1answer
338 views

How to prove a Proposition of Rouquier?

Proposition 7.15 in Rouquier's paper (see Publication paper) "Dimension of triangulated categories J. K-theory, 1 (2008) 193-256" as follows, for details please see Rouquier's paper (or arXiv): ``Let ...
4
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1answer
148 views

Endomorphisms of a maximal ideal of a local ring

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Is it true in general that $\text{Hom}_R(\mathfrak{m},\mathfrak{m})\cong \text{Hom}_R(\mathfrak{m}, R)$? What if the Krull ...
4
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1answer
123 views

$ mult(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
5
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1answer
152 views

Is a projective module of constant finite rank finitely generated?

If $R$ is a (commutative) ring and $P$ is a projective $R$-module, then every localization of $P$ at a prime of $R$ is free by Kaplansky's theorem, and has a well-defined rank. If these ranks are all ...
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45 views

Decomposition of polynomials with three variables

We use $\bigtriangleup _i$ to denote either multiplication or addition. Suppose we have a polynomial $P(x,y,z)$ over some algebraic closed field such that: There are $Q(x), W_1(x,y),W_2(x,z)$ ...
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4answers
555 views

Constructing a space with prescribed cohomology ring

The most general way I can formulate my question is the following: Question 1: Given a Gorenstein quotient ring $S$ of a polynomial ring over a field $K$, can one construct a (topological) space $X$ ...
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40 views

Is there a positive integer k such that any endomorphism of any free module over any commutative ring is a linear combination of k idempotents?

Consider the following Condition (C) on a positive integer $k$: (C) If $R$ is a commutative ring, if $F$ is a free $R$-module, and if $f$ is an endomorphism of $F$, then $f$ is an $R$-linear ...
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0answers
83 views

$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$

Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of ...
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0answers
42 views

Grading on a fractional ideal of Rees ring [closed]

Let $R=K[x_1,\ldots ,x_n]$, $m=<x_1,\ldots,x_n>$ and $I$ be an m-primary ideal. Let $R(I)$ be the Rees ring of $I$. I am trying to understand following: $(R(I):mR(I))/R(I)$ is a finitely ...
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1answer
80 views

Isomorphic quotient of a Module over Noetherian commutative algebra [closed]

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
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1answer
113 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 ...
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0answers
70 views

Recovering fractional ideals from ideals

Let $R$ be a Noetherian domain; let ${I_{(i)}}$ be a set of fractional ideals in $K$, the fraction field of $R$, indexed by a lattice such that $I_{(i)} I_{(j)} = I_{(i + j)}$. Let $J_{(i)} = I_{(i)} ...
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1answer
199 views

Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ ...
2
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1answer
157 views

Scheme vs $A$-scheme morphisms

Let $A=S^{-1}\mathbb Z$ be a localization of a multiplicative set $S\subset \mathbb Z$. Question 1: Let $X$ be an arbitrary $A$-scheme, and view $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the ...
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0answers
159 views

Zariski open set of linear forms

Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$. I am looking for a Zariski open ...
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114 views

Open covering of the Hilbert functor of points

Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then ...
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1answer
47 views

Degenerations and spanning monomials

Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
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0answers
53 views

Tensor product of commutators vs. commutator in a tensor product

Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
2
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2answers
279 views

Canonical Sheaf of Projective Space

I am stuck on one step that occurs without explanation in several Algebraic geometry books. Starting from the exact sequence $$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow ...
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2answers
213 views

commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained). Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...
1
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1answer
116 views

General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset. Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...
2
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0answers
87 views

generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question. Let ...
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120 views

Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here. Geometrically, this asks if every Noetherian connected affine scheme can be ...
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224 views

When is $I \otimes_A \hat{A} \cong I\hat{A}$?

Let $A$ be a commutative ring and $I$ a (finitely generated) ideal in $A$. We denote by $\hat{A}$ the $I$-adic completion of $A$, i.e. $\hat{A} = \varprojlim(A/I \leftarrow A/I^2 \leftarrow \ldots)$. ...
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2answers
252 views

Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
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1answer
62 views

Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
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1answer
157 views

Classification (and automorphisms) of torsion-free modules/sheaves

I would like to know what can be said about the classification of torsion-free modules. For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...
2
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0answers
148 views

why do we care about the irreducibility of parameter ideals?

It is well known that a local commutative unital ring $R$ is Gorenstein if and only if every parameter ideal is irreducible. Why the irreducibility of parameter ideals in a Gorenstein local ring is ...
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1answer
75 views

Formally smooth map from a regular ring

Let $A,B$ be two commutative noetherian rings. Let $f:A\to B$ be a formally smooth homomorphism. If $A$ is a regular ring (in the sense that all its localizations are regular local rings), does this ...
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142 views

A question about the unbounded derived category of the polynomial ring in infinitely many variables

In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that ...
3
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79 views

When does a commutative DGA have a finitely generated quasi-free resolution?

Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question ...
3
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2answers
169 views

Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
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0answers
19 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 ...
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0answers
131 views

Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact: If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
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191 views

R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module. Would $R$ noetherian ...
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1answer
114 views

Is the completion of an arbitrary ring w.r.p. a maximal ideal a local ring? [closed]

Give an arbitrary commutative ring (not necessarily noetherian) $A$ with $\mathfrak{m}$ a maximal ideal. Is the completion $\hat{A}$ w.r.t. the ideal $\mathfrak{m}$ a local ring? If so, is the maximal ...
3
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1answer
74 views

Regular commutative Banach algebras which are not closed under complex conjugate

Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some ...
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40 views

When does the Spectrum of a Commutative Hopf Algebra Separate Points?

Let $H$ be a (finitely generated) commutative Hopf algebra over the complex numbers. When is it true that, for every $g \in H$, we can always find an algebra map $f_g:H \to \mathbb{C}$ such that ...
5
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108 views

Vanishing of Andre-Quillen homology and injective dimension

Let $(A,m,k)$ be a commutative local ring. Assume that for all $n\ge 3$, the Andre-Quillen homology modules $H_n(A,k,k)$ vanish. Does this imply that $A$ has finite injective dimension over itself? ...
5
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2answers
202 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 ...
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67 views

Completion of Bezout Domain a Bezout Domain?

Let $R$ be a Bezout domain, and $I$ any ideal inside of $R$. Is the $I$-adic completion $$ \varprojlim_i R/I^i $$ necessarily a Bezout domain? If not, what conditions (on $R$ or $I$) might ensure that ...
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54 views

stability of discriminant curve of nonsingular nets of quadrics

The following are from this question: A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian ...
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142 views

A doubt from the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring”

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
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2answers
124 views

Ideals in the I-adic completion of a ring

Let $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$ $$ \varprojlim_i R/I^i $$ arise as the $I$-adic completion of some ideal inside of the original ring $R$?