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

learn more… | top users | synonyms (1)

16
votes
3answers
1k views

Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
3
votes
1answer
190 views

Spec of an injective ring map contains minimal primes in its image?

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the ...
1
vote
0answers
42 views

a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...
3
votes
0answers
52 views

survival of a prime ideal in its Nagata transform

Let $R$ be a Noetherian normal domain with fraction field $K$. Recall that for any ideal $I \subseteq R$, its Nagata transform $T(I)$ is defined as the set of elements $f\in K$ such that $I^n f ...
1
vote
0answers
228 views

Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
0
votes
0answers
71 views

F-splitting and F-purity from commutative algebra viewpoint

First I define two terms: Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...
2
votes
0answers
64 views

Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...
1
vote
0answers
104 views

preservation of localness among certain Krull domains

The following question essentially appeared (http://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything ...
7
votes
0answers
275 views

Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end: Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...
0
votes
0answers
113 views

Does exterior product commute functor Hom?

Let $M$ be an module over the commutative ring $R$. I'd like to ask do we have the following isomorphism? $$Hom_R(\wedge^n_RM,R)\simeq \wedge^n_R Hom_R(M,R)$$ We can obviously see it's true for the ...
3
votes
0answers
139 views

Categorical characterization of closed imbeddings

Let $f\colon X\to Y$ be a morphism of schemes. Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. ...
13
votes
2answers
291 views

Flag complexes that are shellable but not vertex decomposable

As the title suggests, I was wondering if anyone can point me to any examples in the literature to flag complexes that are shellable but not vertex decomposable. It is well-known that if a ...
0
votes
1answer
73 views

Depth formula in CM-ring involving canonical module

In this article by Iyama and Wemyss there is the following formula: Let $R$ be a Cohen-Macaulay ring with canonical module $\omega$, let $X$ be a finitely generated $R$-module. Then ...
3
votes
2answers
194 views

Jordan-Holder vs Harder-Narasimhan

Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration: $F^0M=M$; $F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is ...
17
votes
2answers
453 views

Stability of real polynomials with positive coefficients

Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive. For $f$ a ...
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 ...
5
votes
0answers
162 views

Dimension of totally reflexive modules

Let $R$ be a commutative Noetherian ring and let $M$ be a finitely generated $R$-module. Definition. $M$ is called totally reflexive (or $G-\dim_RM = 0$) if it satisfies the following conditions: ...
1
vote
0answers
64 views

Orders of certain quotients of power series rings

Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible ...
0
votes
0answers
165 views

integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting. Let $M$ be a smooth curve over an algebraically closed field $k$. Let ...
1
vote
1answer
145 views

Bounds for Betti numbers

Why the graded Betti numbers of ideal $I \subset k[x_1 , \cdots , x_n]$, are bounded by the graded Betti numbers of $\mathrm{gin}(I)$? (Where $\mathrm{gin}(I)$, is the generic initial ideal of $I$ ...
1
vote
0answers
407 views

Submodule embeddable in a finitely generated module

I have a terminology question. For a commutative ring $A$ (not necessarily Noetherian), $A$-modules that are isomorphic to an $A$-submodule of a finitely generated $A$-module form a fairly good class ...
15
votes
1answer
362 views

If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective?

(In this question, all rings and algebras are commutative with identity.) I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ ...
6
votes
1answer
165 views

Coloring summands of given n-partition with given weights of colors

Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$ Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...
0
votes
0answers
72 views

Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action $$ f: X_i \to (1+X_i)^{m} - 1 $$ on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
1
vote
1answer
76 views

Algorithm for Polynomial Reduction in a Quotient Ring

Any reference or suggestion for the following problem would be greatly appreciated. I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...
2
votes
0answers
124 views

Irreducibility of $x^m-g(y)$

Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?
0
votes
1answer
82 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, ...
2
votes
1answer
353 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 ...
1
vote
1answer
125 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 ...
-1
votes
1answer
137 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 ...
4
votes
0answers
122 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 ...
4
votes
1answer
360 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
votes
1answer
169 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 ...
5
votes
1answer
227 views

Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be $$ \text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!, $$ where $ P_{M} $ denotes the Hilbert ...
5
votes
1answer
189 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 ...
1
vote
0answers
53 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 $Q(x), W_1(x,y),W_2(x,z)$ ...
7
votes
4answers
587 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 $X$ ...
3
votes
0answers
46 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 ...
0
votes
0answers
94 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 ...
-3
votes
1answer
100 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 ...
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 ...
1
vote
0answers
75 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)} ...
2
votes
1answer
222 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
votes
1answer
160 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 ...
0
votes
0answers
163 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 ...
2
votes
1answer
216 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 ...
2
votes
1answer
55 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 ...
2
votes
0answers
61 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
votes
2answers
322 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 ...
1
vote
2answers
221 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 ...