**4**

votes

**2**answers

275 views

### Variety determined by interior product of the determinant?

Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., ...

**0**

votes

**0**answers

45 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 ...

**1**

vote

**0**answers

71 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 ...

**6**

votes

**0**answers

203 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 ...

**2**

votes

**0**answers

55 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{ ...

**17**

votes

**2**answers

392 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

**0**answers

87 views

### Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$? [on hold]

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...

**0**

votes

**0**answers

101 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

**1**answer

114 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

**1**answer

209 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 ...

**7**

votes

**3**answers

1k views

### For which fields K is every subring of K…?

This question was inspired by
How to prove that the subrings of the rational numbers are noetherian?
which some people found too routine to be of interest. So I have decided to liven things up a ...

**2**

votes

**1**answer

162 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

**1**answer

62 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
...

**2**

votes

**2**answers

159 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 ...

**3**

votes

**1**answer

121 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

**0**answers

148 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:
...

**0**

votes

**1**answer

115 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 ...

**3**

votes

**1**answer

174 views

### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...

**7**

votes

**2**answers

454 views

### Faithfully flat ring homomorphism and annihilator

Suppose that the ring homomorphism $R\rightarrow S$ is faithfully flat ($R$ and $S$ are Noetherian commutative rings). Let $A$ be an Artinian $R$-module. Do we have $0:_S(A\bigotimes_R S)=(0:_RA)S$.
...

**0**

votes

**0**answers

52 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

**0**answers

155 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 ...

**2**

votes

**1**answer

156 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 ...

**2**

votes

**0**answers

118 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

**1**answer

182 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 $m$? ...

**2**

votes

**1**answer

121 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 ...

**4**

votes

**1**answer

553 views

### Is there a non-Gorenstein ring but locally Gorenstein?

A commutative noetherian ring $R$ is Gorenstein if $R$ has finite injective dimension.
Obviously, if $R$ is Gorenstein, then $R$ localized at any prime ideal $P$ is also Gorenstein. But I don't know ...

**12**

votes

**4**answers

722 views

### Injective dimension of graded-injective modules

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra":
Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...

**6**

votes

**1**answer

547 views

### When quotients of polynomial rings are isomorphic to polynomial rings?

This might be a silly question but is there a criterion for when the quotient of $k[X_1,\ldots, X_n]$ by some ideal is isomorphic to a polynomial ring?
For instance $k[X,Y]/\langle Y \rangle ...

**10**

votes

**0**answers

260 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 $A$ ...

**0**

votes

**1**answer

135 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 $I$ ...

**5**

votes

**2**answers

223 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 ...

**5**

votes

**1**answer

128 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 ...

**1**

vote

**0**answers

362 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 ...

**0**

votes

**0**answers

48 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 variables $X_1,\dots,X_N$. Consider the analytic manifold $V(I)$ defined by the ideal $I$ in
...

**5**

votes

**0**answers

124 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**3**

votes

**0**answers

118 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 ...

**0**

votes

**1**answer

164 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

**1**answer

163 views

### reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration
1) if $m\geq n$ then $I_m\subset I_n.$
2) for all $m,n,$ ...

**1**

vote

**1**answer

68 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 ...

**21**

votes

**1**answer

829 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 ...

**3**

votes

**3**answers

716 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

**0**answers

43 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, ...

**-1**

votes

**0**answers

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 ...

**2**

votes

**1**answer

321 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

**1**answer

119 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

**1**answer

133 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

**1**answer

139 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) ...

**4**

votes

**1**answer

352 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 ...

**5**

votes

**1**answer

167 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 ...

**7**

votes

**4**answers

577 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 $X$ ...