**1**

vote

**1**answer

90 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

121 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

121 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) ...

**3**

votes

**1**answer

334 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 ...

**3**

votes

**0**answers

85 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 ...

**5**

votes

**1**answer

151 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

553 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 $X$ ...

**4**

votes

**1**answer

147 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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**1**answer

548 views

### common roots of bivariate polynomial equations

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...

**1**

vote

**1**answer

225 views

### Initial ideal of k-th power of an ideal

Hi,
Let $I$ be an ideal in a polynomial ring $S = k[x_1, \ldots, x_n]$, where $k$ is an algebraically closed field of characteristic zero. Fix a term order on
$S$ (e.g. a lexicographic order) and ...

**2**

votes

**1**answer

156 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 ...

**1**

vote

**0**answers

43 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 $Q(x), W_1(x,y),W_2(x,z)$ ...

**3**

votes

**0**answers

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 ...

**0**

votes

**2**answers

223 views

### Computing the minimal free resolution of a coherent sheaf on projective space

Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...

**5**

votes

**1**answer

474 views

### Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference.
Let $A$ be a regular commutative noetherian ring (and ...

**0**

votes

**0**answers

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 ...

**4**

votes

**1**answer

356 views

### Algebraic closure of a polynomial ring

What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $k$, ie ...

**0**

votes

**1**answer

88 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

**0**answers

38 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 ...

**2**

votes

**1**answer

153 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 ...

**3**

votes

**1**answer

423 views

### working with local rings: “abstract” vs “geometric” proofs

Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.
Suppose $R$ happens to be the ring of ...

**0**

votes

**1**answer

158 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

111 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 ...

**2**

votes

**2**answers

511 views

### How to find the generic initial ideal?

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27
Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a generic forms of degree $d$ and $e$,the generic initial ideal of $I=\langle ...

**1**

vote

**1**answer

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 ...

**-3**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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)} ...

**1**

vote

**1**answer

197 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
$$
...

**5**

votes

**2**answers

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 ...

**3**

votes

**2**answers

357 views

### Unramified (finite) extensions of fields complete with respect to a discrete valuation

Hello,
I've been reading the excellent online book on Algebraic Number Theory by J.S.Milne. In the section described above there is a footnote maintaining that the separability of the residue field ...

**4**

votes

**2**answers

366 views

### Algorithm to decide if ideal is principal

Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not ...

**1**

vote

**1**answer

156 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**

votes

**1**answer

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 ...

**21**

votes

**2**answers

1k views

### Discriminant and Different

First some context. In most algebraic number theory textbooks, the notion of
discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their ...

**0**

votes

**0**answers

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 ...

**0**

votes

**1**answer

152 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

135 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 $m,n,$ ...

**4**

votes

**1**answer

523 views

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

A commutative neotherian 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 whether ...

**1**

vote

**0**answers

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 ...

**2**

votes

**0**answers

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 ...

**11**

votes

**4**answers

707 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

**2**answers

352 views

### Powers of elements in an Artinian Ring

Let $R$ be an local Artinian ring, with maximal ideal $\mathfrak{m}$.
Let $e$ be the smallest positive integer for which $\mathfrak{m}^e=(0)$.
Let $t$ be the smallest positive integer for which ...

**1**

vote

**2**answers

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 ...

**28**

votes

**5**answers

2k views

### How to think about CM rings?

There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to ...

**2**

votes

**2**answers

278 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 ...

**3**

votes

**3**answers

454 views

### Height of ideal in graded ring

I have asked this question on MS however I did not receive any answer. Please help me to solve it. Thank you very much!
Let $R$ is a commutative Noetherian graded ring and $I$ is an ideal of $R$, ...

**1**

vote

**1**answer

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 ...

**5**

votes

**1**answer

474 views

### quotients of 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 \simeq k ...

**7**

votes

**0**answers

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. $\hat{A} = \varprojlim(A/I \leftarrow A/I^2 \leftarrow \ldots)$.
...