**3**

votes

**1**answer

277 views

### Invertibility of a matrix whose entries are certain binomial coefficients

Let $l$ be a positive integer. Does the matrix
$$
M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?

**5**

votes

**0**answers

71 views

### How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$

Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...

**1**

vote

**1**answer

124 views

### Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...

**1**

vote

**0**answers

114 views

### A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$

The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...

**0**

votes

**1**answer

108 views

### Canonical module of rees algebra

[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...

**2**

votes

**2**answers

72 views

### Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein?

$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder ...

**3**

votes

**1**answer

346 views

### What is the explicit ideal (wrt the Lazard ring) generated by the associativity of formal group laws?

Quick Preliminaries:
A commutative formal group law is a formal power series $F(x,y)=\sum_{ij}c_{ij}x^iy^j$ that satisfies:
Commutativity: $F(x,y) = F(y,x)$
Identity: $F(x,0)=x=F(0,x)$
...

**2**

votes

**1**answer

112 views

### A criterion for complete intersection in terms of the Hilbert series?

Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...

**1**

vote

**2**answers

210 views

### Algebraic characterization of commutative rings of Krull dimension 1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of ...

**10**

votes

**2**answers

349 views

### Computing intersection of subrings

Let $R$ be a finitely generated commutative ring over a field, for concreteness.
If $S,T \leq R$ are two finitely generated subrings, is their intersection
also finitely generated?
(Certainly ...

**1**

vote

**0**answers

78 views

### Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve

I want to follow this construction of a normal model of a curve:
Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. ...

**0**

votes

**0**answers

111 views

### How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties].
$X$ : an algebraic variety over a field $k$.
$(R,m)$ : a DVR of the quotient field $K(X)$ ...

**0**

votes

**1**answer

105 views

### Examples of fractional ideals whose inverse does not commute with the product

Let $R$ be an integral domain, $K$ its field of fractions, and $I,J$ fractional ideals.
If $R$ is a Krull domain, then $(R:_KIJ)=(R:_KI)(R:_KJ)$, or $(IJ)^{-1}=I^{-1}J^{-1}$.
But I can't see any ...

**3**

votes

**1**answer

120 views

### Example of fractional ideal whose inverse does not commute with localization

Let $R$ be an integral domain, and $K$ its field of fractions. It is well known that for a finitely generated fractional ideal $I$ of $R$, and $S$ a multiplicative set we have ...

**1**

vote

**1**answer

164 views

### specific question related to the extension of an integrally closed domain and the residual fields

I really need help !
In a previous thread, I have asked for the solution of a general question, without getting answers. Since this question was posted, I have reduced the problem to the following ...

**0**

votes

**0**answers

68 views

### An embedding of modules by tensor product over a Noetherian domain

I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...

**2**

votes

**1**answer

255 views

### Automorphisms of ideals of $\mathbb{C}[t]$

Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$.
The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following:
For ...

**4**

votes

**0**answers

133 views

### Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...

**0**

votes

**0**answers

46 views

### Extension of canonical surjection to a place of fraction field

Let $R$ be an integrally closed domain, $K$ its field of fractions, and $m$ a maximal ideal of $R$.
By Chevalley theorem, the canonical surjection $R\to R/m$ extends in at least one way to a place ...

**6**

votes

**1**answer

313 views

### Irreducibility of a polynomial

For $n\ge 1$, let $g(x_1,x_2,\ldots,x_n)$ be an irreducible homogeneous polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x_1,x_2,\ldots,x_n))$ ...

**4**

votes

**0**answers

173 views

### Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...

**3**

votes

**2**answers

179 views

### Divisibility of the degree of an extension by the degree of its residual field

Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...

**0**

votes

**0**answers

88 views

### Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d).
I'm more interested in the algebraic point of ...

**4**

votes

**0**answers

89 views

### Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...

**1**

vote

**1**answer

127 views

### Reducedness of a ring with prime nilradical

Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...

**6**

votes

**3**answers

400 views

### Irreducible/prime/indivisible elements

in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is irreducible if ...

**1**

vote

**0**answers

68 views

### ramification index generalized

I am trying to rewrite the theory of decomposition/inertia/ramification groups independently of the theory of Dedekind or valuation rings (I believe this has been done elsewhere, but I found only few ...

**6**

votes

**2**answers

590 views

### Is being reduced a generic property of schemes?

(Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. ...

**8**

votes

**0**answers

243 views

### What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in ...

**8**

votes

**3**answers

436 views

### Ring of differential operators of a quotient ring

All rings are assumed to have unity.
Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$:
...

**1**

vote

**1**answer

104 views

### question about a particular Polynomial ring [closed]

Let K be a field, let $T = K[X_1, X_2,...]$ be a polynomial ring, let $R=K[X_1^{2}, X_1X_2,..,X_i X_j,..]$, and let $L = Frac(R)$ = field of fractions of R. How can we prove that $R =T \cap L$ ?

**7**

votes

**1**answer

404 views

### Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...

**0**

votes

**0**answers

111 views

### Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?

Let $A$ be a commutative ring. Let $M\subseteq N$ be an extension of positive seminormal monoids. Is the natural map $Pic A[M] \rightarrow Pic A[N]$ injective?

**1**

vote

**1**answer

68 views

### betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b):
can you help please?
...

**2**

votes

**1**answer

197 views

### finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...

**4**

votes

**1**answer

354 views

### Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field.
What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...

**5**

votes

**0**answers

143 views

### cohomology algebra of unordered configuration space with coefficients the finite fields

in the paper The cohomology algebra of unordered configuration spaces (Y. Félix, D. Tanré, J. London Math. Soc., 2005), Theorem 4:
Let $M$ be an odd-dimensional, compact, closed, oriented manifold. ...

**2**

votes

**1**answer

178 views

### When do we get free modules from Noether normalization

Let $X \subseteq \mathbb{P}_{\mathbb{C}}^n$ be an irreducible, projective, Cohen-Macaulay variety of dimension $k$. Let $L \subseteq \mathbb{P}_{\mathbb{C}}^n$ be a linear space of dimension $n-k-1$ ...

**1**

vote

**0**answers

113 views

### Can you always find a regular sequence consisting of monomials?

Let $\mathbb{k}$ be a field, and let $S=\mathbb{k}[x_1,x_2,\ldots,x_n]$. Let $M$ be an $S$-module. A sequence $$f_1,f_2,\ldots,f_r$$ of polynomials in the maximal ideal $\langle x_1,\ldots,x_n\rangle$ ...

**5**

votes

**1**answer

292 views

### automorphisms of local rings vs local change of coordinates

Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can ...

**3**

votes

**0**answers

114 views

### On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...

**3**

votes

**1**answer

230 views

### Non-commutative normalization

Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that
a) $Z$ is a noetherian domain
b) $A$ is a finitely generated module over $Z$.
...

**9**

votes

**3**answers

725 views

### Completion of a local ring of a curve

Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...

**2**

votes

**1**answer

193 views

### Hochschild cohomology of commutative quotients

Notation:
Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$.
If ...

**3**

votes

**1**answer

189 views

### Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$.
...

**0**

votes

**1**answer

136 views

### if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?

definition from Bruns-Herzog:
It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true:
if $ \lambda (I)= \dim R$, can one ...

**0**

votes

**1**answer

111 views

### Connected curve

Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...

**2**

votes

**0**answers

66 views

### variants of ramification groups - need terminology and sources

I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question.
Let $L/K$ be a Galois extension, and $w$ be a valuation of ...

**1**

vote

**1**answer

180 views

### Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...

**3**

votes

**3**answers

259 views

### Injective map between two schemes

Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X_y| =1$, then can we say $f$ is ...