**1**

vote

**0**answers

65 views

### Failure of little lemma in non-separable case

A nice little lemma in commutative algebra says the following (see for instance proposition 5.17 in [Atiyah-MacDonald]):
If $A$ is a Noetherian integrally closed domain, $K$ its field of fractions ...

**0**

votes

**1**answer

269 views

### Number of elements in a fiber

Let $A\subseteq B$ be normal affine domains over an algebraically closed field of characteristic 0. If it is given that the corresponding morphism of schemes Spec $B\rightarrow$ Spec $A$ is ...

**2**

votes

**1**answer

72 views

### Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over ...

**0**

votes

**0**answers

112 views

### Cohen-Macaulay fibers

Let $Y$ be a set of points in $\mathbb{P}^n$. Then we can write a resolution
$$0\rightarrow P_n \rightarrow \cdots \rightarrow P_0\rightarrow \mathcal{O}_Y$$
where each ...

**3**

votes

**1**answer

101 views

### Existence of Factor rings of UFDs which are UFDs

Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$.
I am interested in hypersurfaces of $X$ which again satisfy this ...

**0**

votes

**1**answer

106 views

### Samuel multiplicity

Let $X$ be the hyper-surface defined by
$$f:=\sum_{i=1}^k x_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x_1^{n-1},\dots , x_k^{n-1}) $$
What is ...

**0**

votes

**0**answers

42 views

### t-linked extension

Let $A\subseteq B$ be an extension of commutative integral domains. the extension is t-linked if it satisfies the following property:
If P is a finitely generated ideal of A such that $P^{-1}=A$ than ...

**5**

votes

**1**answer

353 views

### Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?

Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?

**3**

votes

**0**answers

127 views

### going down theorem

Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral.
Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian ...

**0**

votes

**0**answers

56 views

### Is any affine domain J-1?

Can someone please suggest me a reference for the fact(!) that any affine domain over a field $k$ is $J-1$, i.e., its regular locus is open (I hope the result holds even if $k$ is of finite ...

**3**

votes

**1**answer

120 views

### When does a faithful module have an element with zero annihilator?

This is a follow up of Example of a finitely generated faithful torsion module over a commutative ring.
Let $M$ be a finitely generated module over a commutative ring $R$ with the property that ...

**1**

vote

**0**answers

77 views

### What is the reduction number of ideal $\langle t^a,t^b,t^c\rangle$?

Let $K[[t^a,t^b,t^c]]$ be formal power series (where $K$ is a field), $0<a<b<c$ and $\gcd(a,b,c)=1$.
$I=\langle t^a,t^b,t^c\rangle$, $J=\langle t^a\rangle $.
What $n$(least) is satisfy ...

**2**

votes

**1**answer

200 views

### Presentation of the tautological bundle of the Grassmannian

Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the ...

**0**

votes

**0**answers

143 views

### For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo ...

**1**

vote

**0**answers

119 views

### Intersection Multiplicity

Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...

**1**

vote

**1**answer

47 views

### Bound on the weight of the minimum weight generator of [n,k] cyclic codes?

I'm looking at creating sparse generator matrices for cyclic codes of a given length and dimension. A generator matrix of an [n,k] cyclic code can be expressed as
$G = \begin{bmatrix}g_0 & g_1 ...

**0**

votes

**1**answer

103 views

### Scalar restriction and scalar extension

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf ...

**0**

votes

**0**answers

83 views

### Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials

Let $R=\mathbb{R}[X_1,\dots,X_n]$, and
$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$
...

**16**

votes

**0**answers

245 views

### Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions ...

**0**

votes

**0**answers

58 views

### Vanishing of top local cohomology when $R$ is domain

Let ${\rm R}$ be a Noetherian domain, $\frak a$ an ideal of ${\rm R}$ and $c:=\operatorname{cd}(\frak a, {\rm R})$ is finite. Is it true that $\operatorname{Ann}_R(H^c_{\frak a}({\rm R}))=0$. Note ...

**0**

votes

**0**answers

55 views

### annihilators of top local cohomology modules

Let $R$ be a commutative Noetherian ring. Let $\frak a$ be an ideal of $R$ and ${\rm M}$ be a f.g $R$- module such that $c:=cd(\frak{a},{\rm M})$ is finite and $x\in R$. Is it true that $xH^c_{\frak ...

**1**

vote

**1**answer

129 views

### Annihilator of tensor product when $R$ is domain

Let $R$ be a Noetherian domain and $M$ and $N$ be two faithful $R$-modules. Is it true that $\operatorname{Ann}_R(M\otimes_R N)=0$?

**0**

votes

**2**answers

122 views

### Hochster-Roberts Theorem reciprocal

Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$
a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$
is also Cohen-Macaulay. This is known as ...

**1**

vote

**1**answer

75 views

### $IM=mM$. can we say that $I$ is a reduction ideal of $m$?

Question. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite faithful $R$-module. Let $I$ be an ideal of $R$ such that $IM=mM$. Can we say that $I$ is a reduction ideal of $m$? Recall that $I$ ...

**0**

votes

**0**answers

74 views

### regular locus of an affine domain

Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always ...

**1**

vote

**0**answers

127 views

### local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?
For example, we know that the completion ...

**2**

votes

**1**answer

107 views

### Relation between intersection multiplicities

Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that:
i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and
ii) $f_1g_1+\dots+f_ng_n\equiv0$.
...

**3**

votes

**1**answer

184 views

### Does the ring of invariants inherit normality?

Let $A$ be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group $G$ acts on $A$ by ring automorphisms. Form the subring $A^G \subset ...

**2**

votes

**2**answers

113 views

### Ascending chain condition on radical ideals

There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition ...

**4**

votes

**1**answer

290 views

### Is the ring of invariants Noetherian?

Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...

**6**

votes

**0**answers

111 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**0**

votes

**1**answer

159 views

### Properties of Integral Closure [closed]

Definition(Integral closure): Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisﬁes a monic equation
$x^n + i_1x^{n−1} + ··· + i_n = 0$ such ...

**6**

votes

**2**answers

204 views

### Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...

**10**

votes

**1**answer

323 views

### When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?

Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...

**1**

vote

**1**answer

184 views

### Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...

**1**

vote

**1**answer

145 views

### Noetherianess of a finite module over a noethrian ring without Axiom of Choice

All rings are assumed to be commutative with 1.
We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...

**1**

vote

**2**answers

153 views

### if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is “Gorenstein”

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...

**1**

vote

**0**answers

64 views

### The kernel of the residue map before passing to Milnor's K-theory

Let $F$ be a field of zero characteristic. All groups are taken modulo torsion.
Consider a residue map from the exterior algebra of the multiplicative group of the function field of the projective ...

**2**

votes

**0**answers

72 views

### Are all (graded) Artinian complete intersections like this?

I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...

**1**

vote

**0**answers

76 views

### “Exceptional components” of the exceptional divisor of a blow up

Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ ...

**6**

votes

**0**answers

405 views

### Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$

Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...

**0**

votes

**0**answers

43 views

### normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field.
For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal?
Same question for ...

**8**

votes

**0**answers

107 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**4**

votes

**1**answer

194 views

### Generators vs minimal degree polynomials of ideals

Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that
$p_1$ is of degree 2 and up to a ...

**3**

votes

**3**answers

181 views

### canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...

**3**

votes

**1**answer

275 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

123 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 ...