**1**

vote

**1**answer

79 views

### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...

**2**

votes

**0**answers

84 views

### Irreducibility of a general fibre

Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact?
If $A$ is algebraically ...

**8**

votes

**1**answer

190 views

### Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$

Dimension refers to the Krull dimension of a commutative ring.
In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring:
Let $k$ be a field and $K=k(t)$ a ...

**0**

votes

**0**answers

44 views

### How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian
Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite?
I'm looking for equivalent ...

**1**

vote

**1**answer

110 views

### Arbitrary chains of prime ideals in $R[X]$

For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q_1\subset Q_2\subset Q_3$ is a chain of prime ideals of ...

**2**

votes

**0**answers

142 views

### Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?

In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...

**3**

votes

**1**answer

278 views

### UFD and fundamental group

Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus ...

**2**

votes

**0**answers

63 views

### Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:
Let $(A,\mu)$ be a commutative associative algebra ...

**1**

vote

**0**answers

67 views

### Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...

**0**

votes

**0**answers

30 views

### J not being subset of integral closure of I

Definition. Let $I$ and $J$ be ideals in a ring $R$. An element $r \in R$ is said to be
integral over $I$ if there exist an integer $n$ and elements $a_i \in I^i, i = 1, . . . , n$,
such that
$$r^n + ...

**1**

vote

**1**answer

144 views

### On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...

**2**

votes

**0**answers

55 views

### completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion ...

**3**

votes

**1**answer

116 views

### Uncountable chain of prime ideals in an arbitrary direct product of rings

I am only considering commutative rings with $1$. Dimension refers to Krull dimension.
In the paper "Products of commutative rings and zero-dimensionality", Gilmer and Heinzer give necessary and ...

**3**

votes

**3**answers

167 views

### Are linearizations of involutive PDEs locally solvable?

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing ...

**2**

votes

**1**answer

105 views

### complement of an open immersion

Let $A\subseteq B$ be normal affine doamins over a field $k$ with same field of fractions. If the induced morphism of schemes $i^*:Spec\ B \rightarrow Spec\ A$ is an open immersion, how to prove that ...

**1**

vote

**0**answers

59 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

**0**answers

75 views

### is there a relationship between $\ell (R/I^n)$ and $\ell (R/I)$ [closed]

$(R,m)$ is local neotherian cohen-macaulay ring of dimension $d$, and $I$ is an $m$-primary ideal of $R$. since $I$ is an $m$-primary, $\dim R /I=\dim R/I^n =0$. so $\ell(R/I^n)$ and $\ell (R/I)$ are ...

**0**

votes

**1**answer

258 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

47 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

99 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

90 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

81 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

36 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

337 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

107 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

54 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

76 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

64 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 ...

**-1**

votes

**0**answers

55 views

### Rees algebra isomorphism

Let $R$ be a ring, $I$ an ideal and $t$ a variable over $R$. The
Rees algebra of $I$ is the subring of $R[t]$ defined as $R[It]=Rt^0\oplus I^1t^1 \oplus I^2 t^2........$
and extended Rees algebra of ...

**2**

votes

**1**answer

161 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

52 views

### Is there any criterion when an interger solution of the equation $ax^2-by^2=c$ $(a,b,c\in\mathbb{N})$ exists? [migrated]

As title. In particular I'd like to know if it suffices that it's solvable in $\mathbb Z_p $ for $p$ being any prime number.

**-1**

votes

**0**answers

10 views

### intersection of submodule [migrated]

I have question regarding intersection of submodules. Could anyone give example of a commutative ring $R$ with identity and an $R-$module $M$ such that
$$BM\cap CM\nsubseteq (B\cap C)M$$ for some ...

**0**

votes

**0**answers

132 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

104 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

37 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

65 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

77 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
$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$
...

**-1**

votes

**0**answers

181 views

### is it true that $0:_RH^i_{\frak{a}}(M, N)\subseteq 0:_R H^i_{\frak{a}}(Hom_R(M,N))$?

Let $R$ be a Noetherian ring and let $\frak{a}$ be an ideal of $R$. Assume that $M, N$ are two finitely generated $R$-modules. Is the containment $$0:_RH^i_{\frak{a}}(M, N)\subseteq 0:_R ...

**16**

votes

**0**answers

225 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

50 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

46 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**

votes

**0**answers

89 views

### Local cohomology, vanishing of cohomology for sheaves that are not $\mathcal{O}_X$-modules

Let $X$ be a scheme over a field $k$ and x$\in X$ a closed point. Then one can calculate $H^1_x(X,\mathcal{O}_X)$ to be isomorphic to $\mathcal{O}_{X,x}[1/f]/\mathcal{O}_{X,x}$ using the exact ...

**1**

vote

**1**answer

105 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

110 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

65 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

60 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

107 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 ...

**0**

votes

**0**answers

34 views

### Extending and contracting an ideal by a faithfully flat homomorphism [migrated]

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $.
This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...

**0**

votes

**0**answers

99 views

### Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4.
I heard that such a criterion could be possible for a given integer $n$ like
$p = a^2 + n b^2$
...

**2**

votes

**1**answer

101 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) $f_1g_1+\dots+f_ng_n\equiv0$.
...