Questions about commutative rings, their ideals, and their modules.

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6
votes
1answer
38 views

Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, \sqrt{f}]$....
3
votes
2answers
38 views

The content of a polynomial vs the ideal of its values

Let $f(x) = \sum_i a_i x^i$ be a degree $d$ polynomial over some ring $A$. Define the content of $f$ to be the ideal: $$c(f) = (a_0,\dots,a_d).$$ One can ask for the relation of the above ideal to the ...
0
votes
1answer
53 views

A be an affine K-algebra and f be a non-zero divisor of A then can one say that dim A=dim A_f

Let $A$ be an affine $K$-algebra and $f$ be a non-zero divisor of $A$ then can one say that $\dim A=\dim A_f $ ? What I proved that if $A$ is an affine domain and $f$ is a non-zero element in $...
0
votes
1answer
18 views

Sum of Hilbert functions of a finite exact sequence of finitely generated graded modules

Let $A = \bigoplus_{n\geq 0} A_n$ be a graded ring that is generated as an $A_0$-algebra by a finite collection of elements of $A_1$, where $A_0$ is artinian. I wish to show that if $$ 0 \to M(1) \...
2
votes
2answers
43 views

The localization is localization of some affine domain.

Let $A$ be a finitely generated $K$-algebra, and let $\mathfrak p$ be a prime ideal of $A$ such that $A_{\mathfrak p}$ is an integral domain. Then have to show that $A_{\mathfrak p}$ is a localization ...
2
votes
0answers
47 views

Question about the rational normal curve and different representations of it.

I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ...
7
votes
1answer
54 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...
0
votes
1answer
124 views

anillo noetheriano y de generación finita. [closed]

Sea $A=\oplus{A_i}$, $i\geq{0}$ anillo graduado. Si $A$ es anillo noetheriano entonces $A_0$ es noetheriano y $A$ es de generación finita como $A_0$-álgebra. Dm: Defino $I=\oplus{A_i}$ donde $i\geq{...
0
votes
0answers
21 views

Nullstellensatz theorem- find vector not orthogonal to a given set of vectors in $\Bbb{Z}^n$

Let $v_1,v_2,...,v_r \in \Bbb{Z}^n-\{0\}$. Show there exists $w\in \Bbb{Z}^n$ such that $w_1=\langle w,e_1\rangle=0$ and $\langle w,v_i\rangle \neq 0$ for all $1 \le i \le r$. I've tried to ...
2
votes
2answers
41 views

Question about S.Lang's proof of Kummer's Lemma

I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let $K = \mathbf{Q}(\xi_p)$ the $p$-th cyclotomic field extension of $\mathbf{Q}$. Let $u$ be ...
3
votes
1answer
66 views

When does a f.g. algebra over a field $F$ make it “look like $F$ is algebraically closed?”

Let $F$ be a field, and let $A$ be a finitely generated algebra over $F$. If $\mathfrak m$ is a maximal ideal of $A$, then $A/\mathfrak m$ is an algebraic extension of $F$, although it is in general ...
1
vote
1answer
35 views

Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
1
vote
1answer
57 views

localized at associated prime of an ideal [duplicate]

The problem is as follows: Let $I\subseteq J$ be ideals in a Noetherian ring. Show that if $I_{p}=J_{p}$ for every associated prime $p$ of $I$,then $I=J$. It seems reasonable to consider $J/I\...
2
votes
1answer
83 views

To prove that an ideal cannot generated by two elements [duplicate]

Let $k$ be an algebraically closed field and let $\ Y\subset \mathbb{A}^n(k)$ be the curve given parametrically by $x=t^3, y=t^4,z=t^5$ I want to show (i) $I(Y)$ is a prime ideal of height 2 (ii) $...
2
votes
1answer
26 views

Dimension of an Artin $K$-algebra and cardinal of its spectrum

Let $A$ be an Artin ring that is also a finitely generated $K$-algebra. In particular, the krull dimension of $A$ is $0$. By Noether's Normalisation Lemma we have that $A$ is a $K$-vector space of ...
2
votes
1answer
35 views

Is there an adjoint functor to the contravariant hom functor in the category of A-modules.

I should start by saying that I don't know any category theory. However, I am reading Atiyah-MacDonald and have just learned that in the category of A-modules (where here A is a commutative unital ...
2
votes
1answer
42 views

A counterexample to a statement

Question Give an example that $z\in\mathbb{Z}[\sqrt{-d}]$, $d\geq1$, $|z|^2$ is a prime number in $\mathbb{Z}$ but $z$ is not prime in $\mathbb{Z}[\sqrt{-d}]$. Problem I understand that if $\mathbb{...
-1
votes
1answer
46 views

F-rationality of a ring.

Given $R = \dfrac{k[x,y,z]}{(x^2 - y^3 -z^5)} $ where $\operatorname{char}k>5$. Check whether $R$ is $F$-rational or not. ($F$ = Frobenius map) I know, by the theorem of Karen Smith, we have to ...
1
vote
1answer
29 views

A uniqueness theorem for primary decomposition

"Let $R$ be an arbitrary ring and $\mathfrak a$ an ideal of $R$ admitting an irredundant primary representation $\mathfrak a=\bigcap_{i}\mathfrak q_{i}$ and let $\mathfrak p_i=\sqrt{\mathfrak q_i}$. ...
4
votes
1answer
133 views

Defining the set $\{(t^3,t^4,t^5) : t \in \mathbb{C}\}\subset \mathbb{C}^3$ by two polynomial equations

What are two polynomials $f,g \in \mathbb{C}[x,y,z]$ such that $$\{(x,y,z): f(x,y,z)=g(x,y,z)=0\}\;=\;\{(t^3,t^4,t^5): t \in \mathbb{C}\}$$ holds as an equality of subset of $\mathbb{C}^2$? This ...
1
vote
0answers
63 views

Separable morphism and smooth fibers

Let $f:X \to Y$ be a separable, dominant morphism of finite type between noetherian $k$-schemes for $k$ algebraically closed. Does it mean that For a closed point $x \in X$, $f^{-1}(f(x))$ is smooth ...
1
vote
1answer
39 views

Necessary and sufficient condition for a regular sequence.

$f_1, \ldots, f_r$ is a regular sequence in $S/I$ (where $S$ is a polynomial ring in $n$ variables, and $I$ its ideal) iff $$(I, f_1, \ldots, f_{i-1}): (f_i)= (I, f_1, \ldots, f_{i-1}) \quad i \ge 2.$$...
0
votes
0answers
35 views

Difference between parameters and system of parameters in a local commutative ring

Can you please tell me the difference between the 'parameters' and the 'system of parameters' of a commutative local ring? Also, is there any relation between parameters and associated primes of the ...
0
votes
0answers
33 views

Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
0
votes
1answer
56 views

Projective module with non-zero annihilator [closed]

Let $M$ be a projective module. Suppose $\operatorname{Ann}_{R} \left(M \right) \neq 0$, where $\operatorname{Ann}_{R} \left( M \right) =\{r\in R : mr = 0, \ \forall m \in M \}$. Then there exists an ...
0
votes
1answer
13 views

Find $g\in I$ such that $LT(g)\notin \langle LT(g_1),LT(g_2),LT(g_3)\rangle$.

Let $I=\langle g_1,g_2,g_3\rangle\subset \Bbb R[x,y,z]$ where $$g_1=xy^2-xy+y,\qquad g_2=xy-z^2, \text{ and } g_3=x-yz^4$$ Using lexicographic order find $g\in I$ such that $LT(g)\notin \langle LT(...
4
votes
1answer
40 views

Criterion for the integral closure of an domain in a finite field extension being a finitely generated algebra

$A$ is an integral domain, $K=\operatorname{Frac}A$, $L/K$ finite field extension (not necessarily separable), $B$ is the integral closure of $A$ in $L$. Question: with some extra conditions on $A$, ...
1
vote
1answer
102 views

Line bundle trivial on fibers then isomorphic to the pullback of a line bundle

$\require{AMScd}$ I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let $\...
1
vote
1answer
27 views

$M_{1} \oplus M_{2}$ is a cyclic $A$-module $\iff \rm{Ann}(M_1)+\rm{Ann}(M_2)=A$ [duplicate]

Let $A$ be a commutative ring with an identity element $1$. An element $x$ in an $A$-module $M$ is called cyclic if $Ax=M$. An $A$-module which has a cyclic element is called cyclic $A$-module. Let $...
1
vote
1answer
45 views

do formal group laws induce group structures on schemes (as opposed to formal schemes)

Let $R$ be a ring and $f \in R[[x]]$ a commutative formal group law over $R$, meaning $f(f(x, y), z)=f(x, f(y, z))$, $\ f(x, y)=f(y, x)$ and $f(x, y)=x+y + \text{higher order terms}$. Let $G=\...
0
votes
0answers
32 views

An algebraic set is called defined over $k$ if its ideal can be generated by polynomials in $k[x]$. [duplicate]

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves. An algebraic set (in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in $K[X]=K[...
0
votes
1answer
60 views

Same kernels for homomorphisms of free modules

Let $f: R^n \rightarrow R^m$ be an isomorphism of free $R$-modules ($R$ commutative with unity) and $\pi_1: R^n \rightarrow R^n/\mathfrak m^n$, $\pi_2: R^m \rightarrow R^m/\mathfrak m^m$ the canonical ...
3
votes
2answers
58 views

Triviality of $\mathrm{Ann}(\mathfrak m)$

This question is regarding the first paragraph of the proof of Proposition 2.4 from this paper. QUESTION: Is it true that if $(0)$ is irreducible, then $\mathrm{Soc}(R)=\mathrm{Ann}(\mathfrak m)=(0:...
2
votes
1answer
35 views

Example of associated ideal in primary decomposition

Let $I$ be a decomposable ideal of a commutative ring $R$ with minimal primary decomposition $I=\bigcap_{i=1}^n\mathfrak q_i$. The first uniqueness theorem shows that $\{\sqrt {\mathfrak q_i}:1\le i\...
1
vote
0answers
34 views

Let a,b have the same divisor (content) in an integral domain A. When can I deduce $a/b\in A^\times$?

Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be $div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P]$, where the sum ranges ...
1
vote
1answer
65 views

Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
0
votes
1answer
22 views

Reference on a result about integral closures.

Could you please give a reference or a sketch of a proof for the following proposition? Proposition: The integral closure of a complete local Noetherian domain $R$ is module-finite over $R$ You ...
0
votes
0answers
49 views

Independent set of variables modulo ideal and Krull dimension

Let $\mathfrak{a}\subseteq \Bbbk[x_1,\ldots,x_n]$ be an ideal, where $\Bbbk$ is a field. Let the maximal set of indeterminates independent modulo $\mathfrak{a}$ be of cardinality $k$. There is a ...
0
votes
1answer
45 views

Fiber of morphism induced by map on stalks

Given a morphism of schemes $f\colon X\to Y$ and a point $x\in X$, the map on the stalks induces a morphism $\operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)} $. Is it ...
3
votes
1answer
83 views

Exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book

I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's The Geometry of Syzygies book.The problem is as follows: Let $X$ be the union of two disjoint lines in $\mathbb P^3$, or a conic ...
0
votes
2answers
53 views

Is the ideal of a variety the annihilator of a subspace of the symmetric algebra?

Let $V$ be a vector space over an algebraically closed field $K$. Let $\mathrm{Sym}(V^*)=\mathrm{Sym}(V)^*$ be the symmetric algebra on $V$, i.e. if we give a basis $e_1,...,e_n$ of $V$ and let $x_1,...
2
votes
1answer
96 views

Annihilator of a flat ideal

Let $R$ be a commutative ring and let $I$ be a finitely generated flat ideal of $R$. Let $J=\mathrm{Ann}(I)$. How can one prove that $I\cap J=0$? This can be found as a remark in the paper of ...
1
vote
1answer
36 views

Characterization of Groebner Bases in terms of uniqueness of remainders

Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to ...
0
votes
1answer
24 views

Fraction rings ideals members

Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why? Many thanks.
1
vote
1answer
32 views

System of parameters in Noetherian local rings

I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says: Let $R$ be a Noetherian local ring with maximal ideal $m$. Then there exists an $m$-primary ideal ...
1
vote
1answer
40 views

Regularity and Short Exact Sequence

Suppose $ 0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of finitely generated graded $k[x_0,...,x_r]$-modules. Then show that $\mathrm{reg}(M_1) \leq\max(\mathrm{reg}(M_2),\mathrm{reg}(...
2
votes
0answers
29 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
2
votes
0answers
35 views

Trying to Compute Regularity and degree

Definition: For a finite subset $X \subset \mathbb P^r$,the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in X,usually called the degree of $X$. Let $...
1
vote
1answer
55 views

Filling in Proof: Well-definedness of depth(I,M).

From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let $M$ be a finitely generated $R$-module, where $R$ is Noetherian. If $$r= \min \{i : H^i(M\otimes K(x_1,...
0
votes
0answers
33 views

Some ideal property in a local ring

If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question? Again, the new local ring ...