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

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2
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1answer
28 views

Is the localization of an injective cogenerator an injective cogenerator?

We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?
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vote
1answer
30 views

Regular functions extension to normal points of varieties

I am doing the exercise 3.20 in Robin Hartshorne's Algebraic Geometry, Chapter 1. Let $Y$ be a variety of dimension $\geq2$, and let $P\in Y$ be a normal point. Let $f$ be a regular function on $Y-...
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1answer
44 views

Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
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0answers
33 views

Proof of Theorem 4.2.1 in Herzog-Hibi, “Monomial Ideals”

The Theorem and its proof can be found here. Specifically, i am stuck at the fourth paragraph of the proof. Let me give some context: Let $I$ be a graded ideal over a polynomial ring $S=K[x_1,\dots,...
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1answer
23 views

A possible characterization of divisible modules

According to mathworld: Definition. Let $R$ denote a commutative ring and $M$ denote a module over $R$. Then $M$ is divisible iff for every $a \in R$, if $a$ is not a zero-divisor, then for all $x ...
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2answers
89 views

Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
2
votes
1answer
85 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
2
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1answer
32 views

Correspondence between prime ideals and irreducible algebraic sets

Let $k$ be an algebraic closed field. The Nullstellensatz theorem prove that $$I(V(J))=\sqrt{J}$$ and we have $$V(J)\text{ irreducible }\iff I(V(J)) \text{ prime }$$ So if $J$ is prime, $I(V(J))=J$ is ...
0
votes
1answer
55 views

Extension of DVRs and uniformizers

Let $(A,\mathfrak m)$ be a regular, Noetherian, local, domain of dimension $2$ and consider a prime ideal $\mathfrak p\subset A$ of height $1$. Moreover let $\hat{A}$ be the completion of $A$ with ...
1
vote
1answer
41 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
3
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0answers
59 views

Some questions about local rings and chain rings.

(All my rings are commutative with $1$.) The notion of a field spews forth many derived concepts. For example: An integral domain is a ring that can be homomorphically injected into a field. A ...
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0answers
37 views

Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = \operatorname{...
1
vote
1answer
42 views

Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
1
vote
1answer
97 views

Can I use Krull dimension to test if a sequence of polynomials is regular?

A sequence $(f_1, \ldots, f_n)$ of elements of a commutative ring $R$ is said to be regular if for each $i$, $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$. Call a sequence dimension ...
2
votes
1answer
237 views

Does such localization of integral extension preserve inclusion?

Let $R\subset T$ be two commutative rings, and $T$ is integral over $R$. Let $\mathfrak m\in \operatorname{Max} R,\mathfrak n\in\operatorname{Max}T$ such that $\mathfrak m=\mathfrak n\cap R$. Show ...
0
votes
1answer
44 views

Relationship between modules and maximal ideals of a commutative ring [closed]

Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
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0answers
21 views

Maximal ideal of subalgebra over a field [duplicate]

Let $A$ a finite $k$-algebra (with $k$ a field) and $B$ a subalgebra of $A$. Prove that if $\mathfrak{m}$ is a maximal ideal of $A$ then $\mathfrak{m}\cap B$ is a maximal ideal of $B$. It is easy to ...
0
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1answer
30 views

Compute a Gröbner basis for $I=\langle f_1,f_2,f_3\rangle$.

Using lexicographic order compute a Gröbner basis for $$I=\langle f_1=xy^2-xy+y,f_2=xy-z^2,f_3=x-yz^4\rangle\subset \Bbb R[x,y,z]$$ I was strictly using these notes to compute a Gröbner basis. ...
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0answers
43 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. 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 ...
0
votes
1answer
50 views

Commutative Algebra “mess”: recover the complete local rings of the normalization

Premise and main idea: I'm not an expert in the field of commutative algebra and when I encounter problems regarding local rings I try to solve them by following a sort of geometric intuition. It was ...
4
votes
1answer
36 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
37 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
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1answer
51 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
17 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) \...
0
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0answers
28 views

A nilpotent Jacobson radical?

If each ideal of a commutative ring $R$ could be written as a sum of a nilpotent ideal $N$ and an idempotent ideal $I$, is the Jacobson radical $J(R)$ of $R$ necessarily nilpotent (or T-nilpotent)? ...
2
votes
2answers
42 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 ...
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0answers
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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
50 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
120 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{...
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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
38 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
65 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 ...
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1answer
30 views

Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
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vote
1answer
56 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
74 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
33 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{...
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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 ...
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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
127 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 ...
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0answers
56 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 ...
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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 $$(I, f_1, \ldots, f_{i-1}): (f_i)= (I, f_1, \ldots, f_{i-1}) \quad i \ge 2.$$...
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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 ...
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votes
0answers
31 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 $\...