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

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2
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1answer
24 views

Motivation and example for ramification?

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
1
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0answers
18 views

Localization at a prime and direct limits

Let $R$ be a commutative ring with $1 \neq 0$ and let $P \subset R$ be a prime ideal. Apparently we have $$\varinjlim\limits_{f \in R \setminus P} R_f \cong R_P$$ where $R_f$ the the localization of ...
0
votes
1answer
10 views

How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$?

I'm trying to solve exercise 4.7 in Reid's UCA: "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$." I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus ...
-1
votes
0answers
30 views

Some problems with Castelnuovo-Mumford regularity

I have some problems to complete my thesis. In this paper: "Upper bound for the Castelnuovo-Mumford regularity of associated graded modules" - Cao Huy Linh Lemma 4.1 Please explain for me about ...
1
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1answer
54 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
3
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0answers
31 views

If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: I is generated by a (finite) number of polynomials which are ...
1
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1answer
47 views

Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle?

Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} ...
1
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3answers
38 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
2
votes
1answer
35 views

Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints. ...
1
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1answer
42 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
0
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1answer
29 views

A condition that an algebraic set is irreducible.

From the book by Kenji Ueno, Algebraic Geometry 1. From Algebraic Varieties to Schemes: "If an algebraic set $V(J)$ is reducible, it can be expressed as: $$(1.8)\quad V(J)= V(J_1)\cup V(J_2), \ ...
1
vote
1answer
48 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
-2
votes
2answers
55 views

Ring with nested prime ideals [on hold]

If $n>1$ is there a (commutative with identity) ring with Krull dimension $n$ and only $n+1$ prime ideals?
0
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0answers
31 views

Computing homomorphisms between extensions of modules

Suppose we have two exact sequences of $R$-modules ($R$ is a commutative ring) $$0\rightarrow M_0\rightarrow F\rightarrow M_1\rightarrow0$$ $$0\rightarrow N_0\rightarrow G\rightarrow ...
0
votes
0answers
23 views

For a ring homomorphism, why does $f$ induces a homeomorphism from $SpecB$ onto the closed subset $V(\ker f)$ of $SpecA$.

Let $\varphi : A \rightarrow B$ be a ring homomorphism. Then we have a map of sets $Spec(\varphi):Spec(B) \rightarrow Spec(A)$ defined by $p \mapsto \varphi^{-1}(p)$ for every $p \in SpecB$. ...
1
vote
1answer
182 views

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$?

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.) Thanks for any help.
2
votes
3answers
171 views

Commutative integral domain does not finitely generate its field of fractions

I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that ...
8
votes
3answers
411 views

A fraction field is not finitely generated over its subdomain

I'm looking for proofs of the following fact. Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as $R$-module. I know this fact is ...
0
votes
0answers
75 views

About the multiplicity of a semigroup ring.

Let $A=K[X^{n_1}, \dots, X^{n_s}]$, $S=\langle n_1, \dots, n_s \rangle$ ($n_1 < \cdots < n_s$) a numerical semigroup and let $\mathfrak m$ be the maximal ideal $(X^{n_1}, \dots, X^{n_s})$. We ...
0
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0answers
19 views

How to calculate the multiplicity of semigroup ring of dimension one?

Let $k$ be a field and $R=k[t^{a_1},...,t^{a_n}]$ such that $0<a_1<a_2<\cdots<a_n$ are integers. Is $a_1$ the multiplicity of $R$? Why?
1
vote
1answer
25 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
1
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1answer
21 views

$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
1
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1answer
38 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
1
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3answers
147 views
4
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2answers
155 views

Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura

Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $JA_P\subseteq IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subseteq I$. I'm reading Matsumura's Commutative Ring Theory book ...
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0answers
23 views

Why does the following reduction holds? [on hold]

Let $I=(a,b,c)$ be an ideal of local ring $R$. Then $(a^n,b^n,c^n)$ is a reduction of $I^n$ for all $n$
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0answers
24 views

Why does the following length holds? [on hold]

Let $R=k[x,y]$, $0=a_n<a_{n-1}<...<a_1$ and $0=b_1<b_2<...<b_n$ be integers. Set $I=(x^{a_i}y^{b_i}: i=1,2,...,n)$. Why $\ell(R/I)=\sum_{i=1}^{n-1}a_i(b_{i+1}-b_i)$
0
votes
0answers
21 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
1
vote
1answer
91 views

ideals in rings of algebraic integers are finitely generated

I am trying to write about rings of algebraic integers $\mathcal{O}_K$ in a number field $K$ without introducing to much field theory. I want to show that these rings are Dedekind. First of all I want ...
1
vote
1answer
29 views

Presentation of a local complete intersection

What is the simplest example of a local (noetherian) complete intersection ring $R$ that can not be presented as $R=S/I$, where $S$ is a regular local ring and $I$ is an ideal generated by a regular ...
2
votes
1answer
256 views

Proving a Certain $\mathbb{C}$-Algebra is a Domain Using a Specified Method

I would like to prove that $A = \mathbb{C}[x,y,z]/(x^2+z^2-1, x^2-y^2-z^2)$ is an integral domain. I feel that it would be easy enough to prove the denominator is prime using the techniques of ...
-1
votes
1answer
34 views

Question about notation on ideals

If $R$ is a commutative ring and $a,b \in R$ then $(a)+(b)=(a,b)= \{xa+yb : x,y \in R \}$, however if $I$ is an ideal of $R$ then what is $(I,a)$? My guess is $(I,a)=\{hg + xa : h,x \in R, g \in I ...
0
votes
1answer
20 views

Comparing an ideal and its saturation

Let $S = k[x_0,x_1,\ldots,x_n]$ with its usual grading and let $I \subset S$ be a homogeneous ideal not containing $S_+ = (x_0,x_1,\ldots,x_n)$. We define the saturation of $I$ to be the homogeneous ...
1
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0answers
24 views

proof of Proposition 3.3.18 in Bruns and Herzog

This set of questions pertains to the proof of Proposition 3.3.18(b) in Bruns and Herzog, Cohen-Macaulay Rings: Question 1: It seems to me that under the hypothesis (a) of the theorem, the ...
2
votes
1answer
48 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
1
vote
1answer
47 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
0
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0answers
42 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
1
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1answer
33 views
1
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1answer
33 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...
5
votes
1answer
603 views

What is the connection between the definition of complete intersection variety and complete intersection ring?

An algebraic variety is called a complete intersection if its defining ideal is generated by codimension many polynomials. A Noetherian local ring $R$ is called a complete intersection if its ...
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0answers
32 views

Relation about prime ideals in $B$ and invariant subring $B^G$

Suppose $B$ is a commutative ring, $G$ is a finite group acting on $B$, $A=B^G$ is the invariant subring. Suppose $P$ is a prime ideal in $A$, $Q_1,...,Q_s$ are all the prime ideals in $B$ such that ...
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0answers
42 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
2
votes
2answers
126 views

Do we have $\dim A = \sup_{\operatorname{ht}( \mathfrak{p})=0} \dim A/\mathfrak{p}$?

Let $A$ be a ring (assume Noetherian if necessary). Then it is clear to me that we have $$ \sup_{\operatorname{ht}(\mathfrak{p}) = 0} \dim A/\mathfrak{p}\leq \dim A.$$ However, I can't seem to prove ...
0
votes
1answer
20 views

Krull dimension and zero divisors of $k[x,y,z]/(x^ay,x^bz)$

I found the primary decomposition of $(0)$ in the ring $k[x,y,z]/(x^ay,x^bz)$, where $a\geq b \geq 1$, $k$ is alg. closed, to be $(x^b) \cap (x^a,z) \cap (y,z)$ (is this correct?). Now I am now ...
1
vote
1answer
32 views

closed and open subscheme of affine scheme

Let $X=Spec(A)$ be a noetherian affine scheme. Let $I_1, \ldots, I_n$ be ideals of $A$ such that $I_i + I_j = 1$ for all $i \neq j$. Define $X_i = Spec(A/I_i)$ so that X is the disjoint union of the ...
2
votes
1answer
34 views

Dimension of irreducible variety

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?
0
votes
1answer
45 views

Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
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0answers
46 views

When is a holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
0
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0answers
24 views

On different versions of Schwarz Zippel

Theorem (Schwartz, Zippel). Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d≥0$ over a Field $F$. Let $S$ be a finite subset of $F$ and let $r_1,r_2,...,r_n$ be selected at ...
0
votes
1answer
53 views

Maximal among some ideals is prime

I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy ...