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

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1
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1answer
29 views

Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$

Suppose $R$ is integral domain and $K$ is the fraction field of $R$. Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$.
2
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1answer
40 views

Proving $C\otimes_A\Omega^1_{A/R} \cong \Omega^1_{C/B}$

I am completely stuck on this so any help would be great. Let $R$ be a commutative ring and let $A$ and $B$ be $R$-algebras. Let $C:=A\otimes_RB$. Show that $C\otimes_A\Omega^1_{A/R} \cong ...
2
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2answers
44 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
3
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1answer
54 views

Motivation and examples 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 ...
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0answers
34 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 ...
1
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1answer
15 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 ...
4
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1answer
64 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: $I_n$ is generated by a (finite) number of polynomials which ...
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0answers
34 views

a math-software that can compute analytic spread

I want to compute "analytic spread" . So I need a math-software that can compute it. can anyone help please? Here is the definition:
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1answer
51 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} ...
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2answers
55 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
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1answer
38 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. ...
2
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1answer
52 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 ...
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2answers
69 views

Ring with nested prime ideals [closed]

If $n>1$ is there a (commutative with identity) ring with Krull dimension $n$ and only $n+1$ prime ideals?
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1answer
30 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), \ ...
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0answers
36 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 ...
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0answers
33 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
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1answer
55 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 ...
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0answers
29 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?
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1answer
23 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
48 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 ...
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1answer
26 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 ...
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0answers
24 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 ...
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1answer
38 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 ...
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1answer
24 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 ...
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0answers
30 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 ...
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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 ...
2
votes
1answer
51 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 ...
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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 ...
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1answer
55 views

Example of Gorenstein local ring of dimension 1

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
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1answer
51 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 = ...
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0answers
43 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 ...
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1answer
36 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
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1answer
31 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 ...
1
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1answer
34 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 ...
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1answer
49 views

A question about varieties (proposition from Miles Reid undergraduate comm algebra)

In Miles Reid, undergraduate commutative algebra, I read the following: "Suppose that $k$ is an algebraically closed field and that $A=k[x_1,...,x_n]$ is a finitely generated $k$-algebra of form ...
2
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1answer
34 views

When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
2
votes
1answer
36 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 $m-1$?
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37 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 ...
0
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1answer
49 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
50 views

When is a holomorphy ring a PID? [duplicate]

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 ...
1
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1answer
61 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 ...
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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 ...
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0answers
21 views

Find some differential operators in the D-modules theory

I begin with the algebraic $D$-modules, and here are my questions: 1) What is the ring of $\mathbb C$-linear differential operators on $\mathbb C[[x]]$? 2) Let $S=k[s,t,s^{-1},t^{-1}]$. Prove ...
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1answer
50 views

How is this method of finding a maximal ideal specific to finite algebras over a field?

Let $A$ be a finitely generated $K$-algebra over a field $K$. A typical problem is to find a maximal ideal $\frak{m}$ such that $f\notin\mathfrak{m}$ and it does not coincide (or contains) another ...
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1answer
68 views

Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...
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0answers
86 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
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0answers
46 views

Cardinal of a linearly independent subset of $R$-module

Let $R$ be a commutative ring, and consider $R$ as an $R$-module with the action given by the product of $R$. Prove that if $B\subset R$ is linearly independent, then $\operatorname{card}(B)=1.$ ...
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1answer
25 views

Open and closed sets for j-Spec $A$.

The following is from Matsumura, Theorem 4.10 Let $A$ be a ring and $M$ a finite $A$-module. (i) For any non-negative integer $r$ set $$U_r = \{p \in \text{Spec} \space A | M_\mathfrak{p} ...
3
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1answer
66 views

Counterexample - modules over non-Noetherian domain

Does anyone know an example of a (necessarily non-Noetherian) domain $A$ and a finitely generated $A$-module $M$ with the property that $M_f$ is not free for any nonzero $f \in A$? This would provide ...
3
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1answer
37 views

Monomial ideal is radical iff it is generated by square-free monomials

I'm trying to prove that if $ K$ is a field and $ I $ is a monomial ideal in $ K[x_1, \dots, x_n] $, then $$\sqrt{I} = I \iff I ~\text{is generated by square-free monomials}$$ So I tried to do the ...