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

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1answer
14 views

Isomorphism between affine varieties

I am working with a ring and I am trying to show it is not isomorphic (as $k$-algebra) to another ring: $k[x,y,z]/\langle xy-z^2\rangle$ and $k[u,w]$. What I tried so far was. I aim for a ...
0
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4answers
48 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
0
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1answer
30 views

Finding a bijective morphism

I am given two Varieties $Z=V(x^2+y^2+1) \subset C^2$ and $W=V(x^2-y^2-1) \subset C^2$. We need to find a bijective morphism f such is an isomorphism with the inverse of f. First how we defined ...
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0answers
25 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
2
votes
1answer
46 views

Isomorphism of Localizations

I believe, though a not sure, that any two ideals $A, B$ of a Dedekind domain $X$ are isomorphic as $X$-modules iff their localizations $A_p, B_p$ are isomorphic for any prime ideal $p$. Could anyone ...
1
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0answers
19 views

Not connected Zariski topology implies the existence of an idempotent element. [duplicate]

I am trying to prove that for a commutative ring $A$ (with the unit) the Zariski topology is not connected if and only if there is an idempotent element different from 0 and 1. I proved "Existence of ...
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1answer
22 views

Commutative Algebra and Monomial orders

So whenever we are doing any problem related to ideals in the polynomial ring $k[x_{1},x_{2},\dots x_{n}]$,(e.g. calculating a grobner basis for instance or doing the division algorithm for a set of ...
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0answers
19 views

Homogeneous prime ideal in $K[x_1,\ldots,x_n]$

Let $K$ be a field and $P$ a homogeneous prime ideal in $K[x_1,\ldots,x_n]$, with height $r$. I want to show that there is a chain of homogeneous primes $P_0\subsetneqq P_1\subsetneqq\cdots\subsetneqq ...
0
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1answer
21 views

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite. $P$ is a maximal ideal of $R/P$ is a field. I ...
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0answers
41 views

Class number of $\mathbb Q(\sqrt{10}) $

I am interested in knowing how to compute the class number of $\mathbb Q(\sqrt{10}) $. I am confused with these class number computations.
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0answers
34 views

Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
1
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2answers
26 views

Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
0
votes
1answer
23 views

MaxSpec of the ring of continuous function on a compact topological space. [on hold]

Consider a compact topological space $X$ and let $A$ be the ring of continuous functions $f: X \to \mathbb{C}$. Let $\mathfrak m_x$ for a point $x \in X$ be the kernel of the evaluation map $f \mapsto ...
2
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0answers
26 views

Ramification group - do you know/can produce a simple proof to this?

Let $(K,v)$ be a valuation field, $L$ a finite extension of $K$, and $w$ a valuation of $L$ above $v$. The ramification group of $w$ in $L$ is the subgroup of ${\rm Gal}(L/K)$ of all automorphisms ...
3
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1answer
28 views

Understanding the $\mathfrak{a}$-adic completion of an $A$-module as a functor

$\require{AMScd}$ I recently read the chapter 10 on Completions in Atiyah-MacDonald. They describe the $\mathfrak{a}$-adic completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
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2answers
71 views

On the minimal set of generators of monomial ideals in $\mathbb{C}[x,y]$.

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
1
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1answer
66 views

relation between vector space and torsion free module [on hold]

Can you please help me to prove this. Let $R$ be a domain, $A$ be an $R$-module, and $Q=Frac(R)$. Then a module $A$ is a vector space over $Q$ if and only if it is torsion-free and divisible. ...
0
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0answers
31 views

Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...
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0answers
38 views

tensor of two vector space [on hold]

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
1
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1answer
20 views

Maximal ideals of finite algebra over a local ring

Let $R$ be a local ring with residue field $k$. Let $A$ be an $R$-algebra which is finitely generated as $R$-module. I want to show that the maximal ideals of $A$ are in one-to-one correspondence ...
1
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2answers
38 views

Prove Kähler Differential is always surjective using universal property.

Let $A$ be an $R$-Algebra. An $R$-linear derivation $d \colon A \to \Omega_{A/R}$ is called universal derivation or Kähler differential if for every $R$-linear derivation $D \colon A \to M$ there is a ...
1
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0answers
38 views

Algebraic closedness in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
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0answers
44 views

Tangent Cone of a Complete Intersection

Can you give me an example of an affine variety $X \subseteq \mathbb{A}^n_{\mathbb{C}}$ over the complex numbers which is a complete intersection such that the reduced tangent cone at some point $p ...
0
votes
0answers
27 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
9
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3answers
66 views

Definition of $\mathfrak{m}$-adic completion.

Let $V$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}$ and let $T$ be a prime element of $V$. Assume that we have a subfield $k\subseteq V$ such that the induced map $ k \to ...
4
votes
2answers
42 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
1
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2answers
81 views

Cohen-Macaulay rings and Normal rings

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
3
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1answer
44 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...
0
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2answers
39 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
1
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1answer
25 views

Is torsion-free equivalent to free for non-finitely generated modules over a PID?

Maybe this is a trivial question. If $A$ is a PID and $M$ is a finitely generated $A$-module, it's well known that $M$ is torsion-free iff $M$ is free. However, if $M$ is not finitely generated, does ...
2
votes
2answers
54 views

Why is it called the category of representations?

Let $A$ be a (Hopf) algebra. Let $C_A$ be a category whose objects are $A$-modules and whose morphisms are $A$-linear maps. This category is called "the category of representations". My question is: ...
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2answers
45 views

How do I find an isomorphism between varieties

Our book defines an isomorphism between varieties when there exist two maps say $\phi: V \rightarrow W$ and $\psi: W \rightarrow V$ both morphisms and $\psi \circ \phi =id_V$ and $\phi \circ \psi =id ...
1
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1answer
17 views

Representation matrix for modules map

Here $S=\mathbb Q[x,y]$, and we define $\oplus Se_i$ to be a $S$-free module with basis $\{e_1,e_2,e_3\}$. Define a map from $\oplus Se_i$ to $S$ by $e_1\to x^2$, $e_2\to xy+y^2$, $e_3\to y^3$. Is the ...
0
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1answer
20 views

Showing $\hat{A} \otimes_{A} M \cong \hat{M}$ when $M$ is a finitely generated free $A$-module.

I had a reading question on Proposition 10.13 from Atiyah-MacDonald. The proposition is the following PROPOSITION. For any ring $A$, if $M$ is finitely-generated, $\hat{A} \otimes_{A} M \rightarrow ...
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1answer
77 views

Short exact sequence of modules over a Noetherian local ring of depth $1$.

I am reading an article in algebraic geometry and am having trouble understanding a particular point that reduces to a problem in commutative algebra. I'm not familiar with the concepts involved so am ...
1
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0answers
92 views

How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
1
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1answer
22 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
4
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0answers
52 views

Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the ...
0
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0answers
39 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
2
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2answers
76 views

One-dimensional Noetherian UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone ...
3
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1answer
50 views

Maximal ideal in a polynomial ring over a field that is not algebraically closed

I want to prove that although $K$ is a field that IS NOT algebraically closed, every maximal ideal in $K[x_1, \ldots, x_n]$ can be generated by $n$ elements. To prove this, I am following the next ...
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0answers
19 views

theorem 31.6 Matsumura [on hold]

Let (A,m) be a formally equidimensional Noetherian local ring and I be an ideal of A If A/I is equidimensional Why A/I is formally equidimensional?
2
votes
1answer
27 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
1
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1answer
37 views

Contraction of a maximal ideal in a polynomial ring

I have two questions: If $K$ is a field, $R=K[x_1,\ldots,x_n]$, the ring of polynomials over $K$ with $n$ indeterminates, and $M$ is a maximal ideal of $R$ why is the contraction $N$ of $M$ to ...
3
votes
0answers
74 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
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1answer
63 views

Question on a property of $\mathrm{Ass}(M)$ for modules over noetherian rings

I got stuck reading a proof of the following lemma (Lemma 0.19 in this file): Lemma Suppose that $M$ is a module over a commutative noetherian ring $R$ and let $m\neq 0 \in M$. Let $S$ be a ...
2
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2answers
63 views

If the localizations of two submodules with respect to any prime ideal are equal then the submodules are equal [closed]

I want to prove the following: Let R be a commutative ring with 1 and let N and L be two submodules of an R-module M. If the localizations of N and L with respect to any prime ideal of R are ...
0
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0answers
63 views

Localization of a regular local ring is regular

Quoting Hartshorne's Algebraic Geometry Definition. We say a scheme $X$ is regular in codimension one if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular. The most ...
4
votes
1answer
72 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
2
votes
1answer
38 views

Relatively prime ideals in Dedekind Domains

I am currently working through Lang's Algebra and have come across an exercise I can not solve (Chapter II, Exercise $19$). Any help would be greatly appreciated. Let $R$ be a Dedekind domain. ...